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id="toc-Entropy_of_an_information_source-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Joint_entropy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Joint_entropy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Joint entropy</span> </div> </a> <ul id="toc-Joint_entropy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conditional_entropy_(equivocation)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conditional_entropy_(equivocation)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Conditional entropy (equivocation)</span> </div> </a> <ul id="toc-Conditional_entropy_(equivocation)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mutual_information_(transinformation)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mutual_information_(transinformation)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Mutual information (transinformation)</span> </div> </a> <ul id="toc-Mutual_information_(transinformation)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kullback–Leibler_divergence_(information_gain)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kullback–Leibler_divergence_(information_gain)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Kullback–Leibler divergence (information gain)</span> </div> </a> <ul id="toc-Kullback–Leibler_divergence_(information_gain)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Directed_Information" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Directed_Information"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Directed Information</span> </div> </a> <ul id="toc-Directed_Information-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_quantities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_quantities"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Other quantities</span> </div> </a> <ul id="toc-Other_quantities-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Coding_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Coding_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Coding theory</span> </div> </a> <button aria-controls="toc-Coding_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Coding theory subsection</span> </button> <ul id="toc-Coding_theory-sublist" class="vector-toc-list"> <li id="toc-Source_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Source_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Source theory</span> </div> </a> <ul id="toc-Source_theory-sublist" class="vector-toc-list"> <li id="toc-Rate" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Rate"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Rate</span> </div> </a> <ul id="toc-Rate-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Channel_capacity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Channel_capacity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Channel capacity</span> </div> </a> <ul id="toc-Channel_capacity-sublist" class="vector-toc-list"> <li id="toc-Capacity_of_particular_channel_models" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Capacity_of_particular_channel_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.1</span> <span>Capacity of particular channel models</span> </div> </a> <ul id="toc-Capacity_of_particular_channel_models-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Channels_with_memory_and_directed_information" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Channels_with_memory_and_directed_information"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.2</span> <span>Channels with memory and directed information</span> </div> </a> <ul id="toc-Channels_with_memory_and_directed_information-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Fungible_information" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fungible_information"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Fungible information</span> </div> </a> <ul id="toc-Fungible_information-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications_to_other_fields" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications_to_other_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applications to other fields</span> </div> </a> <button aria-controls="toc-Applications_to_other_fields-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications to other fields subsection</span> </button> <ul id="toc-Applications_to_other_fields-sublist" class="vector-toc-list"> <li id="toc-Intelligence_uses_and_secrecy_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Intelligence_uses_and_secrecy_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Intelligence uses and secrecy applications</span> </div> </a> <ul id="toc-Intelligence_uses_and_secrecy_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pseudorandom_number_generation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pseudorandom_number_generation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Pseudorandom number generation</span> </div> </a> <ul id="toc-Pseudorandom_number_generation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Seismic_exploration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Seismic_exploration"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Seismic exploration</span> </div> </a> <ul id="toc-Seismic_exploration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semiotics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semiotics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Semiotics</span> </div> </a> <ul id="toc-Semiotics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integrated_process_organization_of_neural_information" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integrated_process_organization_of_neural_information"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Integrated process organization of neural information</span> </div> </a> <ul id="toc-Integrated_process_organization_of_neural_information-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Miscellaneous_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Miscellaneous_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Miscellaneous applications</span> </div> </a> <ul id="toc-Miscellaneous_applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <button aria-controls="toc-See_also-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle See also subsection</span> </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Theory</span> </div> </a> <ul id="toc-Theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Concepts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Concepts"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Concepts</span> </div> </a> <ul id="toc-Concepts-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <button aria-controls="toc-Further_reading-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Further reading subsection</span> </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-The_classic_work" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_classic_work"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>The classic work</span> </div> </a> <ul id="toc-The_classic_work-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_journal_articles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_journal_articles"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Other journal articles</span> </div> </a> <ul id="toc-Other_journal_articles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Textbooks_on_information_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Textbooks_on_information_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Textbooks on information theory</span> </div> </a> <ul id="toc-Textbooks_on_information_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_books" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_books"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>Other books</span> </div> </a> <ul id="toc-Other_books-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Information theory</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 74 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-74" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">74 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Inligtingsteorie" title="Inligtingsteorie – Afrikaans" lang="af" hreflang="af" data-title="Inligtingsteorie" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D9%85%D8%B9%D9%84%D9%88%D9%85%D8%A7%D8%AA" title="نظرية المعلومات – Arabic" lang="ar" hreflang="ar" data-title="نظرية المعلومات" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_de_la_informaci%C3%B3n" title="Teoría de la información – Asturian" lang="ast" hreflang="ast" data-title="Teoría de la información" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0nformasiya_n%C9%99z%C9%99riyy%C9%99si" title="İnformasiya nəzəriyyəsi – Azerbaijani" lang="az" hreflang="az" data-title="İnformasiya nəzəriyyəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A6%A5%E0%A7%8D%E0%A6%AF_%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="তথ্য তত্ত্ব – Bangla" lang="bn" hreflang="bn" data-title="তথ্য তত্ত্ব" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9C%D3%99%D2%93%D0%BB%D2%AF%D0%BC%D3%99%D1%82_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Мәғлүмәт теорияһы – Bashkir" lang="ba" hreflang="ba" data-title="Мәғлүмәт теорияһы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D1%96%D0%BD%D1%84%D0%B0%D1%80%D0%BC%D0%B0%D1%86%D1%8B%D1%96" title="Тэорыя інфармацыі – Belarusian" lang="be" hreflang="be" data-title="Тэорыя інфармацыі" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%B8%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%8F%D1%82%D0%B0" title="Теория на информацията – Bulgarian" lang="bg" hreflang="bg" data-title="Теория на информацията" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Informationstheorie" title="Informationstheorie – Bavarian" lang="bar" hreflang="bar" data-title="Informationstheorie" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_la_informaci%C3%B3" title="Teoria de la informació – Catalan" lang="ca" hreflang="ca" data-title="Teoria de la informació" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Teorie_informace" title="Teorie informace – Czech" lang="cs" hreflang="cs" data-title="Teorie informace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Informationsteori" title="Informationsteori – Danish" lang="da" hreflang="da" data-title="Informationsteori" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Informationstheorie" title="Informationstheorie – German" lang="de" hreflang="de" data-title="Informationstheorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Informatsiooniteooria" title="Informatsiooniteooria – Estonian" lang="et" hreflang="et" data-title="Informatsiooniteooria" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CF%80%CE%BB%CE%B7%CF%81%CE%BF%CF%86%CE%BF%CF%81%CE%AF%CE%B1%CF%82" title="Θεωρία πληροφορίας – Greek" lang="el" hreflang="el" data-title="Θεωρία πληροφορίας" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_de_la_informaci%C3%B3n" title="Teoría de la información – Spanish" lang="es" hreflang="es" data-title="Teoría de la información" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Informteorio" title="Informteorio – Esperanto" lang="eo" hreflang="eo" data-title="Informteorio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Informazio-teoria" title="Informazio-teoria – Basque" lang="eu" hreflang="eu" data-title="Informazio-teoria" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D8%A7%D8%B7%D9%84%D8%A7%D8%B9%D8%A7%D8%AA" title="نظریه اطلاعات – Persian" lang="fa" hreflang="fa" data-title="نظریه اطلاعات" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_de_l%27information" title="Théorie de l'information – French" lang="fr" hreflang="fr" data-title="Théorie de l'information" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Teoiric_na_faisn%C3%A9ise" title="Teoiric na faisnéise – Irish" lang="ga" hreflang="ga" data-title="Teoiric na faisnéise" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_da_informaci%C3%B3n" title="Teoría da información – Galician" lang="gl" hreflang="gl" data-title="Teoría da información" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EB%B3%B4_%EC%9D%B4%EB%A1%A0" title="정보 이론 – Korean" lang="ko" hreflang="ko" data-title="정보 이론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BB%D5%B6%D6%86%D5%B8%D6%80%D5%B4%D5%A1%D6%81%D5%AB%D5%A1%D5%B5%D5%AB_%D5%BF%D5%A5%D5%BD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Ինֆորմացիայի տեսություն – Armenian" lang="hy" hreflang="hy" data-title="Ինֆորմացիայի տեսություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Teorija_informacije" title="Teorija informacije – Croatian" lang="hr" hreflang="hr" data-title="Teorija informacije" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Informo-teorio" title="Informo-teorio – Ido" lang="io" hreflang="io" data-title="Informo-teorio" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teori_informasi" title="Teori informasi – Indonesian" lang="id" hreflang="id" data-title="Teori informasi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Theoria_del_information" title="Theoria del information – Interlingua" lang="ia" hreflang="ia" data-title="Theoria del information" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Umchachiso_wokwaziswa" title="Umchachiso wokwaziswa – Zulu" lang="zu" hreflang="zu" data-title="Umchachiso wokwaziswa" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_dell%27informazione" title="Teoria dell'informazione – Italian" lang="it" hreflang="it" data-title="Teoria dell'informazione" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%90%D7%99%D7%A0%D7%A4%D7%95%D7%A8%D7%9E%D7%A6%D7%99%D7%94" title="תורת האינפורמציה – Hebrew" lang="he" hreflang="he" data-title="תורת האינפורמציה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%98%E1%83%9C%E1%83%A4%E1%83%9D%E1%83%A0%E1%83%9B%E1%83%90%E1%83%AA%E1%83%98%E1%83%98%E1%83%A1_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90" title="ინფორმაციის თეორია – Georgian" lang="ka" hreflang="ka" data-title="ინფორმაციის თეორია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D2%9B%D0%BF%D0%B0%D1%80%D0%B0%D1%82_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Ақпарат теориясы – Kazakh" lang="kk" hreflang="kk" data-title="Ақпарат теориясы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B0%D0%BB%D1%8B%D0%BC%D0%B0%D1%82_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D1%81%D1%8B" title="Маалымат теориясы – Kyrgyz" lang="ky" hreflang="ky" data-title="Маалымат теориясы" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Inform%C4%81cijas_teorija" title="Informācijas teorija – Latvian" lang="lv" hreflang="lv" data-title="Informācijas teorija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Informacijos_teorija" title="Informacijos teorija – Lithuanian" lang="lt" hreflang="lt" data-title="Informacijos teorija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Inform%C3%A1ci%C3%B3elm%C3%A9let" title="Információelmélet – Hungarian" lang="hu" hreflang="hu" data-title="Információelmélet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%BD%D0%B0_%D0%B8%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D0%B8%D1%82%D0%B5" title="Теорија на информациите – Macedonian" lang="mk" hreflang="mk" data-title="Теорија на информациите" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%BF%E0%B4%B5%E0%B4%B0_%E0%B4%B8%E0%B4%BF%E0%B4%A6%E0%B5%8D%E0%B4%A7%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%82" title="വിവര സിദ്ധാന്തം – Malayalam" lang="ml" hreflang="ml" data-title="വിവര സിദ്ധാന്തം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_maklumat" title="Teori maklumat – Malay" lang="ms" hreflang="ms" data-title="Teori maklumat" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Teorie_de_l%27anforma%C3%A7on" title="Teorie de l'anformaçon – Mirandese" lang="mwl" hreflang="mwl" data-title="Teorie de l'anformaçon" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9C%D1%8D%D0%B4%D1%8D%D1%8D%D0%BB%D0%BB%D0%B8%D0%B9%D0%BD_%D0%BE%D0%BD%D0%BE%D0%BB" title="Мэдээллийн онол – Mongolian" lang="mn" hreflang="mn" data-title="Мэдээллийн онол" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9E%E1%80%90%E1%80%84%E1%80%BA%E1%80%B8%E1%80%A1%E1%80%81%E1%80%BB%E1%80%80%E1%80%BA%E1%80%A1%E1%80%9C%E1%80%80%E1%80%BA_%E1%80%9E%E1%80%AE%E1%80%A1%E1%80%AD%E1%80%AF%E1%80%9B%E1%80%AE" title="သတင်းအချက်အလက် သီအိုရီ – Burmese" lang="my" hreflang="my" data-title="သတင်းအချက်အလက် သီအိုရီ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Informatietheorie" title="Informatietheorie – Dutch" lang="nl" hreflang="nl" data-title="Informatietheorie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%83%85%E5%A0%B1%E7%90%86%E8%AB%96" title="情報理論 – Japanese" lang="ja" hreflang="ja" data-title="情報理論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Informasjonsteori" title="Informasjonsteori – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Informasjonsteori" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Informasjonsteori" title="Informasjonsteori – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Informasjonsteori" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Teoria_de_l%27informacion" title="Teoria de l'informacion – Occitan" lang="oc" hreflang="oc" data-title="Teoria de l'informacion" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Informatsiya_nazariyasi" title="Informatsiya nazariyasi – Uzbek" lang="uz" hreflang="uz" data-title="Informatsiya nazariyasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%DB%82_%D8%A7%D8%B7%D9%84%D8%A7%D8%B9%D8%A7%D8%AA" title="نظریۂ اطلاعات – Western Punjabi" lang="pnb" hreflang="pnb" data-title="نظریۂ اطلاعات" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF_%D9%85%D8%B9%D9%84%D9%88%D9%85%D8%A7%D8%AA%D9%88_%D9%86%D8%B8%D8%B1%DB%8C%D9%87" title="د معلوماتو نظریه – Pashto" lang="ps" hreflang="ps" data-title="د معلوماتو نظریه" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Teoria_informacji" title="Teoria informacji – Polish" lang="pl" hreflang="pl" data-title="Teoria informacji" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_da_informa%C3%A7%C3%A3o" title="Teoria da informação – Portuguese" lang="pt" hreflang="pt" data-title="Teoria da informação" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Informaciya_teoriyas%C4%B1" title="Informaciya teoriyası – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Informaciya teoriyası" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teoria_informa%C8%9Biei" title="Teoria informației – Romanian" lang="ro" hreflang="ro" data-title="Teoria informației" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%B8%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D0%B8" title="Теория информации – Russian" lang="ru" hreflang="ru" data-title="Теория информации" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Information_theory" title="Information theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Information theory" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Te%C3%B3ria_inform%C3%A1cie" title="Teória informácie – Slovak" lang="sk" hreflang="sk" data-title="Teória informácie" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Teorija_informacij" title="Teorija informacij – Slovenian" lang="sl" hreflang="sl" data-title="Teorija informacij" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%DB%8C%DB%8C_%D8%B2%D8%A7%D9%86%DB%8C%D8%A7%D8%B1%DB%8C" title="تیۆریی زانیاری – Central Kurdish" lang="ckb" hreflang="ckb" data-title="تیۆریی زانیاری" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%B8%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%98%D0%B5" title="Теорија информације – Serbian" lang="sr" hreflang="sr" data-title="Теорија информације" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Teorija_informacije" title="Teorija informacije – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Teorija informacije" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Informaatioteoria" title="Informaatioteoria – Finnish" lang="fi" hreflang="fi" data-title="Informaatioteoria" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Informationsteori" title="Informationsteori – Swedish" lang="sv" hreflang="sv" data-title="Informationsteori" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Teorya_ng_impormasyon" title="Teorya ng impormasyon – Tagalog" lang="tl" hreflang="tl" data-title="Teorya ng impormasyon" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%95%E0%AE%B5%E0%AE%B2%E0%AF%8D_%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="தகவல் கோட்பாடு – Tamil" lang="ta" hreflang="ta" data-title="தகவல் கோட்பாடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%AA%E0%B8%B2%E0%B8%A3%E0%B8%AA%E0%B8%99%E0%B9%80%E0%B8%97%E0%B8%A8" title="ทฤษฎีสารสนเทศ – Thai" lang="th" hreflang="th" data-title="ทฤษฎีสารสนเทศ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Bilgi_teorisi" title="Bilgi teorisi – Turkish" lang="tr" hreflang="tr" data-title="Bilgi teorisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D1%96%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D1%96%D1%97" title="Теорія інформації – Ukrainian" lang="uk" hreflang="uk" data-title="Теорія інформації" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%DB%82_%D8%A7%D8%B7%D9%84%D8%A7%D8%B9%D8%A7%D8%AA" title="نظریۂ اطلاعات – Urdu" lang="ur" hreflang="ur" data-title="نظریۂ اطلاعات" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C3%BD_thuy%E1%BA%BFt_th%C3%B4ng_tin" title="Lý thuyết thông tin – Vietnamese" lang="vi" hreflang="vi" data-title="Lý thuyết thông tin" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E4%BF%A1%E6%81%AF%E8%AE%BA" title="信息论 – Wu" lang="wuu" hreflang="wuu" data-title="信息论" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="資訊理論 – Cantonese" lang="yue" hreflang="yue" data-title="資訊理論" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BF%A1%E6%81%AF%E8%AE%BA" title="信息论 – Chinese" lang="zh" hreflang="zh" data-title="信息论" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q131222#sitelinks-wikipedia" title="Edit 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Scientific study of digital information</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Information_science" title="Information science">Information science</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist 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a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><th class="sidebar-title"><a class="mw-selflink selflink">Information theory</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Binaryerasurechannel.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Binaryerasurechannel.png/100px-Binaryerasurechannel.png" decoding="async" width="100" height="92" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Binaryerasurechannel.png/150px-Binaryerasurechannel.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/Binaryerasurechannel.png/200px-Binaryerasurechannel.png 2x" data-file-width="666" data-file-height="610" /></a></span></td></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">Entropy</a></li> <li><a href="/wiki/Differential_entropy" title="Differential entropy">Differential entropy</a></li> <li><a href="/wiki/Conditional_entropy" title="Conditional entropy">Conditional entropy</a></li> <li><a href="/wiki/Joint_entropy" title="Joint entropy">Joint entropy</a></li> <li><a href="/wiki/Mutual_information" title="Mutual information">Mutual information</a></li> <li><a href="/wiki/Directed_information" title="Directed information">Directed information</a></li> <li><a href="/wiki/Conditional_mutual_information" title="Conditional mutual information">Conditional mutual information</a></li> <li><a href="/wiki/Relative_entropy" class="mw-redirect" title="Relative entropy">Relative entropy</a></li> <li><a href="/wiki/Entropy_rate" title="Entropy rate">Entropy rate</a></li> <li><a href="/wiki/Limiting_density_of_discrete_points" title="Limiting density of discrete points">Limiting density of discrete points</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Asymptotic_equipartition_property" title="Asymptotic equipartition property">Asymptotic equipartition property</a></li> <li><a href="/wiki/Rate%E2%80%93distortion_theory" title="Rate–distortion theory">Rate–distortion theory</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Shannon%27s_source_coding_theorem" title="Shannon's source coding theorem">Shannon's source coding theorem</a></li> <li><a href="/wiki/Channel_capacity" title="Channel capacity">Channel capacity</a></li> <li><a href="/wiki/Noisy-channel_coding_theorem" title="Noisy-channel coding theorem">Noisy-channel coding theorem</a></li> <li><a href="/wiki/Shannon%E2%80%93Hartley_theorem" title="Shannon–Hartley theorem">Shannon–Hartley theorem</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Information_theory" title="Template:Information theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Information_theory" title="Template talk:Information theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Information_theory" title="Special:EditPage/Template:Information theory"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Information theory</b> is the mathematical study of the <a href="/wiki/Quantification_(science)" title="Quantification (science)">quantification</a>, <a href="/wiki/Data_storage" title="Data storage">storage</a>, and <a href="/wiki/Telecommunications" title="Telecommunications">communication</a> of <a href="/wiki/Information" title="Information">information</a>. The field was established and formalized by <a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a> in the 1940s,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> though early contributions were made in the 1920s through the works of <a href="/wiki/Harry_Nyquist" title="Harry Nyquist">Harry Nyquist</a> and <a href="/wiki/Ralph_Hartley" title="Ralph Hartley">Ralph Hartley</a>. It is at the intersection of <a href="/wiki/Electronic_engineering" title="Electronic engineering">electronic engineering</a>, <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Computer_science" title="Computer science">computer science</a>, <a href="/wiki/Neuroscience" title="Neuroscience">neurobiology</a>, <a href="/wiki/Physics" title="Physics">physics</a>, and <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a>.<sup id="cite_ref-:2_2-0" class="reference"><a href="#cite_note-:2-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>A key measure in information theory is <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">entropy</a>. Entropy quantifies the amount of uncertainty involved in the value of a <a href="/wiki/Random_variable" title="Random variable">random variable</a> or the outcome of a <a href="/wiki/Random_process" class="mw-redirect" title="Random process">random process</a>. For example, identifying the outcome of a <a href="/wiki/Fair_coin" title="Fair coin">fair</a> <a href="/wiki/Coin_flip" class="mw-redirect" title="Coin flip">coin flip</a> (which has two equally likely outcomes) provides less information (lower entropy, less uncertainty) than identifying the outcome from a roll of a <a href="/wiki/Dice" title="Dice">die</a> (which has six equally likely outcomes). Some other important measures in information theory are <a href="/wiki/Mutual_information" title="Mutual information">mutual information</a>, <a href="/wiki/Channel_capacity" title="Channel capacity">channel capacity</a>, <a href="/wiki/Error_exponent" title="Error exponent">error exponents</a>, and <a href="/wiki/Relative_entropy" class="mw-redirect" title="Relative entropy">relative entropy</a>. Important sub-fields of information theory include <a href="/wiki/Source_coding" class="mw-redirect" title="Source coding">source coding</a>, <a href="/wiki/Algorithmic_complexity_theory" class="mw-redirect" title="Algorithmic complexity theory">algorithmic complexity theory</a>, <a href="/wiki/Algorithmic_information_theory" title="Algorithmic information theory">algorithmic information theory</a> and <a href="/wiki/Information-theoretic_security" title="Information-theoretic security">information-theoretic security</a>. </p><p>Applications of fundamental topics of information theory include source coding/<a href="/wiki/Data_compression" title="Data compression">data compression</a> (e.g. for <a href="/wiki/ZIP_(file_format)" title="ZIP (file format)">ZIP files</a>), and channel coding/<a href="/wiki/Error_detection_and_correction" title="Error detection and correction">error detection and correction</a> (e.g. for <a href="/wiki/Digital_subscriber_line" title="Digital subscriber line">DSL</a>). Its impact has been crucial to the success of the <a href="/wiki/Voyager_program" title="Voyager program">Voyager</a> missions to deep space,<sup id="cite_ref-:22_4-0" class="reference"><a href="#cite_note-:22-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> the invention of the <a href="/wiki/Compact_disc" title="Compact disc">compact disc</a>, the feasibility of mobile phones and the development of the <a href="/wiki/Internet" title="Internet">Internet</a> and <a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The theory has also found applications in other areas, including <a href="/wiki/Statistical_inference" title="Statistical inference">statistical inference</a>,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Cryptography" title="Cryptography">cryptography</a>, <a href="/wiki/Neurobiology" class="mw-redirect" title="Neurobiology">neurobiology</a>,<sup id="cite_ref-Spikes_8-0" class="reference"><a href="#cite_note-Spikes-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Perception" title="Perception">perception</a>,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>,<sup id="cite_ref-:2_2-1" class="reference"><a href="#cite_note-:2-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Linguistics" title="Linguistics">linguistics</a>, the evolution<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and function<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> of molecular codes (<a href="/wiki/Bioinformatics" title="Bioinformatics">bioinformatics</a>), <a href="/wiki/Thermal_physics" title="Thermal physics">thermal physics</a>,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a>,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Black_hole" title="Black hole">black holes</a>, <a href="/wiki/Quantum_computing" title="Quantum computing">quantum computing</a>, <a href="/wiki/Information_retrieval" title="Information retrieval">information retrieval</a>, <a href="/wiki/Intelligence_(Information_Gathering)" class="mw-redirect" title="Intelligence (Information Gathering)">intelligence gathering</a>, <a href="/wiki/Plagiarism_detection" class="mw-redirect" title="Plagiarism detection">plagiarism detection</a>,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Pattern_recognition" title="Pattern recognition">pattern recognition</a>, <a href="/wiki/Anomaly_detection" title="Anomaly detection">anomaly detection</a>,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> the analysis of <a href="/wiki/Music" title="Music">music</a>,<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Art" title="Art">art creation</a>,<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Imaging_system" class="mw-redirect" title="Imaging system">imaging system</a> design,<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> study of <a href="/wiki/Outer_space" title="Outer space">outer space</a>,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> the dimensionality of <a href="/wiki/Space" title="Space">space</a>,<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Epistemology" title="Epistemology">epistemology</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Information theory studies the transmission, processing, extraction, and utilization of <a href="/wiki/Information" title="Information">information</a>. Abstractly, information can be thought of as the resolution of uncertainty. In the case of communication of information over a noisy channel, this abstract concept was formalized in 1948 by <a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a> in a paper entitled <i><a href="/wiki/A_Mathematical_Theory_of_Communication" title="A Mathematical Theory of Communication">A Mathematical Theory of Communication</a></i>, in which information is thought of as a set of possible messages, and the goal is to send these messages over a noisy channel, and to have the receiver reconstruct the message with low probability of error, in spite of the channel noise. Shannon's main result, the <a href="/wiki/Noisy-channel_coding_theorem" title="Noisy-channel coding theorem">noisy-channel coding theorem</a>, showed that, in the limit of many channel uses, the rate of information that is asymptotically achievable is equal to the channel capacity, a quantity dependent merely on the statistics of the channel over which the messages are sent.<sup id="cite_ref-Spikes_8-1" class="reference"><a href="#cite_note-Spikes-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Coding theory is concerned with finding explicit methods, called <i>codes</i>, for increasing the efficiency and reducing the error rate of data communication over noisy channels to near the channel capacity. These codes can be roughly subdivided into data compression (source coding) and <a href="/wiki/Error-correction" class="mw-redirect" title="Error-correction">error-correction</a> (channel coding) techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2024)">citation needed</span></a></i>]</sup> </p><p>A third class of information theory codes are <a href="/wiki/Cryptographic_algorithm" class="mw-redirect" title="Cryptographic algorithm">cryptographic algorithms</a> (both <a href="/wiki/Code_(cryptography)" title="Code (cryptography)">codes</a> and <a href="/wiki/Cipher" title="Cipher">ciphers</a>). Concepts, methods and results from coding theory and information theory are widely used in cryptography and <a href="/wiki/Cryptanalysis" title="Cryptanalysis">cryptanalysis</a>, such as the <a href="/wiki/Ban_(unit)" class="mw-redirect" title="Ban (unit)">unit ban</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Historical_background">Historical background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=2" title="Edit section: Historical background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_information_theory" title="History of information theory">History of information theory</a></div> <p>The landmark event <i>establishing</i> the discipline of information theory and bringing it to immediate worldwide attention was the publication of Claude E. Shannon's classic paper "A Mathematical Theory of Communication" in the <i><a href="/wiki/Bell_System_Technical_Journal" class="mw-redirect" title="Bell System Technical Journal">Bell System Technical Journal</a></i> in July and October 1948. Historian <a href="/wiki/James_Gleick" title="James Gleick">James Gleick</a> rated the paper as the most important development of 1948, noting that the paper was "even more profound and more fundamental" than the <a href="/wiki/Transistor" title="Transistor">transistor</a>.<sup id="cite_ref-FOOTNOTEGleick20113–4_23-0" class="reference"><a href="#cite_note-FOOTNOTEGleick20113–4-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> He came to be known as the "father of information theory".<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_26-0" class="reference"><a href="#cite_note-:1-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Shannon outlined some of his initial ideas of information theory as early as 1939 in a letter to <a href="/wiki/Vannevar_Bush" title="Vannevar Bush">Vannevar Bush</a>.<sup id="cite_ref-:1_26-1" class="reference"><a href="#cite_note-:1-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>Prior to this paper, limited information-theoretic ideas had been developed at <a href="/wiki/Bell_Labs" title="Bell Labs">Bell Labs</a>, all implicitly assuming events of equal probability. <a href="/wiki/Harry_Nyquist" title="Harry Nyquist">Harry Nyquist</a>'s 1924 paper, <i>Certain Factors Affecting Telegraph Speed</i>, contains a theoretical section quantifying "intelligence" and the "line speed" at which it can be transmitted by a communication system, giving the relation <span class="texhtml"><i>W</i> = <i>K</i> log <i>m</i></span> (recalling the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>), where <i>W</i> is the speed of transmission of intelligence, <i>m</i> is the number of different voltage levels to choose from at each time step, and <i>K</i> is a constant. <a href="/wiki/Ralph_Hartley" title="Ralph Hartley">Ralph Hartley</a>'s 1928 paper, <i>Transmission of Information</i>, uses the word <i>information</i> as a measurable quantity, reflecting the receiver's ability to distinguish one <a href="/wiki/Sequence_of_symbols" class="mw-redirect" title="Sequence of symbols">sequence of symbols</a> from any other, thus quantifying information as <span class="texhtml"><i>H</i> = log <i>S</i><sup><i>n</i></sup> = <i>n</i> log <i>S</i></span>, where <i>S</i> was the number of possible symbols, and <i>n</i> the number of symbols in a transmission. The unit of information was therefore the <a href="/wiki/Decimal_digit" class="mw-redirect" title="Decimal digit">decimal digit</a>, which since has sometimes been called the <a href="/wiki/Hartley_(unit)" title="Hartley (unit)">hartley</a> in his honor as a unit or scale or measure of information. <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a> in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war <a href="/wiki/Cryptanalysis_of_the_Enigma" title="Cryptanalysis of the Enigma">Enigma</a> ciphers.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2024)">citation needed</span></a></i>]</sup> </p><p>Much of the mathematics behind information theory with events of different probabilities were developed for the field of <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> by <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a> and <a href="/wiki/J._Willard_Gibbs" class="mw-redirect" title="J. Willard Gibbs">J. Willard Gibbs</a>. Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by <a href="/wiki/Rolf_Landauer" title="Rolf Landauer">Rolf Landauer</a> in the 1960s, are explored in <i><a href="/wiki/Entropy_in_thermodynamics_and_information_theory" title="Entropy in thermodynamics and information theory">Entropy in thermodynamics and information theory</a></i>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2024)">citation needed</span></a></i>]</sup> </p><p>In Shannon's revolutionary and groundbreaking paper, the work for which had been substantially completed at Bell Labs by the end of 1944, Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory, opening with the assertion: </p> <dl><dd>"<i>The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point.</i>"</dd></dl> <p>With it came the ideas of: </p> <ul><li>the <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">information entropy</a> and <a href="/wiki/Redundancy_(information_theory)" title="Redundancy (information theory)">redundancy</a> of a source, and its relevance through the <a href="/wiki/Source_coding_theorem" class="mw-redirect" title="Source coding theorem">source coding theorem</a>;</li> <li>the <a href="/wiki/Mutual_information" title="Mutual information">mutual information</a>, and the channel capacity of a noisy channel, including the promise of perfect loss-free communication given by the noisy-channel coding theorem;</li> <li>the practical result of the <a href="/wiki/Shannon%E2%80%93Hartley_law" class="mw-redirect" title="Shannon–Hartley law">Shannon–Hartley law</a> for the channel capacity of a <a href="/wiki/Gaussian_channel" class="mw-redirect" title="Gaussian channel">Gaussian channel</a>; as well as</li> <li>the <a href="/wiki/Bit" title="Bit">bit</a>—a new way of seeing the most fundamental unit of information.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2024)">citation needed</span></a></i>]</sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Quantities_of_information">Quantities of information</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=3" title="Edit section: Quantities of information"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Information_theory" title="Special:EditPage/Information theory">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">April 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quantities_of_information" title="Quantities of information">Quantities of information</a></div> <p>Information theory is based on <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and statistics, where <a href="/wiki/Quantities_of_information" title="Quantities of information">quantified information</a> is usually described in terms of bits. Information theory often concerns itself with measures of information of the distributions associated with random variables. One of the most important measures is called <a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">entropy</a>, which forms the building block of many other measures. Entropy allows quantification of measure of information in a single random variable.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> Another useful concept is mutual information defined on two random variables, which describes the measure of information in common between those variables, which can be used to describe their correlation. The former quantity is a property of the probability distribution of a random variable and gives a limit on the rate at which data generated by independent samples with the given distribution can be reliably compressed. The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy <a href="/wiki/Communication_channel" title="Communication channel">channel</a> in the limit of long block lengths, when the channel statistics are determined by the joint distribution. </p><p>The choice of logarithmic base in the following formulae determines the <a href="/wiki/Units_of_measurement" class="mw-redirect" title="Units of measurement">unit</a> of information entropy that is used. A common unit of information is the bit or <a href="/wiki/Shannon_(unit)" title="Shannon (unit)">shannon</a>, based on the <a href="/wiki/Binary_logarithm" title="Binary logarithm">binary logarithm</a>. Other units include the <a href="/wiki/Nat_(unit)" title="Nat (unit)">nat</a>, which is based on the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>, and the <a href="/wiki/Deciban" class="mw-redirect" title="Deciban">decimal digit</a>, which is based on the <a href="/wiki/Common_logarithm" title="Common logarithm">common logarithm</a>. </p><p>In what follows, an expression of the form <span class="texhtml"><i>p</i> log <i>p</i></span> is considered by convention to be equal to zero whenever <span class="texhtml"><i>p</i> = 0</span>. This is justified because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{p\rightarrow 0+}p\log p=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mo>+</mo> </mrow> </munder> <mi>p</mi> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{p\rightarrow 0+}p\log p=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/182a05385b07bc194d5aa35023abee98dbc5256a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.063ex; width:15.367ex; height:4.176ex;" alt="{\displaystyle \lim _{p\rightarrow 0+}p\log p=0}"></span> for any logarithmic base. </p> <div class="mw-heading mw-heading3"><h3 id="Entropy_of_an_information_source">Entropy of an information source</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=4" title="Edit section: Entropy of an information source"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Based on the <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass function</a> of each source symbol to be communicated, the Shannon <a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">entropy</a> <span class="texhtml"><i>H</i></span>, in units of bits (per symbol), is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=-\sum _{i}p_{i}\log _{2}(p_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=-\sum _{i}p_{i}\log _{2}(p_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a7673eba0df7fb924d538fb6b1c94174f812af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.26ex; height:5.509ex;" alt="{\displaystyle H=-\sum _{i}p_{i}\log _{2}(p_{i})}"></span></dd></dl> <p>where <span class="texhtml"><i>p<sub>i</sub></i></span> is the probability of occurrence of the <span class="texhtml"><i>i</i></span>-th possible value of the source symbol. This equation gives the entropy in the units of "bits" (per symbol) because it uses a logarithm of base 2, and this base-2 measure of entropy has sometimes been called the <a href="/wiki/Shannon_(unit)" title="Shannon (unit)">shannon</a> in his honor. Entropy is also commonly computed using the natural logarithm (base <span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a></span>, where <span class="texhtml mvar" style="font-style:italic;">e</span> is Euler's number), which produces a measurement of entropy in nats per symbol and sometimes simplifies the analysis by avoiding the need to include extra constants in the formulas. Other bases are also possible, but less commonly used. For example, a logarithm of base <span class="nowrap">2<sup>8</sup> = 256</span> will produce a measurement in <a href="/wiki/Byte" title="Byte">bytes</a> per symbol, and a logarithm of base 10 will produce a measurement in decimal digits (or <a href="/wiki/Hartley_(unit)" title="Hartley (unit)">hartleys</a>) per symbol. </p><p>Intuitively, the entropy <span class="texhtml"><i>H<sub>X</sub></i></span> of a discrete random variable <span class="texhtml"><i>X</i></span> is a measure of the amount of <i>uncertainty</i> associated with the value of <span class="texhtml"><i>X</i></span> when only its distribution is known. </p><p>The entropy of a source that emits a sequence of <span class="texhtml"><i>N</i></span> symbols that are <a href="/wiki/Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">independent and identically distributed</a> (iid) is <span class="texhtml"><i>N</i> ⋅ <i>H</i></span> bits (per message of <span class="texhtml"><i>N</i></span> symbols). If the source data symbols are identically distributed but not independent, the entropy of a message of length <span class="texhtml"><i>N</i></span> will be less than <span class="texhtml"><i>N</i> ⋅ <i>H</i></span>. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Binary_entropy_plot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Binary_entropy_plot.svg/200px-Binary_entropy_plot.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Binary_entropy_plot.svg/300px-Binary_entropy_plot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Binary_entropy_plot.svg/400px-Binary_entropy_plot.svg.png 2x" data-file-width="300" data-file-height="300" /></a><figcaption>The entropy of a <a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a> as a function of success probability, often called the <em><a href="/wiki/Binary_entropy_function" title="Binary entropy function">binary entropy function</a></em>, <span class="texhtml"><i>H</i><sub>b</sub>(<i>p</i>)</span>. The entropy is maximized at 1 bit per trial when the two possible outcomes are equally probable, as in an unbiased coin toss.</figcaption></figure> <p>If one transmits 1000 bits (0s and 1s), and the value of each of these bits is known to the receiver (has a specific value with certainty) ahead of transmission, it is clear that no information is transmitted. If, however, each bit is independently equally likely to be 0 or 1, 1000 shannons of information (more often called bits) have been transmitted. Between these two extremes, information can be quantified as follows. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {X} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c0954f67c8e841542d7ac6a8472c4ea739824b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {X} }"></span> is the set of all messages <span class="texhtml">{<i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>n</i></sub>}</span> that <span class="texhtml"><i>X</i></span> could be, and <span class="texhtml"><i>p</i>(<i>x</i>)</span> is the probability of some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {X} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93b167a7fd856749b6844408ead71c3667da3358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {X} }"></span>, then the entropy, <span class="texhtml"><i>H</i></span>, of <span class="texhtml"><i>X</i></span> is defined:<sup id="cite_ref-FOOTNOTEReza1994_28-0" class="reference"><a href="#cite_note-FOOTNOTEReza1994-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(X)=\mathbb {E} _{X}[I(x)]=-\sum _{x\in \mathbb {X} }p(x)\log p(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">X</mi> </mrow> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(X)=\mathbb {E} _{X}[I(x)]=-\sum _{x\in \mathbb {X} }p(x)\log p(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971ffd75b32f284123036d4ae8fc3dd6e377e030" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.784ex; height:5.676ex;" alt="{\displaystyle H(X)=\mathbb {E} _{X}[I(x)]=-\sum _{x\in \mathbb {X} }p(x)\log p(x).}"></span></dd></dl> <p>(Here, <span class="texhtml"><i>I</i>(<i>x</i>)</span> is the <a href="/wiki/Self-information" class="mw-redirect" title="Self-information">self-information</a>, which is the entropy contribution of an individual message, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} _{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0703f8f752bd96aa7e4aa1b49d2b49ee3713114" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.183ex; height:2.509ex;" alt="{\displaystyle \mathbb {E} _{X}}"></span> is the <a href="/wiki/Expected_value" title="Expected value">expected value</a>.) A property of entropy is that it is maximized when all the messages in the message space are equiprobable <span class="texhtml"><i>p</i>(<i>x</i>) = 1/<i>n</i></span>; i.e., most unpredictable, in which case <span class="texhtml"><i>H</i>(<i>X</i>) = log <i>n</i></span>. </p><p>The special case of information entropy for a random variable with two outcomes is the binary entropy function, usually taken to the logarithmic base 2, thus having the <a href="/wiki/Shannon_(unit)" title="Shannon (unit)">shannon</a> (Sh) as unit: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\mathrm {b} }(p)=-p\log _{2}p-(1-p)\log _{2}(1-p).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>p</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>p</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\mathrm {b} }(p)=-p\log _{2}p-(1-p)\log _{2}(1-p).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9430bcaab47b309a20d10f8b9a299b35a90d3bd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.965ex; height:2.843ex;" alt="{\displaystyle H_{\mathrm {b} }(p)=-p\log _{2}p-(1-p)\log _{2}(1-p).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Joint_entropy">Joint entropy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=5" title="Edit section: Joint entropy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <em><a href="/wiki/Joint_entropy" title="Joint entropy">joint entropy</a></em> of two discrete random variables <span class="texhtml"><i>X</i></span> and <span class="texhtml"><i>Y</i></span> is merely the entropy of their pairing: <span class="texhtml">(<i>X</i>, <i>Y</i>)</span>. This implies that if <span class="texhtml"><i>X</i></span> and <span class="texhtml"><i>Y</i></span> are <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a>, then their joint entropy is the sum of their individual entropies. </p><p>For example, if <span class="texhtml">(<i>X</i>, <i>Y</i>)</span> represents the position of a chess piece—<span class="texhtml"><i>X</i></span> the row and <span class="texhtml"><i>Y</i></span> the column, then the joint entropy of the row of the piece and the column of the piece will be the entropy of the position of the piece. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(X,Y)=\mathbb {E} _{X,Y}[-\log p(x,y)]=-\sum _{x,y}p(x,y)\log p(x,y)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(X,Y)=\mathbb {E} _{X,Y}[-\log p(x,y)]=-\sum _{x,y}p(x,y)\log p(x,y)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a10572b1a21187833168e50a998f66856658674d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:56.163ex; height:5.843ex;" alt="{\displaystyle H(X,Y)=\mathbb {E} _{X,Y}[-\log p(x,y)]=-\sum _{x,y}p(x,y)\log p(x,y)\,}"></span></dd></dl> <p>Despite similar notation, joint entropy should not be confused with <em><a href="/wiki/Cross-entropy" title="Cross-entropy">cross-entropy</a></em>. </p> <div class="mw-heading mw-heading3"><h3 id="Conditional_entropy_(equivocation)"><span id="Conditional_entropy_.28equivocation.29"></span>Conditional entropy (equivocation)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=6" title="Edit section: Conditional entropy (equivocation)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <em><a href="/wiki/Conditional_entropy" title="Conditional entropy">conditional entropy</a></em> or <i>conditional uncertainty</i> of <span class="texhtml"><i>X</i></span> given random variable <span class="texhtml"><i>Y</i></span> (also called the <i>equivocation</i> of <span class="texhtml"><i>X</i></span> about <span class="texhtml"><i>Y</i></span>) is the average conditional entropy over <span class="texhtml"><i>Y</i></span>:<sup id="cite_ref-FOOTNOTEAsh1990_29-0" class="reference"><a href="#cite_note-FOOTNOTEAsh1990-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(X|Y)=\mathbb {E} _{Y}[H(X|y)]=-\sum _{y\in Y}p(y)\sum _{x\in X}p(x|y)\log p(x|y)=-\sum _{x,y}p(x,y)\log p(x|y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(X|Y)=\mathbb {E} _{Y}[H(X|y)]=-\sum _{y\in Y}p(y)\sum _{x\in X}p(x|y)\log p(x|y)=-\sum _{x,y}p(x,y)\log p(x|y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bdf0f6f2421309afb521010957e1e556669e479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:82.742ex; height:5.843ex;" alt="{\displaystyle H(X|Y)=\mathbb {E} _{Y}[H(X|y)]=-\sum _{y\in Y}p(y)\sum _{x\in X}p(x|y)\log p(x|y)=-\sum _{x,y}p(x,y)\log p(x|y).}"></span></dd></dl> <p>Because entropy can be conditioned on a random variable or on that random variable being a certain value, care should be taken not to confuse these two definitions of conditional entropy, the former of which is in more common use. A basic property of this form of conditional entropy is that: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(X|Y)=H(X,Y)-H(Y).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(X|Y)=H(X,Y)-H(Y).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a800109119215c303f549c60dd667d707ff58c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.552ex; height:2.843ex;" alt="{\displaystyle H(X|Y)=H(X,Y)-H(Y).\,}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Mutual_information_(transinformation)"><span id="Mutual_information_.28transinformation.29"></span>Mutual information (transinformation)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=7" title="Edit section: Mutual information (transinformation)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i><a href="/wiki/Mutual_information" title="Mutual information">Mutual information</a></i> measures the amount of information that can be obtained about one random variable by observing another. It is important in communication where it can be used to maximize the amount of information shared between sent and received signals. The mutual information of <span class="texhtml"><i>X</i></span> relative to <span class="texhtml"><i>Y</i></span> is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(X;Y)=\mathbb {E} _{X,Y}[SI(x,y)]=\sum _{x,y}p(x,y)\log {\frac {p(x,y)}{p(x)\,p(y)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>S</mi> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X;Y)=\mathbb {E} _{X,Y}[SI(x,y)]=\sum _{x,y}p(x,y)\log {\frac {p(x,y)}{p(x)\,p(y)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d86fba391a52b75e1cf36dae6bfaa1ccbccf461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:51.804ex; height:7.176ex;" alt="{\displaystyle I(X;Y)=\mathbb {E} _{X,Y}[SI(x,y)]=\sum _{x,y}p(x,y)\log {\frac {p(x,y)}{p(x)\,p(y)}}}"></span></dd></dl> <p>where <span class="texhtml">SI</span> (<i>S</i>pecific mutual Information) is the <a href="/wiki/Pointwise_mutual_information" title="Pointwise mutual information">pointwise mutual information</a>. </p><p>A basic property of the mutual information is that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(X;Y)=H(X)-H(X|Y).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X;Y)=H(X)-H(X|Y).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff968f71052c9abe1da3e3e414327077de6f2721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.867ex; height:2.843ex;" alt="{\displaystyle I(X;Y)=H(X)-H(X|Y).\,}"></span></dd></dl> <p>That is, knowing <i>Y</i>, we can save an average of <span class="texhtml"><i>I</i>(<i>X</i>; <i>Y</i>)</span> bits in encoding <i>X</i> compared to not knowing <i>Y</i>. </p><p>Mutual information is <a href="/wiki/Symmetric_function" title="Symmetric function">symmetric</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(X;Y)=I(Y;X)=H(X)+H(Y)-H(X,Y).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>;</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X;Y)=I(Y;X)=H(X)+H(Y)-H(X,Y).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c8ca328d60dbd8eaab8833e9ec1e882daaa3b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.608ex; height:2.843ex;" alt="{\displaystyle I(X;Y)=I(Y;X)=H(X)+H(Y)-H(X,Y).\,}"></span></dd></dl> <p>Mutual information can be expressed as the average Kullback–Leibler divergence (information gain) between the <a href="/wiki/Posterior_probability" title="Posterior probability">posterior probability distribution</a> of <i>X</i> given the value of <i>Y</i> and the <a href="/wiki/Prior_probability" title="Prior probability">prior distribution</a> on <i>X</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(X;Y)=\mathbb {E} _{p(y)}[D_{\mathrm {KL} }(p(X|Y=y)\|p(X))].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">[</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Y</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X;Y)=\mathbb {E} _{p(y)}[D_{\mathrm {KL} }(p(X|Y=y)\|p(X))].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d90fc72e685981072e77a3933c7ad6c4c83d13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:41.539ex; height:3.176ex;" alt="{\displaystyle I(X;Y)=\mathbb {E} _{p(y)}[D_{\mathrm {KL} }(p(X|Y=y)\|p(X))].}"></span></dd></dl> <p>In other words, this is a measure of how much, on the average, the probability distribution on <i>X</i> will change if we are given the value of <i>Y</i>. This is often recalculated as the divergence from the product of the marginal distributions to the actual joint distribution: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(X;Y)=D_{\mathrm {KL} }(p(X,Y)\|p(X)p(Y)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X;Y)=D_{\mathrm {KL} }(p(X,Y)\|p(X)p(Y)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03b108331b20bad2137992fa16f9fdd48906072d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.424ex; height:2.843ex;" alt="{\displaystyle I(X;Y)=D_{\mathrm {KL} }(p(X,Y)\|p(X)p(Y)).}"></span></dd></dl> <p>Mutual information is closely related to the <a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">log-likelihood ratio test</a> in the context of contingency tables and the <a href="/wiki/Multinomial_distribution" title="Multinomial distribution">multinomial distribution</a> and to <a href="/wiki/Pearson%27s_chi-squared_test" title="Pearson's chi-squared test">Pearson's χ<sup>2</sup> test</a>: mutual information can be considered a statistic for assessing independence between a pair of variables, and has a well-specified asymptotic distribution. </p> <div class="mw-heading mw-heading3"><h3 id="Kullback–Leibler_divergence_(information_gain)"><span id="Kullback.E2.80.93Leibler_divergence_.28information_gain.29"></span>Kullback–Leibler divergence (information gain)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=8" title="Edit section: Kullback–Leibler divergence (information gain)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i><a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback–Leibler divergence</a></i> (or <i>information divergence</i>, <i>information gain</i>, or <i>relative entropy</i>) is a way of comparing two distributions: a "true" <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7425278aab7b6ceb8ddb54fa5ce71d2cf52d28f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.048ex; height:2.843ex;" alt="{\displaystyle p(X)}"></span>⁠</span>, and an arbitrary probability distribution <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352e294916cf0c2b425f097071daf0da1b47123b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle q(X)}"></span>⁠</span>. If we compress data in a manner that assumes <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352e294916cf0c2b425f097071daf0da1b47123b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle q(X)}"></span>⁠</span> is the distribution underlying some data, when, in reality, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7425278aab7b6ceb8ddb54fa5ce71d2cf52d28f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.048ex; height:2.843ex;" alt="{\displaystyle p(X)}"></span>⁠</span> is the correct distribution, the Kullback–Leibler divergence is the number of average additional bits per datum necessary for compression. It is thus defined </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mathrm {KL} }(p(X)\|q(X))=\sum _{x\in X}-p(x)\log {q(x)}\,-\,\sum _{x\in X}-p(x)\log {p(x)}=\sum _{x\in X}p(x)\log {\frac {p(x)}{q(x)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </munder> <mo>−</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </munder> <mo>−</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\mathrm {KL} }(p(X)\|q(X))=\sum _{x\in X}-p(x)\log {q(x)}\,-\,\sum _{x\in X}-p(x)\log {p(x)}=\sum _{x\in X}p(x)\log {\frac {p(x)}{q(x)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b6882199003743a8f6d6d0d4723b2bd2e496f98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:80.622ex; height:7.009ex;" alt="{\displaystyle D_{\mathrm {KL} }(p(X)\|q(X))=\sum _{x\in X}-p(x)\log {q(x)}\,-\,\sum _{x\in X}-p(x)\log {p(x)}=\sum _{x\in X}p(x)\log {\frac {p(x)}{q(x)}}.}"></span></dd></dl> <p>Although it is sometimes used as a 'distance metric', KL divergence is not a true <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> since it is not symmetric and does not satisfy the <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a> (making it a semi-quasimetric). </p><p>Another interpretation of the KL divergence is the "unnecessary surprise" introduced by a prior from the truth: suppose a number <i>X</i> is about to be drawn randomly from a discrete set with probability distribution <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb7afced134ef75572e5314a5d278c2d644f438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}"></span>⁠</span>. If Alice knows the true distribution <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb7afced134ef75572e5314a5d278c2d644f438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}"></span>⁠</span>, while Bob believes (has a <a href="/wiki/Prior_probability" title="Prior probability">prior</a>) that the distribution is <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c38bbafe34a043d284f19231b946a76c0a4b16b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.209ex; height:2.843ex;" alt="{\displaystyle q(x)}"></span>⁠</span>, then Bob will be more <a href="/wiki/Information_content" title="Information content">surprised</a> than Alice, on average, upon seeing the value of <i>X</i>. The KL divergence is the (objective) expected value of Bob's (subjective) <a href="/wiki/Information_content" title="Information content">surprisal</a> minus Alice's surprisal, measured in bits if the <i>log</i> is in base 2. In this way, the extent to which Bob's prior is "wrong" can be quantified in terms of how "unnecessarily surprised" it is expected to make him. </p> <div class="mw-heading mw-heading3"><h3 id="Directed_Information">Directed Information</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=9" title="Edit section: Directed Information"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i><a href="/wiki/Directed_information" title="Directed information">Directed information</a></i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(X^{n}\to Y^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X^{n}\to Y^{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/779afffca11b308d71883749356d24da2f8e3659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.929ex; height:2.843ex;" alt="{\displaystyle I(X^{n}\to Y^{n})}"></span>, is an information theory measure that quantifies the <a href="/wiki/Information" title="Information">information</a> flow from the random process <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{n}=\{X_{1},X_{2},\dots ,X_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{n}=\{X_{1},X_{2},\dots ,X_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2a01bc29253e3a34d2b4efc98b2ca157877660" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.951ex; height:2.843ex;" alt="{\displaystyle X^{n}=\{X_{1},X_{2},\dots ,X_{n}\}}"></span> to the random process <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y^{n}=\{Y_{1},Y_{2},\dots ,Y_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{n}=\{Y_{1},Y_{2},\dots ,Y_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34055d31664058ce10332b0576dadc2e5cbe779c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.133ex; height:2.843ex;" alt="{\displaystyle Y^{n}=\{Y_{1},Y_{2},\dots ,Y_{n}\}}"></span>. The term <i>directed information</i> was coined by <a href="/wiki/James_Massey" title="James Massey">James Massey</a> and is defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(X^{n}\to Y^{n})\triangleq \sum _{i=1}^{n}I(X^{i};Y_{i}|Y^{i-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>≜</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>I</mi> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>;</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X^{n}\to Y^{n})\triangleq \sum _{i=1}^{n}I(X^{i};Y_{i}|Y^{i-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0c7ae55a98d50e4673f581d644dbeb19cd506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.179ex; height:6.843ex;" alt="{\displaystyle I(X^{n}\to Y^{n})\triangleq \sum _{i=1}^{n}I(X^{i};Y_{i}|Y^{i-1})}"></span>,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(X^{i};Y_{i}|Y^{i-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>;</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X^{i};Y_{i}|Y^{i-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/262d572ffcff0f5ccddb6e850c54d63abbfda28c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.409ex; height:3.176ex;" alt="{\displaystyle I(X^{i};Y_{i}|Y^{i-1})}"></span> is the <a href="/wiki/Conditional_mutual_information" title="Conditional mutual information">conditional mutual information</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(X_{1},X_{2},...,X_{i};Y_{i}|Y_{1},Y_{2},...,Y_{i-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X_{1},X_{2},...,X_{i};Y_{i}|Y_{1},Y_{2},...,Y_{i-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7743dc75d5d8ef30a1e1da83165a5bca2feef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.961ex; height:2.843ex;" alt="{\displaystyle I(X_{1},X_{2},...,X_{i};Y_{i}|Y_{1},Y_{2},...,Y_{i-1})}"></span>. </p><p>In contrast to <i>mutual</i> information, <i>directed</i> information is not symmetric. The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(X^{n}\to Y^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X^{n}\to Y^{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/779afffca11b308d71883749356d24da2f8e3659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.929ex; height:2.843ex;" alt="{\displaystyle I(X^{n}\to Y^{n})}"></span> measures the information bits that are transmitted causally<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="definition of causal transmission? (November 2024)">clarification needed</span></a></i>]</sup> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/268db8293666fefd75cfb00513706171948edf09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.215ex; height:2.343ex;" alt="{\displaystyle X^{n}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e8fe1b832e2088d10ef7c91d815a0d21aad4ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.119ex; height:2.176ex;" alt="{\displaystyle Y^{n}}"></span>. The Directed information has many applications in problems where <a href="/wiki/Causality" title="Causality">causality</a> plays an important role such as <a href="/wiki/Channel_capacity" title="Channel capacity">capacity of channel</a> with feedback,<sup id="cite_ref-massey_30-0" class="reference"><a href="#cite_note-massey-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> capacity of discrete <a href="/wiki/Memoryless" class="mw-redirect" title="Memoryless">memoryless</a> networks with feedback,<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Sports_gambling" class="mw-redirect" title="Sports gambling">gambling</a> with causal side information,<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Data_compression" title="Data compression">compression</a> with causal side information,<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Real-time_control" class="mw-redirect" title="Real-time control">real-time control</a> communication settings,<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> and in statistical physics.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Other_quantities">Other quantities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=10" title="Edit section: Other quantities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Other important information theoretic quantities include the <a href="/wiki/R%C3%A9nyi_entropy" title="Rényi entropy">Rényi entropy</a> and the <a href="/wiki/Tsallis_entropy" title="Tsallis entropy">Tsallis entropy</a> (generalizations of the concept of entropy), <a href="/wiki/Differential_entropy" title="Differential entropy">differential entropy</a> (a generalization of quantities of information to continuous distributions), and the <a href="/wiki/Conditional_mutual_information" title="Conditional mutual information">conditional mutual information</a>. Also, <a href="/wiki/Pragmatic_information" class="mw-redirect" title="Pragmatic information">pragmatic information</a> has been proposed as a measure of how much information has been used in making a decision. </p> <div class="mw-heading mw-heading2"><h2 id="Coding_theory">Coding theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=11" title="Edit section: Coding theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Information_theory" title="Special:EditPage/Information theory">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">April 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Coding_theory" title="Coding theory">Coding theory</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:CDSCRATCHES.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/CDSCRATCHES.jpg/220px-CDSCRATCHES.jpg" decoding="async" width="220" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/5a/CDSCRATCHES.jpg 1.5x" data-file-width="283" data-file-height="130" /></a><figcaption>A picture showing scratches on the readable surface of a CD-R. Music and data CDs are coded using error correcting codes and thus can still be read even if they have minor scratches using <a href="/wiki/Error_detection_and_correction" title="Error detection and correction">error detection and correction</a>.</figcaption></figure> <p>Coding theory is one of the most important and direct applications of information theory. It can be subdivided into source coding theory and channel coding theory. Using a statistical description for data, information theory quantifies the number of bits needed to describe the data, which is the information entropy of the source. </p> <ul><li><a href="/wiki/Data_compression" title="Data compression">Data compression</a> (source coding): There are two formulations for the compression problem: <ul><li><a href="/wiki/Lossless_data_compression" class="mw-redirect" title="Lossless data compression">lossless data compression</a>: the data must be reconstructed exactly;</li> <li><a href="/wiki/Lossy_data_compression" class="mw-redirect" title="Lossy data compression">lossy data compression</a>: allocates bits needed to reconstruct the data, within a specified fidelity level measured by a distortion function. This subset of information theory is called <i><a href="/wiki/Rate%E2%80%93distortion_theory" title="Rate–distortion theory">rate–distortion theory</a></i>.</li></ul></li> <li><a href="/wiki/Error-correcting_code" class="mw-redirect" title="Error-correcting code">Error-correcting codes</a> (channel coding): While data compression removes as much redundancy as possible, an error-correcting code adds just the right kind of redundancy (i.e., error correction) needed to transmit the data efficiently and faithfully across a noisy channel.</li></ul> <p>This division of coding theory into compression and transmission is justified by the information transmission theorems, or source–channel separation theorems that justify the use of bits as the universal currency for information in many contexts. However, these theorems only hold in the situation where one transmitting user wishes to communicate to one receiving user. In scenarios with more than one transmitter (the multiple-access channel), more than one receiver (the <a href="/wiki/Broadcast_channel" class="mw-redirect" title="Broadcast channel">broadcast channel</a>) or intermediary "helpers" (the <a href="/wiki/Relay_channel" title="Relay channel">relay channel</a>), or more general <a href="/wiki/Computer_network" title="Computer network">networks</a>, compression followed by transmission may no longer be optimal. </p> <div class="mw-heading mw-heading3"><h3 id="Source_theory">Source theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=12" title="Edit section: Source theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any process that generates successive messages can be considered a <em><a href="/wiki/Communication_source" title="Communication source">source</a></em> of information. A memoryless source is one in which each message is an <a href="/wiki/Independent_identically_distributed_random_variables" class="mw-redirect" title="Independent identically distributed random variables">independent identically distributed random variable</a>, whereas the properties of <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodicity</a> and <a href="/wiki/Stationary_process" title="Stationary process">stationarity</a> impose less restrictive constraints. All such sources are <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic</a>. These terms are well studied in their own right outside information theory. </p> <div class="mw-heading mw-heading4"><h4 id="Rate">Rate</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=13" title="Edit section: Rate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Information <i><a href="/wiki/Entropy_rate" title="Entropy rate">rate</a></i> is the average entropy per symbol. For memoryless sources, this is merely the entropy of each symbol, while, in the case of a stationary stochastic process, it is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=\lim _{n\to \infty }H(X_{n}|X_{n-1},X_{n-2},X_{n-3},\ldots );}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\lim _{n\to \infty }H(X_{n}|X_{n-1},X_{n-2},X_{n-3},\ldots );}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2344c3096dbc21ce050a9b70a4ddf2062ff41e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.671ex; height:3.843ex;" alt="{\displaystyle r=\lim _{n\to \infty }H(X_{n}|X_{n-1},X_{n-2},X_{n-3},\ldots );}"></span></dd></dl> <p>that is, the conditional entropy of a symbol given all the previous symbols generated. For the more general case of a process that is not necessarily stationary, the <i>average rate</i> is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=\lim _{n\to \infty }{\frac {1}{n}}H(X_{1},X_{2},\dots X_{n});}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\lim _{n\to \infty }{\frac {1}{n}}H(X_{1},X_{2},\dots X_{n});}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/324db7538097abf899fe20ce454b645c2e7ee5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.835ex; height:5.176ex;" alt="{\displaystyle r=\lim _{n\to \infty }{\frac {1}{n}}H(X_{1},X_{2},\dots X_{n});}"></span></dd></dl> <p>that is, the limit of the joint entropy per symbol. For stationary sources, these two expressions give the same result.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Information_rate" class="mw-redirect" title="Information rate">information rate</a> is defined as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=\lim _{n\to \infty }{\frac {1}{n}}I(X_{1},X_{2},\dots X_{n};Y_{1},Y_{2},\dots Y_{n});}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=\lim _{n\to \infty }{\frac {1}{n}}I(X_{1},X_{2},\dots X_{n};Y_{1},Y_{2},\dots Y_{n});}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b4dd61deebad94d93705b144f5350f604105b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.534ex; height:5.176ex;" alt="{\displaystyle r=\lim _{n\to \infty }{\frac {1}{n}}I(X_{1},X_{2},\dots X_{n};Y_{1},Y_{2},\dots Y_{n});}"></span></dd></dl> <p>It is common in information theory to speak of the "rate" or "entropy" of a language. This is appropriate, for example, when the source of information is English prose. The rate of a source of information is related to its redundancy and how well it can be compressed, the subject of <em>source coding</em>. </p> <div class="mw-heading mw-heading3"><h3 id="Channel_capacity">Channel capacity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=14" title="Edit section: Channel capacity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Channel_capacity" title="Channel capacity">Channel capacity</a></div> <p>Communications over a channel is the primary motivation of information theory. However, channels often fail to produce exact reconstruction of a signal; noise, periods of silence, and other forms of signal corruption often degrade quality. </p><p>Consider the communications process over a discrete channel. A simple model of the process is shown below: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\xrightarrow[{\text{Message}}]{W}}{\begin{array}{|c| }\hline {\text{Encoder}}\\f_{n}\\\hline \end{array}}{\xrightarrow[{\mathrm {Encoded \atop sequence} }]{X^{n}}}{\begin{array}{|c| }\hline {\text{Channel}}\\p(y|x)\\\hline \end{array}}{\xrightarrow[{\mathrm {Received \atop sequence} }]{Y^{n}}}{\begin{array}{|c| }\hline {\text{Decoder}}\\g_{n}\\\hline \end{array}}{\xrightarrow[{\mathrm {Estimated \atop message} }]{\hat {W}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset="-.24em"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Message</mtext> </mrow> </mpadded> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mi>W</mi> </mpadded> </munderover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" columnlines="" frame="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Encoder</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset="-.24em"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> </mrow> </mfrac> </mrow> </mrow> </mpadded> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mpadded> </munderover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" columnlines="" frame="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Channel</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset="-.24em"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">e</mi> </mrow> </mfrac> </mrow> </mrow> </mpadded> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mpadded> </munderover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" columnlines="" frame="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Decoder</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset="-.24em"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">e</mi> </mrow> </mfrac> </mrow> </mrow> </mpadded> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>W</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mpadded> </munderover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\xrightarrow[{\text{Message}}]{W}}{\begin{array}{|c| }\hline {\text{Encoder}}\\f_{n}\\\hline \end{array}}{\xrightarrow[{\mathrm {Encoded \atop sequence} }]{X^{n}}}{\begin{array}{|c| }\hline {\text{Channel}}\\p(y|x)\\\hline \end{array}}{\xrightarrow[{\mathrm {Received \atop sequence} }]{Y^{n}}}{\begin{array}{|c| }\hline {\text{Decoder}}\\g_{n}\\\hline \end{array}}{\xrightarrow[{\mathrm {Estimated \atop message} }]{\hat {W}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5366c809cb41ab57f4364b475895f13a9dd328" class="mwe-math-fallback-image-inline mw-invert skin-invert" title="Channel model" aria-hidden="true" style="vertical-align: -3.146ex; margin-bottom: -0.525ex; width:61.869ex; height:7.843ex;" alt="{\displaystyle {\xrightarrow[{\text{Message}}]{W}}{\begin{array}{|c| }\hline {\text{Encoder}}\\f_{n}\\\hline \end{array}}{\xrightarrow[{\mathrm {Encoded \atop sequence} }]{X^{n}}}{\begin{array}{|c| }\hline {\text{Channel}}\\p(y|x)\\\hline \end{array}}{\xrightarrow[{\mathrm {Received \atop sequence} }]{Y^{n}}}{\begin{array}{|c| }\hline {\text{Decoder}}\\g_{n}\\\hline \end{array}}{\xrightarrow[{\mathrm {Estimated \atop message} }]{\hat {W}}}}"></span></dd></dl> <p>Here <i>X</i> represents the space of messages transmitted, and <i>Y</i> the space of messages received during a unit time over our channel. Let <span class="texhtml"><i>p</i>(<i>y</i>|<i>x</i>)</span> be the <a href="/wiki/Conditional_probability" title="Conditional probability">conditional probability</a> distribution function of <i>Y</i> given <i>X</i>. We will consider <span class="texhtml"><i>p</i>(<i>y</i>|<i>x</i>)</span> to be an inherent fixed property of our communications channel (representing the nature of the <i><a href="/wiki/Signal_noise" class="mw-redirect" title="Signal noise">noise</a></i> of our channel). Then the joint distribution of <i>X</i> and <i>Y</i> is completely determined by our channel and by our choice of <span class="texhtml"><i>f</i>(<i>x</i>)</span>, the marginal distribution of messages we choose to send over the channel. Under these constraints, we would like to maximize the rate of information, or the <i><a href="/wiki/Signal_(electrical_engineering)" class="mw-redirect" title="Signal (electrical engineering)">signal</a></i>, we can communicate over the channel. The appropriate measure for this is the mutual information, and this maximum mutual information is called the <em>channel capacity</em> and is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=\max _{f}I(X;Y).\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </munder> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\max _{f}I(X;Y).\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6698ae5326003e7530add14e24f875f834fb68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-right: -0.204ex; width:17.809ex; height:4.343ex;" alt="{\displaystyle C=\max _{f}I(X;Y).\!}"></span></dd></dl> <p>This capacity has the following property related to communicating at information rate <i>R</i> (where <i>R</i> is usually bits per symbol). For any information rate <i>R</i> < <i>C</i> and coding error <i>ε</i> > 0, for large enough <i>N</i>, there exists a code of length <i>N</i> and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ <i>ε</i>; that is, it is always possible to transmit with arbitrarily small block error. In addition, for any rate <i>R</i> > <i>C</i>, it is impossible to transmit with arbitrarily small block error. </p><p><i><a href="/wiki/Channel_code" class="mw-redirect" title="Channel code">Channel coding</a></i> is concerned with finding such nearly optimal codes that can be used to transmit data over a noisy channel with a small coding error at a rate near the channel capacity. </p> <div class="mw-heading mw-heading4"><h4 id="Capacity_of_particular_channel_models">Capacity of particular channel models</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=15" title="Edit section: Capacity of particular channel models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A continuous-time analog communications channel subject to <a href="/wiki/Gaussian_noise" title="Gaussian noise">Gaussian noise</a>—see <a href="/wiki/Shannon%E2%80%93Hartley_theorem" title="Shannon–Hartley theorem">Shannon–Hartley theorem</a>.</li> <li>A <a href="/wiki/Binary_symmetric_channel" title="Binary symmetric channel">binary symmetric channel</a> (BSC) with crossover probability <i>p</i> is a binary input, binary output channel that flips the input bit with probability <i>p</i>. The BSC has a capacity of <span class="texhtml">1 − <i>H</i><sub>b</sub>(<i>p</i>)</span> bits per channel use, where <span class="texhtml"><i>H</i><sub>b</sub></span> is the binary entropy function to the base-2 logarithm:</li></ul> <dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Binary_symmetric_channel.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Binary_symmetric_channel.svg/156px-Binary_symmetric_channel.svg.png" decoding="async" width="156" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Binary_symmetric_channel.svg/234px-Binary_symmetric_channel.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Binary_symmetric_channel.svg/312px-Binary_symmetric_channel.svg.png 2x" data-file-width="156" data-file-height="167" /></a></span></dd></dl></dd></dl> <ul><li>A <a href="/wiki/Binary_erasure_channel" title="Binary erasure channel">binary erasure channel</a> (BEC) with erasure probability <i>p</i> is a binary input, ternary output channel. The possible channel outputs are 0, 1, and a third symbol 'e' called an erasure. The erasure represents complete loss of information about an input bit. The capacity of the BEC is <span class="nowrap">1 − <i>p</i></span> bits per channel use.</li></ul> <dl><dd><dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Binary_erasure_channel.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Binary_erasure_channel.svg/156px-Binary_erasure_channel.svg.png" decoding="async" width="156" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Binary_erasure_channel.svg/234px-Binary_erasure_channel.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Binary_erasure_channel.svg/312px-Binary_erasure_channel.svg.png 2x" data-file-width="156" data-file-height="167" /></a></span></dd></dl></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Channels_with_memory_and_directed_information">Channels with memory and directed information</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=16" title="Edit section: Channels with memory and directed information"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In practice many channels have memory. Namely, at time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> the channel is given by the conditional probability<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(y_{i}|x_{i},x_{i-1},x_{i-2},...,x_{1},y_{i-1},y_{i-2},...,y_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(y_{i}|x_{i},x_{i-1},x_{i-2},...,x_{1},y_{i-1},y_{i-2},...,y_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5485007c8ac5f922152674775f7c876c4bcb73d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.861ex; height:2.843ex;" alt="{\displaystyle P(y_{i}|x_{i},x_{i-1},x_{i-2},...,x_{1},y_{i-1},y_{i-2},...,y_{1})}"></span>. It is often more comfortable to use the notation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{i}=(x_{i},x_{i-1},x_{i-2},...,x_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{i}=(x_{i},x_{i-1},x_{i-2},...,x_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d94c204f6ab6c6678a43f8bff7b1f3c9091895a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.248ex; height:3.176ex;" alt="{\displaystyle x^{i}=(x_{i},x_{i-1},x_{i-2},...,x_{1})}"></span> and the channel become <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(y_{i}|x^{i},y^{i-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(y_{i}|x^{i},y^{i-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8789df9402aa82b9f55eb6cd52bccfef98ae3ce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.364ex; height:3.176ex;" alt="{\displaystyle P(y_{i}|x^{i},y^{i-1})}"></span>. In such a case the capacity is given by the <a href="/wiki/Mutual_information" title="Mutual information">mutual information</a> rate when there is no feedback available and the <a href="/wiki/Directed_information" title="Directed information">Directed information</a> rate in the case that either there is feedback or not<sup id="cite_ref-massey_30-1" class="reference"><a href="#cite_note-massey-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> (if there is no feedback the directed information equals the mutual information). </p> <div class="mw-heading mw-heading3"><h3 id="Fungible_information">Fungible information</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=17" title="Edit section: Fungible information"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Fungible information</b> is the <a href="/wiki/Information" title="Information">information</a> for which the means of <a href="/wiki/Encoding" class="mw-redirect" title="Encoding">encoding</a> is not important.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> Classical information theorists and computer scientists are mainly concerned with information of this sort. It is sometimes referred as speakable information.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications_to_other_fields">Applications to other fields</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=18" title="Edit section: Applications to other fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Intelligence_uses_and_secrecy_applications">Intelligence uses and secrecy applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=19" title="Edit section: Intelligence uses and secrecy applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Information_theory" title="Special:EditPage/Information theory">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">April 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Information theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the <a href="/wiki/Ban_(unit)" class="mw-redirect" title="Ban (unit)">ban</a>, was used in the <a href="/wiki/Ultra_(cryptography)" title="Ultra (cryptography)">Ultra</a> project, breaking the German <a href="/wiki/Enigma_machine" title="Enigma machine">Enigma machine</a> code and hastening the <a href="/wiki/Victory_in_Europe_Day" title="Victory in Europe Day">end of World War II in Europe</a>. Shannon himself defined an important concept now called the <a href="/wiki/Unicity_distance" title="Unicity distance">unicity distance</a>. Based on the redundancy of the <a href="/wiki/Plaintext" title="Plaintext">plaintext</a>, it attempts to give a minimum amount of <a href="/wiki/Ciphertext" title="Ciphertext">ciphertext</a> necessary to ensure unique decipherability. </p><p>Information theory leads us to believe it is much more difficult to keep secrets than it might first appear. A <a href="/wiki/Brute_force_attack" class="mw-redirect" title="Brute force attack">brute force attack</a> can break systems based on <a href="/wiki/Public-key_cryptography" title="Public-key cryptography">asymmetric key algorithms</a> or on most commonly used methods of <a href="/wiki/Symmetric-key_algorithm" title="Symmetric-key algorithm">symmetric key algorithms</a> (sometimes called secret key algorithms), such as <a href="/wiki/Block_cipher" title="Block cipher">block ciphers</a>. The security of all such methods comes from the assumption that no known attack can break them in a practical amount of time. </p><p><a href="/wiki/Information_theoretic_security" class="mw-redirect" title="Information theoretic security">Information theoretic security</a> refers to methods such as the <a href="/wiki/One-time_pad" title="One-time pad">one-time pad</a> that are not vulnerable to such brute force attacks. In such cases, the positive conditional mutual information between the plaintext and ciphertext (conditioned on the <a href="/wiki/Key_(cryptography)" title="Key (cryptography)">key</a>) can ensure proper transmission, while the unconditional mutual information between the plaintext and ciphertext remains zero, resulting in absolutely secure communications. In other words, an eavesdropper would not be able to improve his or her guess of the plaintext by gaining knowledge of the ciphertext but not of the key. However, as in any other cryptographic system, care must be used to correctly apply even information-theoretically secure methods; the <a href="/wiki/Venona_project" title="Venona project">Venona project</a> was able to crack the one-time pads of the Soviet Union due to their improper reuse of key material. </p> <div class="mw-heading mw-heading3"><h3 id="Pseudorandom_number_generation">Pseudorandom number generation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=20" title="Edit section: Pseudorandom number generation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Information_theory" title="Special:EditPage/Information theory">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">April 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><a href="/wiki/Pseudorandom_number_generator" title="Pseudorandom number generator">Pseudorandom number generators</a> are widely available in computer language libraries and application programs. They are, almost universally, unsuited to cryptographic use as they do not evade the deterministic nature of modern computer equipment and software. A class of improved random number generators is termed <a href="/wiki/Cryptographically_secure_pseudorandom_number_generator" title="Cryptographically secure pseudorandom number generator">cryptographically secure pseudorandom number generators</a>, but even they require <a href="/wiki/Random_seed" title="Random seed">random seeds</a> external to the software to work as intended. These can be obtained via <a href="/wiki/Extractor_(mathematics)" title="Extractor (mathematics)">extractors</a>, if done carefully. The measure of sufficient randomness in extractors is <a href="/wiki/Min-entropy" title="Min-entropy">min-entropy</a>, a value related to Shannon entropy through <a href="/wiki/R%C3%A9nyi_entropy" title="Rényi entropy">Rényi entropy</a>; Rényi entropy is also used in evaluating randomness in cryptographic systems. Although related, the distinctions among these measures mean that a random variable with high Shannon entropy is not necessarily satisfactory for use in an extractor and so for cryptography uses. </p> <div class="mw-heading mw-heading3"><h3 id="Seismic_exploration">Seismic exploration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=21" title="Edit section: Seismic exploration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One early commercial application of information theory was in the field of seismic oil exploration. Work in this field made it possible to strip off and separate the unwanted noise from the desired seismic signal. Information theory and <a href="/wiki/Digital_signal_processing" title="Digital signal processing">digital signal processing</a> offer a major improvement of resolution and image clarity over previous analog methods.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Semiotics">Semiotics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=22" title="Edit section: Semiotics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Semiotics" title="Semiotics">Semioticians</a> <a href="/w/index.php?title=Doede_Nauta&action=edit&redlink=1" class="new" title="Doede Nauta (page does not exist)">Doede Nauta</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://nl.wikipedia.org/wiki/Doede_Nauta" class="extiw" title="nl:Doede Nauta">nl</a>]</span> and <a href="/wiki/Winfried_N%C3%B6th" title="Winfried Nöth">Winfried Nöth</a> both considered <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a> as having created a theory of information in his works on semiotics.<sup id="cite_ref-Nauta_1972_43-0" class="reference"><a href="#cite_note-Nauta_1972-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 171">: 171 </span></sup><sup id="cite_ref-Nöth_2012_44-0" class="reference"><a href="#cite_note-Nöth_2012-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 137">: 137 </span></sup> Nauta defined semiotic information theory as the study of "<i>the internal processes of coding, filtering, and information processing.</i>"<sup id="cite_ref-Nauta_1972_43-1" class="reference"><a href="#cite_note-Nauta_1972-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 91">: 91 </span></sup> </p><p>Concepts from information theory such as redundancy and code control have been used by semioticians such as <a href="/wiki/Umberto_Eco" title="Umberto Eco">Umberto Eco</a> and <a href="/w/index.php?title=Ferruccio_Rossi-Landi&action=edit&redlink=1" class="new" title="Ferruccio Rossi-Landi (page does not exist)">Ferruccio Rossi-Landi</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://it.wikipedia.org/wiki/Ferruccio_Rossi-Landi" class="extiw" title="it:Ferruccio Rossi-Landi">it</a>]</span> to explain ideology as a form of message transmission whereby a dominant social class emits its message by using signs that exhibit a high degree of redundancy such that only one message is decoded among a selection of competing ones.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Integrated_process_organization_of_neural_information">Integrated process organization of neural information</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=23" title="Edit section: Integrated process organization of neural information"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quantitative information theoretic methods have been applied in <a href="/wiki/Cognitive_science" title="Cognitive science">cognitive science</a> to analyze the integrated process organization of neural information in the context of the <a href="/wiki/Binding_problem" title="Binding problem">binding problem</a> in <a href="/wiki/Cognitive_neuroscience" title="Cognitive neuroscience">cognitive neuroscience</a>.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> In this context, either an information-theoretical measure, such as <em>functional clusters</em> (<a href="/wiki/Gerald_Edelman" title="Gerald Edelman">Gerald Edelman</a> and <a href="/wiki/Giulio_Tononi" title="Giulio Tononi">Giulio Tononi</a>'s functional clustering model and dynamic core hypothesis (DCH)<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup>) or <em>effective information</em> (Tononi's <a href="/wiki/Integrated_information_theory" title="Integrated information theory">integrated information theory</a> (IIT) of consciousness<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup>), is defined (on the basis of a reentrant process organization, i.e. the synchronization of neurophysiological activity between groups of neuronal populations), or the measure of the minimization of free energy on the basis of statistical methods (<a href="/wiki/Karl_J._Friston" title="Karl J. Friston">Karl J. Friston</a>'s <a href="/wiki/Free_energy_principle" title="Free energy principle">free energy principle</a> (FEP), an information-theoretical measure which states that every adaptive change in a self-organized system leads to a minimization of free energy, and the <a href="/wiki/Bayesian_brain" class="mw-redirect" title="Bayesian brain">Bayesian brain</a> hypothesis<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup>). </p> <div class="mw-heading mw-heading3"><h3 id="Miscellaneous_applications">Miscellaneous applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=24" title="Edit section: Miscellaneous applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Information theory also has applications in the <a href="/wiki/Search_for_extraterrestrial_intelligence" title="Search for extraterrestrial intelligence">search for extraterrestrial intelligence</a>,<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Black_hole_information_paradox" title="Black hole information paradox">black holes</a>,<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Bioinformatics" title="Bioinformatics">bioinformatics</a>,<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Gambling_and_information_theory" title="Gambling and information theory">gambling</a>.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=25" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 21em;"> <ul><li><a href="/wiki/Algorithmic_probability" title="Algorithmic probability">Algorithmic probability</a></li> <li><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></li> <li><a href="/wiki/Communication_theory" title="Communication theory">Communication theory</a></li> <li><a href="/wiki/Constructor_theory" title="Constructor theory">Constructor theory</a> – a generalization of information theory that includes quantum information</li> <li><a href="/wiki/Formal_science" title="Formal science">Formal science</a></li> <li><a href="/wiki/Inductive_probability" title="Inductive probability">Inductive probability</a></li> <li><a href="/wiki/Info-metrics" title="Info-metrics">Info-metrics</a></li> <li><a href="/wiki/Minimum_message_length" title="Minimum message length">Minimum message length</a></li> <li><a href="/wiki/Minimum_description_length" title="Minimum description length">Minimum description length</a></li> <li><a href="/wiki/Philosophy_of_information" title="Philosophy of information">Philosophy of information</a></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Applications">Applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=26" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col"> <ul><li><a href="/wiki/Active_networking" title="Active networking">Active networking</a></li> <li><a href="/wiki/Cryptanalysis" title="Cryptanalysis">Cryptanalysis</a></li> <li><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a></li> <li><a href="/wiki/Cybernetics" title="Cybernetics">Cybernetics</a></li> <li><a href="/wiki/Entropy_in_thermodynamics_and_information_theory" title="Entropy in thermodynamics and information theory">Entropy in thermodynamics and information theory</a></li> <li><a href="/wiki/Gambling" title="Gambling">Gambling</a></li> <li><a href="/wiki/Intelligence_(information_gathering)" class="mw-redirect" title="Intelligence (information gathering)">Intelligence (information gathering)</a></li> <li><a href="/wiki/Reflection_seismology" title="Reflection seismology">Seismic exploration</a></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="History">History</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=27" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ralph_Hartley" title="Ralph Hartley">Hartley, R.V.L.</a></li> <li><a href="/wiki/History_of_information_theory" title="History of information theory">History of information theory</a></li> <li><a href="/wiki/Claude_Elwood_Shannon" class="mw-redirect" title="Claude Elwood Shannon">Shannon, C.E.</a></li> <li><a href="/wiki/Timeline_of_information_theory" title="Timeline of information theory">Timeline of information theory</a></li> <li><a href="/wiki/Hubert_Yockey" title="Hubert Yockey">Yockey, H.P.</a></li> <li><a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Andrey Kolmogorov</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Theory">Theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=28" title="Edit section: Theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col" style="column-width: 16em;"> <ul><li><a href="/wiki/Coding_theory" title="Coding theory">Coding theory</a></li> <li><a href="/wiki/Detection_theory" title="Detection theory">Detection theory</a></li> <li><a href="/wiki/Estimation_theory" title="Estimation theory">Estimation theory</a></li> <li><a href="/wiki/Fisher_information" title="Fisher information">Fisher information</a></li> <li><a href="/wiki/Information_algebra" title="Information algebra">Information algebra</a></li> <li><a href="/wiki/Information_asymmetry" title="Information asymmetry">Information asymmetry</a></li> <li><a href="/wiki/Information_field_theory" title="Information field theory">Information field theory</a></li> <li><a href="/wiki/Information_geometry" title="Information geometry">Information geometry</a></li> <li><a href="/wiki/Information_theory_and_measure_theory" title="Information theory and measure theory">Information theory and measure theory</a></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/List_of_unsolved_problems_in_information_theory" title="List of unsolved problems in information theory">List of unsolved problems in information theory</a></li> <li><a href="/wiki/Logic_of_information" title="Logic of information">Logic of information</a></li> <li><a href="/wiki/Network_coding" class="mw-redirect" title="Network coding">Network coding</a></li> <li><a href="/wiki/Philosophy_of_information" title="Philosophy of information">Philosophy of information</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li> <li><a href="/wiki/Source_coding" class="mw-redirect" title="Source coding">Source coding</a></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Concepts">Concepts</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=29" title="Edit section: Concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col" style="column-width: 16em;"> <ul><li><a href="/wiki/Ban_(unit)" class="mw-redirect" title="Ban (unit)">Ban (unit)</a></li> <li><a href="/wiki/Channel_capacity" title="Channel capacity">Channel capacity</a></li> <li><a href="/wiki/Communication_channel" title="Communication channel">Communication channel</a></li> <li><a href="/wiki/Communication_source" title="Communication source">Communication source</a></li> <li><a href="/wiki/Conditional_entropy" title="Conditional entropy">Conditional entropy</a></li> <li><a href="/wiki/Covert_channel" title="Covert channel">Covert channel</a></li> <li><a href="/wiki/Data_compression" title="Data compression">Data compression</a></li> <li>Decoder</li> <li><a href="/wiki/Differential_entropy" title="Differential entropy">Differential entropy</a></li> <li><a href="/wiki/Fungible_information" class="mw-redirect" title="Fungible information">Fungible information</a></li> <li><a href="/wiki/Information_fluctuation_complexity" title="Information fluctuation complexity">Information fluctuation complexity</a></li> <li><a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">Information entropy</a></li> <li><a href="/wiki/Joint_entropy" title="Joint entropy">Joint entropy</a></li> <li><a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback–Leibler divergence</a></li> <li><a href="/wiki/Mutual_information" title="Mutual information">Mutual information</a></li> <li><a href="/wiki/Pointwise_mutual_information" title="Pointwise mutual information">Pointwise mutual information</a> (PMI)</li> <li><a href="/wiki/Receiver_(information_theory)" title="Receiver (information theory)">Receiver (information theory)</a></li> <li><a href="/wiki/Redundancy_(information_theory)" title="Redundancy (information theory)">Redundancy</a></li> <li><a href="/wiki/R%C3%A9nyi_entropy" title="Rényi entropy">Rényi entropy</a></li> <li><a href="/wiki/Self-information" class="mw-redirect" title="Self-information">Self-information</a></li> <li><a href="/wiki/Unicity_distance" title="Unicity distance">Unicity distance</a></li> <li><a href="/wiki/Variety_(cybernetics)" title="Variety (cybernetics)">Variety</a></li> <li><a href="/wiki/Hamming_distance" title="Hamming distance">Hamming distance</a></li> <li><a href="/wiki/Perplexity" title="Perplexity">Perplexity</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=30" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSchneider2006" class="citation journal cs1">Schneider, Thomas D. 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+D+%28The+Statistician%29&rft.atitle=The+Kelly+Criterion+and+Bet+Comparisons+in+Spread+Betting&rft.volume=49&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E531-%3C%2Fspan%3E539&rft.date=2000&rft_id=info%3Adoi%2F10.1111%2F1467-9884.00251&rft.issn=1467-9884&rft.aulast=Haigh&rft.aufirst=John&rft_id=https%3A%2F%2Frss.onlinelibrary.wiley.com%2Fdoi%2F10.1111%2F1467-9884.00251&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInformation+theory" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=31" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="The_classic_work">The classic work</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=32" title="Edit section: The classic work"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><a href="/wiki/Claude_Elwood_Shannon" class="mw-redirect" title="Claude Elwood Shannon">Shannon, C.E.</a> (1948), "<a href="/wiki/A_Mathematical_Theory_of_Communication" title="A Mathematical Theory of Communication">A Mathematical Theory of Communication</a>", <i>Bell System Technical Journal</i>, 27, pp. 379–423 & 623–656, July & October, 1948. <a rel="nofollow" class="external text" href="http://math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf">PDF.</a> <br /><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150409204946/http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html">Notes and other formats.</a></li> <li>R.V.L. Hartley, <a rel="nofollow" class="external text" href="http://www.dotrose.com/etext/90_Miscellaneous/transmission_of_information_1928b.pdf">"Transmission of Information"</a>, <i>Bell System Technical Journal</i>, July 1928</li> <li><a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Andrey Kolmogorov</a> (1968), "<a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/pdf/10.1080/00207166808803030">Three approaches to the quantitative definition of information</a>" in <i><a href="/wiki/International_Journal_of_Computer_Mathematics" title="International Journal of Computer Mathematics">International Journal of Computer Mathematics</a></i>, 2, pp. 157–168.</li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Other_journal_articles">Other journal articles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=33" title="Edit section: Other journal articles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li>J. 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Dev</i>. <b>5</b> (3): <span class="nowrap">183–</span>191. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1147%2Frd.53.0183">10.1147/rd.53.0183</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IBM+J.+Res.+Dev.&rft.atitle=Irreversibility+and+Heat+Generation+in+the+Computing+Process&rft.volume=5&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E183-%3C%2Fspan%3E191&rft.date=1961&rft_id=info%3Adoi%2F10.1147%2Frd.53.0183&rft.aulast=Landauer&rft.aufirst=R.&rft_id=http%3A%2F%2Fwww.research.ibm.com%2Fjournal%2Frd%2F441%2Flandauerii.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInformation+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTimmeAlfordFleckerBeggs2012" class="citation arxiv cs1">Timme, Nicholas; Alford, Wesley; Flecker, Benjamin; Beggs, John M. (2012). "Multivariate information measures: an experimentalist's perspective". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1111.6857">1111.6857</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.IT">cs.IT</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Multivariate+information+measures%3A+an+experimentalist%27s+perspective&rft.date=2012&rft_id=info%3Aarxiv%2F1111.6857&rft.aulast=Timme&rft.aufirst=Nicholas&rft.au=Alford%2C+Wesley&rft.au=Flecker%2C+Benjamin&rft.au=Beggs%2C+John+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInformation+theory" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Textbooks_on_information_theory">Textbooks on information theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=34" title="Edit section: Textbooks on information theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li>Alajaji, F. and Chen, P.N. An Introduction to Single-User Information Theory. Singapore: Springer, 2018. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-10-8000-5" title="Special:BookSources/978-981-10-8000-5">978-981-10-8000-5</a></li> <li>Arndt, C. <i>Information Measures, Information and its Description in Science and Engineering</i> (Springer Series: Signals and Communication Technology), 2004, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-40855-0" title="Special:BookSources/978-3-540-40855-0">978-3-540-40855-0</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAsh1990" class="citation book cs1">Ash, Robert B. (1990) [1965]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ngZhvUfF0UIC&q=intitle:information+intitle:theory+inauthor:ash+conditional+uncertainty&pg=PA16"><i>Information Theory</i></a>. New York: Dover Publications, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-66521-6" title="Special:BookSources/0-486-66521-6"><bdi>0-486-66521-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Information+Theory&rft.place=New+York&rft.pub=Dover+Publications%2C+Inc.&rft.date=1990&rft.isbn=0-486-66521-6&rft.aulast=Ash&rft.aufirst=Robert+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DngZhvUfF0UIC%26q%3Dintitle%3Ainformation%2Bintitle%3Atheory%2Binauthor%3Aash%2Bconditional%2Buncertainty%26pg%3DPA16&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInformation+theory" class="Z3988"></span></li> <li><a href="/wiki/Gallager,_R" class="mw-redirect" title="Gallager, R">Gallager, R</a>. <i>Information Theory and Reliable Communication.</i> New York: John Wiley and Sons, 1968. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-29048-3" title="Special:BookSources/0-471-29048-3">0-471-29048-3</a></li> <li>Goldman, S. <i>Information Theory</i>. New York: Prentice Hall, 1953. New York: Dover 1968 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-62209-6" title="Special:BookSources/0-486-62209-6">0-486-62209-6</a>, 2005 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-44271-3" title="Special:BookSources/0-486-44271-3">0-486-44271-3</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoverThomas2006" class="citation book cs1"><a href="/wiki/Thomas_M._Cover" title="Thomas M. Cover">Cover, Thomas</a>; Thomas, Joy A. (2006). <i>Elements of information theory</i> (2nd ed.). New York: <a href="/wiki/Wiley-Interscience" class="mw-redirect" title="Wiley-Interscience">Wiley-Interscience</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-24195-4" title="Special:BookSources/0-471-24195-4"><bdi>0-471-24195-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+information+theory&rft.place=New+York&rft.edition=2nd&rft.pub=Wiley-Interscience&rft.date=2006&rft.isbn=0-471-24195-4&rft.aulast=Cover&rft.aufirst=Thomas&rft.au=Thomas%2C+Joy+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInformation+theory" class="Z3988"></span></li> <li><a href="/wiki/Csiszar,_I" class="mw-redirect" title="Csiszar, I">Csiszar, I</a>, Korner, J. <i>Information Theory: Coding Theorems for Discrete Memoryless Systems</i> Akademiai Kiado: 2nd edition, 1997. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/963-05-7440-3" title="Special:BookSources/963-05-7440-3">963-05-7440-3</a></li> <li><a href="/wiki/David_J._C._MacKay" title="David J. C. MacKay">MacKay, David J. C.</a> <i><a rel="nofollow" class="external text" href="http://www.inference.phy.cam.ac.uk/mackay/itila/book.html">Information Theory, Inference, and Learning Algorithms</a></i> Cambridge: Cambridge University Press, 2003. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-64298-1" title="Special:BookSources/0-521-64298-1">0-521-64298-1</a></li> <li>Mansuripur, M. <i>Introduction to Information Theory</i>. New York: Prentice Hall, 1987. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-484668-0" title="Special:BookSources/0-13-484668-0">0-13-484668-0</a></li> <li><a href="/wiki/Robert_McEliece" title="Robert McEliece">McEliece, R</a>. <i>The Theory of Information and Coding</i>. Cambridge, 2002. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0521831857" title="Special:BookSources/978-0521831857">978-0521831857</a></li> <li><a href="/wiki/John_R._Pierce" title="John R. Pierce">Pierce, JR</a>. "An introduction to information theory: symbols, signals and noise". Dover (2nd Edition). 1961 (reprinted by Dover 1980).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReza1994" class="citation book cs1"><a href="/wiki/Fazlollah_Reza" title="Fazlollah Reza">Reza, Fazlollah M.</a> (1994) [1961]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RtzpRAiX6OgC&q=intitle:%22An+Introduction+to+Information+Theory%22++%22entropy+of+a+simple+source%22&pg=PA8"><i>An Introduction to Information Theory</i></a>. New York: Dover Publications, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-68210-2" title="Special:BookSources/0-486-68210-2"><bdi>0-486-68210-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Information+Theory&rft.place=New+York&rft.pub=Dover+Publications%2C+Inc.&rft.date=1994&rft.isbn=0-486-68210-2&rft.aulast=Reza&rft.aufirst=Fazlollah+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRtzpRAiX6OgC%26q%3Dintitle%3A%2522An%2BIntroduction%2Bto%2BInformation%2BTheory%2522%2B%2B%2522entropy%2Bof%2Ba%2Bsimple%2Bsource%2522%26pg%3DPA8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInformation+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShannonWeaver1949" class="citation book cs1"><a href="/wiki/Claude_Shannon" title="Claude Shannon">Shannon, Claude</a>; <a href="/wiki/Warren_Weaver" title="Warren Weaver">Weaver, Warren</a> (1949). <a rel="nofollow" class="external text" href="https://monoskop.org/images/b/be/Shannon_Claude_E_Weaver_Warren_The_Mathematical_Theory_of_Communication_1963.pdf"><i>The Mathematical Theory of Communication</i></a> <span class="cs1-format">(PDF)</span>. <a href="/wiki/Urbana,_Illinois" title="Urbana, Illinois">Urbana, Illinois</a>: <a href="/wiki/University_of_Illinois_Press" title="University of Illinois Press">University of Illinois Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-252-72548-4" title="Special:BookSources/0-252-72548-4"><bdi>0-252-72548-4</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/49-11922">49-11922</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical+Theory+of+Communication&rft.place=Urbana%2C+Illinois&rft.pub=University+of+Illinois+Press&rft.date=1949&rft_id=info%3Alccn%2F49-11922&rft.isbn=0-252-72548-4&rft.aulast=Shannon&rft.aufirst=Claude&rft.au=Weaver%2C+Warren&rft_id=http%3A%2F%2Fmonoskop.org%2Fimages%2Fb%2Fbe%2FShannon_Claude_E_Weaver_Warren_The_Mathematical_Theory_of_Communication_1963.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInformation+theory" class="Z3988"></span></li> <li>Stone, JV. Chapter 1 of book <a rel="nofollow" class="external text" href="https://jamesstone.sites.sheffield.ac.uk/books/information-theory-2nd-edition">"Information Theory: A Tutorial Introduction"</a>, University of Sheffield, England, 2014. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0956372857" title="Special:BookSources/978-0956372857">978-0956372857</a>.</li> <li>Yeung, RW. <i><a rel="nofollow" class="external text" href="http://iest2.ie.cuhk.edu.hk/~whyeung/book/">A First Course in Information Theory</a></i> Kluwer Academic/Plenum Publishers, 2002. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-306-46791-7" title="Special:BookSources/0-306-46791-7">0-306-46791-7</a>.</li> <li>Yeung, RW. <i><a rel="nofollow" class="external text" href="http://iest2.ie.cuhk.edu.hk/~whyeung/book2/">Information Theory and Network Coding</a></i> Springer 2008, 2002. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-79233-0" title="Special:BookSources/978-0-387-79233-0">978-0-387-79233-0</a></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Other_books">Other books</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=35" title="Edit section: Other books"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li>Leon Brillouin, <i>Science and Information Theory</i>, Mineola, N.Y.: Dover, [1956, 1962] 2004. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-43918-6" title="Special:BookSources/0-486-43918-6">0-486-43918-6</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGleick2011" class="citation book cs1"><a href="/wiki/James_Gleick" title="James Gleick">Gleick, James</a> (2011). <a href="/w/index.php?title=The_Information:_A_History,_a_Theory,_a_Flood_Pantheon_Books&action=edit&redlink=1" class="new" title="The Information: A History, a Theory, a Flood Pantheon Books (page does not exist)"><i>The Information: A History, a Theory, a Flood</i></a> (1st ed.). New York: Pantheon Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4000-9623-7" title="Special:BookSources/978-1-4000-9623-7"><bdi>978-1-4000-9623-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/607975727">607975727</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Information%3A+A+History%2C+a+Theory%2C+a+Flood&rft.place=New+York&rft.edition=1st&rft.pub=Pantheon+Books&rft.date=2011&rft_id=info%3Aoclcnum%2F607975727&rft.isbn=978-1-4000-9623-7&rft.aulast=Gleick&rft.aufirst=James&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInformation+theory" class="Z3988"></span></li> <li>A. I. Khinchin, <i>Mathematical Foundations of Information Theory</i>, New York: Dover, 1957. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-60434-9" title="Special:BookSources/0-486-60434-9">0-486-60434-9</a></li> <li>H. S. Leff and A. F. Rex, Editors, <i>Maxwell's Demon: Entropy, Information, Computing</i>, Princeton University Press, Princeton, New Jersey (1990). <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08727-X" title="Special:BookSources/0-691-08727-X">0-691-08727-X</a></li> <li><a href="/wiki/Robert_K._Logan" title="Robert K. Logan">Robert K. Logan</a>. <i>What is Information? - Propagating Organization in the Biosphere, the Symbolosphere, the Technosphere and the Econosphere</i>, Toronto: DEMO Publishing.</li> <li>Tom Siegfried, <i>The Bit and the Pendulum</i>, Wiley, 2000. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-32174-5" title="Special:BookSources/0-471-32174-5">0-471-32174-5</a></li> <li><a href="/wiki/Charles_Seife" title="Charles Seife">Charles Seife</a>, <i><a href="/wiki/Decoding_the_Universe" title="Decoding the Universe">Decoding the Universe</a></i>, Viking, 2006. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-670-03441-X" title="Special:BookSources/0-670-03441-X">0-670-03441-X</a></li> <li>Jeremy Campbell, <i><a href="/wiki/Grammatical_Man" title="Grammatical Man">Grammatical Man</a></i>, Touchstone/Simon & Schuster, 1982, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-671-44062-4" title="Special:BookSources/0-671-44062-4">0-671-44062-4</a></li> <li>Henri Theil, <i>Economics and Information Theory</i>, Rand McNally & Company - Chicago, 1967.</li> <li>Escolano, Suau, Bonev, <i><a rel="nofollow" class="external text" href="https://www.springer.com/computer/image+processing/book/978-1-84882-296-2">Information Theory in Computer Vision and Pattern Recognition</a></i>, Springer, 2009. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84882-296-2" title="Special:BookSources/978-1-84882-296-2">978-1-84882-296-2</a></li> <li>Vlatko Vedral, <i>Decoding Reality: The Universe as Quantum Information</i>, Oxford University Press 2010. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-923769-7" title="Special:BookSources/0-19-923769-7">0-19-923769-7</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Information_theory&action=edit&section=36" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox 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class="side-box-text plainlist"><ul><li><a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=wp&su=Information+theory">Resources in your library</a></li> <li><a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=wp&su=Information+theory&library=0CHOOSE0">Resources in other libraries</a></li> </ul></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Information">"Information"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Information&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DInformation&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInformation+theory" class="Z3988"></span></li> <li>Lambert F. L. (1999), "<a rel="nofollow" class="external text" href="http://jchemed.chem.wisc.edu/Journal/Issues/1999/Oct/abs1385.html">Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms - Examples of Entropy Increase? Nonsense!</a>", <i>Journal of Chemical Education</i></li> <li><a rel="nofollow" class="external text" href="http://www.itsoc.org/">IEEE Information Theory Society</a> and <a rel="nofollow" class="external text" href="https://www.itsoc.org/resources/surveys">ITSOC Monographs, Surveys, and Reviews</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180612144016/https://www.itsoc.org/resources/surveys">Archived</a> 2018-06-12 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output 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class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Cybernetics" title="Template:Cybernetics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Cybernetics" title="Template talk:Cybernetics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Cybernetics" title="Special:EditPage/Template:Cybernetics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Subfields_of_and_cyberneticians_involved_in_cybernetics100" style="font-size:114%;margin:0 4em">Subfields of and cyberneticians involved in <a href="/wiki/Cybernetics" title="Cybernetics">cybernetics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Subfields</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</a></li> <li><a href="/wiki/Biocybernetics" title="Biocybernetics">Biological cybernetics</a></li> <li><a href="/wiki/Biomedical_cybernetics" title="Biomedical cybernetics">Biomedical cybernetics</a></li> <li><a href="/wiki/Biorobotics" title="Biorobotics">Biorobotics</a></li> <li><a href="/wiki/Biosemiotics" title="Biosemiotics">Biosemiotics</a></li> <li><a href="/wiki/Brain%E2%80%93computer_interface" title="Brain–computer interface">Neurocybernetics</a></li> <li><a href="/wiki/Catastrophe_theory" title="Catastrophe theory">Catastrophe theory</a></li> <li><a href="/wiki/Computational_neuroscience" title="Computational neuroscience">Computational neuroscience</a></li> <li><a href="/wiki/Connectionism" title="Connectionism">Connectionism</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li> <li><a href="/wiki/Conversation_theory" title="Conversation theory">Conversation theory</a></li> <li><a href="/wiki/Cybernetics_in_the_Soviet_Union" title="Cybernetics in the Soviet Union">Cybernetics in the Soviet Union</a></li> <li><a href="/wiki/Decision_theory" title="Decision theory">Decision theory</a></li> <li><a href="/wiki/Emergence" title="Emergence">Emergence</a></li> <li><a href="/wiki/Engineering_cybernetics" title="Engineering cybernetics">Engineering cybernetics</a></li> <li><a href="/wiki/Homeostasis" title="Homeostasis">Homeostasis</a></li> <li><a class="mw-selflink selflink">Information theory</a></li> <li><a href="/wiki/Management_cybernetics" title="Management cybernetics">Management cybernetics</a></li> <li><a href="/wiki/Medical_cybernetics" title="Medical cybernetics">Medical cybernetics</a></li> <li><a href="/wiki/Second-order_cybernetics" title="Second-order cybernetics">Second-order cybernetics</a></li> <li><a href="/wiki/Cybersemiotics" class="mw-redirect" title="Cybersemiotics">Cybersemiotics</a></li> <li><a href="/wiki/Sociocybernetics" title="Sociocybernetics">Sociocybernetics</a></li> <li><a href="/wiki/Synergetics_(Haken)" title="Synergetics (Haken)">Synergetics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Cyberneticist" title="Cyberneticist">Cyberneticians</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexander_Lerner" title="Alexander Lerner">Alexander Lerner</a></li> <li><a href="/wiki/Alexey_Lyapunov" title="Alexey Lyapunov">Alexey Lyapunov</a></li> <li><a href="/wiki/Alfred_Radcliffe-Brown" title="Alfred Radcliffe-Brown">Alfred Radcliffe-Brown</a></li> <li><a href="/wiki/Allenna_Leonard" title="Allenna Leonard">Allenna Leonard</a></li> <li><a href="/wiki/Anthony_Wilden" title="Anthony Wilden">Anthony Wilden</a></li> <li><a href="/wiki/Buckminster_Fuller" title="Buckminster Fuller">Buckminster Fuller</a></li> <li><a href="/wiki/Charles_Fran%C3%A7ois_(systems_scientist)" title="Charles François (systems scientist)">Charles François</a></li> <li><a href="/wiki/Genevieve_Bell" title="Genevieve Bell">Genevieve Bell</a></li> <li><a href="/wiki/Margaret_Boden" title="Margaret Boden">Margaret Boden</a></li> <li><a href="/wiki/Claude_Bernard" title="Claude Bernard">Claude Bernard</a></li> <li><a href="/wiki/Cliff_Joslyn" title="Cliff Joslyn">Cliff Joslyn</a></li> <li><a href="/wiki/Erich_von_Holst" title="Erich von Holst">Erich von Holst</a></li> <li><a href="/wiki/Ernst_von_Glasersfeld" title="Ernst von Glasersfeld">Ernst von Glasersfeld</a></li> <li><a href="/wiki/Francis_Heylighen" title="Francis Heylighen">Francis Heylighen</a></li> <li><a href="/wiki/Francisco_Varela" title="Francisco Varela">Francisco Varela</a></li> <li><a href="/wiki/Frederic_Vester" title="Frederic Vester">Frederic Vester</a></li> <li><a href="/wiki/Geoffrey_Vickers" title="Geoffrey Vickers">Charles Geoffrey Vickers</a></li> <li><a href="/wiki/Gordon_Pask" title="Gordon Pask">Gordon Pask</a></li> <li><a href="/wiki/Gordon_S._Brown" title="Gordon S. Brown">Gordon S. Brown</a></li> <li><a href="/wiki/Gregory_Bateson" title="Gregory Bateson">Gregory Bateson</a></li> <li><a href="/wiki/Heinz_von_Foerster" title="Heinz von Foerster">Heinz von Foerster</a></li> <li><a href="/wiki/Humberto_Maturana" title="Humberto Maturana">Humberto Maturana</a></li> <li><a href="/wiki/I._A._Richards" title="I. A. Richards">I. A. Richards</a></li> <li><a href="/wiki/Igor_Aleksander" title="Igor Aleksander">Igor Aleksander</a></li> <li><a href="/wiki/Jacque_Fresco" title="Jacque Fresco">Jacque Fresco</a></li> <li><a href="/wiki/Jakob_Johann_von_Uexk%C3%BCll" title="Jakob Johann von Uexküll">Jakob von Uexküll</a></li> <li><a href="/wiki/Jason_Jixuan_Hu" title="Jason Jixuan Hu">Jason Jixuan Hu</a></li> <li><a href="/wiki/Jay_Wright_Forrester" title="Jay Wright Forrester">Jay Wright Forrester</a></li> <li><a href="/wiki/Jennifer_Wilby" title="Jennifer Wilby">Jennifer Wilby</a></li> <li><a href="/wiki/John_N._Warfield" title="John N. Warfield">John N. Warfield</a></li> <li><a href="/wiki/Kevin_Warwick" title="Kevin Warwick">Kevin Warwick</a></li> <li><a href="/wiki/Ludwig_von_Bertalanffy" title="Ludwig von Bertalanffy">Ludwig von Bertalanffy</a></li> <li><a href="/wiki/Maleyka_Abbaszadeh" title="Maleyka Abbaszadeh">Maleyka Abbaszadeh</a></li> <li><a href="/wiki/Manfred_Clynes" title="Manfred Clynes">Manfred Clynes</a></li> <li><a href="/wiki/Margaret_Mead" title="Margaret Mead">Margaret Mead</a></li> <li><a href="/wiki/Marian_Mazur" title="Marian Mazur">Marian Mazur</a></li> <li><a href="/wiki/N._Katherine_Hayles" title="N. Katherine Hayles">N. Katherine Hayles</a></li> <li><a href="/wiki/Natalia_Bekhtereva" title="Natalia Bekhtereva">Natalia Bekhtereva</a></li> <li><a href="/wiki/Niklas_Luhmann" title="Niklas Luhmann">Niklas Luhmann</a></li> <li><a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a></li> <li><a href="/wiki/Petro_Grigorenko" title="Petro Grigorenko">Pyotr Grigorenko</a></li> <li><a href="/wiki/Qian_Xuesen" title="Qian Xuesen">Qian Xuesen</a></li> <li><a href="/wiki/Ranulph_Glanville" title="Ranulph Glanville">Ranulph Glanville</a></li> <li><a href="/wiki/Robert_Trappl" title="Robert Trappl">Robert Trappl</a></li> <li><a href="/wiki/Sergei_P._Kurdyumov" class="mw-redirect" title="Sergei P. Kurdyumov">Sergei P. Kurdyumov</a></li> <li><a href="/wiki/Stafford_Beer" title="Stafford Beer">Anthony Stafford Beer</a></li> <li><a href="/wiki/Stuart_Kauffman" title="Stuart Kauffman">Stuart Kauffman</a></li> <li><a href="/wiki/Stuart_Umpleby" title="Stuart Umpleby">Stuart Umpleby</a></li> <li><a href="/wiki/Talcott_Parsons" title="Talcott Parsons">Talcott Parsons</a></li> <li><a href="/wiki/Ulla_Mitzdorf" title="Ulla Mitzdorf">Ulla Mitzdorf</a></li> <li><a href="/wiki/Valentin_Turchin" title="Valentin Turchin">Valentin Turchin</a></li> <li><a href="/wiki/Valentino_Braitenberg" title="Valentino Braitenberg">Valentin Braitenberg</a></li> <li><a href="/wiki/W._Ross_Ashby" title="W. Ross Ashby">William Ross Ashby</a></li> <li><a href="/wiki/Walter_Bradford_Cannon" title="Walter Bradford Cannon">Walter Bradford Cannon</a></li> <li><a href="/wiki/Walter_Pitts" title="Walter Pitts">Walter Pitts</a></li> <li><a href="/wiki/Warren_Sturgis_McCulloch" title="Warren Sturgis McCulloch">Warren McCulloch</a></li> <li><a href="/wiki/William_Grey_Walter" title="William Grey Walter">William Grey Walter</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Information_processing141" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Informatics" title="Template:Informatics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Informatics" title="Template talk:Informatics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Informatics" title="Special:EditPage/Template:Informatics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Information_processing141" style="font-size:114%;margin:0 4em"><a href="/wiki/Information_technology" title="Information technology">Information processing</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Information processes</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">information processes by function</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Perception" title="Perception">perception</a></li> <li><a href="/wiki/Attention" title="Attention">attention</a></li> <li><a href="/wiki/Social_influence" title="Social influence">influence</a></li> <li><a href="/wiki/Remote_control" title="Remote control">operating</a></li> <li><a href="/wiki/Communication" title="Communication">communication</a></li> <li><a href="/wiki/Reason" title="Reason">reasoning</a></li> <li><a href="/wiki/Learning" title="Learning">learning</a></li> <li><a href="/wiki/Information_storage" class="mw-redirect" title="Information storage">storing</a></li> <li><a href="/wiki/Decision-making" title="Decision-making">decision-making</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">information processing abstractions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Event_processing" class="mw-redirect" title="Event processing">event processing</a></li> <li><a href="/wiki/Semiotics" title="Semiotics">sign processesing</a></li> <li><a href="/wiki/Signal" title="Signal">signal processing</a></li> <li><a href="/wiki/Data_processing" title="Data processing">data processing</a></li> <li><a href="/wiki/Stream_processing" title="Stream processing">stream processing</a></li> <li><a href="/wiki/Multi-agent_system" title="Multi-agent system">agent processing</a></li> <li><a href="/wiki/State_(computer_science)" title="State (computer science)">state processing</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Information processors</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">natural</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Natural_computing#Nature_as_information_processing" title="Natural computing">nature as information processing</a></li> <li><a href="/wiki/Information_processing_theory#Humans_as_Information_Processing_Systems" title="Information processing theory">humans as information processing systems</a></li> <li><a href="/wiki/Social_information_processing" title="Social information processing">society as information processing system</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">mixed</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mixed_reality" title="Mixed reality">mixed reality</a></li> <li><a href="/wiki/Brain%E2%80%93computer_interface" title="Brain–computer interface">brain–computer interface</a></li> <li><a href="/wiki/Physical_computing" title="Physical computing">physical computing</a></li> <li><a href="/wiki/Human%E2%80%93computer_interaction" title="Human–computer interaction">human–computer interaction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">artificial</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Processor_(computing)" title="Processor (computing)">processors</a> and <a href="/wiki/Process_(computing)" title="Process (computing)">processes</a></li> <li><a href="/wiki/Bio-inspired_computing" title="Bio-inspired computing">bio-inspired computing</a></li> <li><a href="/wiki/Ubiquitous_computing" title="Ubiquitous computing">ubiquitous computing</a></li> <li><a href="/wiki/Artificial_brain" title="Artificial brain">artificial brain</a> and <a href="/wiki/Mind_uploading" title="Mind uploading">mind uploading</a></li> <li><a href="/wiki/Virtual_reality" title="Virtual reality">virtual reality</a></li> <li><a href="/wiki/Virtual_world" title="Virtual world">virtual world</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Information processing <br />theories and concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">in biology</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computational_biology" title="Computational biology">computational</a> and <a href="/wiki/Systems_biology" title="Systems biology">systems biology</a></li> <li><a href="/wiki/Genome_informatics" title="Genome informatics">genetic informatics</a> and <a href="/wiki/Cellular_computing" class="mw-redirect" title="Cellular computing">cellular computing</a></li> <li><a href="/wiki/Computational_neuroscience" title="Computational neuroscience">computational neuroscience</a> and <a href="/wiki/Neurocomputing" class="mw-redirect" title="Neurocomputing">neurocomputing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">in cognitive psychology</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Information_processing_theory" title="Information processing theory">information processing theory</a></li> <li><a href="/wiki/Mind" title="Mind">mind</a> and <a href="/wiki/Intelligence" title="Intelligence">intelligence</a></li> <li><a href="/wiki/Cognitive_informatics" class="mw-redirect" title="Cognitive informatics">cognitive informatics</a> and <a href="/wiki/Neuroinformatics" title="Neuroinformatics">neuroinformatics</a></li> <li><a href="/wiki/Behavior_informatics" title="Behavior informatics">behavior informatics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">in computer science</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Neural_computation" title="Neural computation">neural computation</a></li> <li><a href="/wiki/Theory_of_computation" title="Theory of computation">computation theory</a></li> <li><a href="/wiki/Algorithms" class="mw-redirect" title="Algorithms">algorithms</a> and <a href="/wiki/Information_structure" title="Information structure">information structures</a></li> <li><a href="/wiki/Circuit_(computer_science)" title="Circuit (computer science)">computational circuits</a></li> <li><a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">in philosophy</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computational_theory_of_mind" title="Computational theory of mind">computational theory of mind</a></li> <li><a href="/wiki/Philosophy_of_information" title="Philosophy of information">philosophy of information</a></li> <li><a href="/wiki/Philosophy_of_artificial_intelligence" title="Philosophy of artificial intelligence">philosophy of artificial intelligence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">interdisciplinary</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">information theory</a></li> <li><a href="/wiki/Decision_theory" title="Decision theory">decision theory</a></li> <li><a href="/wiki/Systems_theory" title="Systems theory">systems theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Infosphere" title="Infosphere">infosphere</a></li> <li><a href="/wiki/Inforg" title="Inforg">inforg</a></li> <li><i><a href="/wiki/Decoding_the_Universe" title="Decoding the Universe">Decoding the Universe</a></i></li> <li><a href="/wiki/Information_overload" title="Information overload">information overload</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Data_compression_methods223" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Compression_methods" title="Template:Compression methods"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Compression_methods" title="Template talk:Compression methods"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Compression_methods" title="Special:EditPage/Template:Compression methods"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Data_compression_methods223" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_compression" title="Data compression">Data compression</a> methods</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lossless_compression" title="Lossless compression">Lossless</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Entropy_coding" title="Entropy coding">Entropy type</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_coding" title="Adaptive coding">Adaptive coding</a></li> <li><a href="/wiki/Arithmetic_coding" title="Arithmetic coding">Arithmetic</a></li> <li><a href="/wiki/Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric numeral systems</a></li> <li><a href="/wiki/Golomb_coding" title="Golomb coding">Golomb</a></li> <li><a href="/wiki/Huffman_coding" title="Huffman coding">Huffman</a> <ul><li><a href="/wiki/Adaptive_Huffman_coding" title="Adaptive Huffman coding">Adaptive</a></li> <li><a href="/wiki/Canonical_Huffman_code" title="Canonical Huffman code">Canonical</a></li> <li><a href="/wiki/Modified_Huffman_coding" title="Modified Huffman coding">Modified</a></li></ul></li> <li><a href="/wiki/Range_coding" title="Range coding">Range</a></li> <li><a href="/wiki/Shannon_coding" title="Shannon coding">Shannon</a></li> <li><a href="/wiki/Shannon%E2%80%93Fano_coding" title="Shannon–Fano coding">Shannon–Fano</a></li> <li><a href="/wiki/Shannon%E2%80%93Fano%E2%80%93Elias_coding" title="Shannon–Fano–Elias coding">Shannon–Fano–Elias</a></li> <li><a href="/wiki/Tunstall_coding" title="Tunstall coding">Tunstall</a></li> <li><a href="/wiki/Unary_coding" title="Unary coding">Unary</a></li> <li><a href="/wiki/Universal_code_(data_compression)" title="Universal code (data compression)">Universal</a> <ul><li><a href="/wiki/Exponential-Golomb_coding" title="Exponential-Golomb coding">Exp-Golomb</a></li> <li><a href="/wiki/Fibonacci_coding" title="Fibonacci coding">Fibonacci</a></li> <li><a href="/wiki/Elias_gamma_coding" title="Elias gamma coding">Gamma</a></li> <li><a href="/wiki/Levenshtein_coding" title="Levenshtein coding">Levenshtein</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Dictionary_coder" title="Dictionary coder">Dictionary type</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Byte_pair_encoding" title="Byte pair encoding">Byte pair encoding</a></li> <li><a href="/wiki/LZ77_and_LZ78" title="LZ77 and LZ78">Lempel–Ziv</a> <ul><li><a href="/wiki/842_(compression_algorithm)" title="842 (compression algorithm)">842</a></li> <li><a href="/wiki/LZ4_(compression_algorithm)" title="LZ4 (compression algorithm)">LZ4</a></li> <li><a href="/wiki/LZJB" class="mw-redirect" title="LZJB">LZJB</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Oberhumer" title="Lempel–Ziv–Oberhumer">LZO</a></li> <li><a href="/wiki/LZRW" title="LZRW">LZRW</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Storer%E2%80%93Szymanski" title="Lempel–Ziv–Storer–Szymanski">LZSS</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Welch" title="Lempel–Ziv–Welch">LZW</a></li> <li><a href="/wiki/LZWL" title="LZWL">LZWL</a></li> <li><a href="/wiki/Snappy_(compression)" title="Snappy (compression)">Snappy</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Other types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burrows%E2%80%93Wheeler_transform" title="Burrows–Wheeler transform">BWT</a></li> <li><a href="/wiki/Context_tree_weighting" title="Context tree weighting">CTW</a></li> <li><a href="/wiki/Context_mixing" title="Context mixing">CM</a></li> <li><a href="/wiki/Delta_encoding" title="Delta encoding">Delta</a> <ul><li><a href="/wiki/Incremental_encoding" title="Incremental encoding">Incremental</a></li></ul></li> <li><a href="/wiki/Dynamic_Markov_compression" title="Dynamic Markov compression">DMC</a></li> <li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a></li> <li><a href="/wiki/Grammar-based_code" title="Grammar-based code">Grammar</a> <ul><li><a href="/wiki/Re-Pair" title="Re-Pair">Re-Pair</a></li> <li><a href="/wiki/Sequitur_algorithm" title="Sequitur algorithm">Sequitur</a></li></ul></li> <li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">LDCT</a></li> <li><a href="/wiki/Move-to-front_transform" title="Move-to-front transform">MTF</a></li> <li><a href="/wiki/PAQ" title="PAQ">PAQ</a></li> <li><a href="/wiki/Prediction_by_partial_matching" title="Prediction by partial matching">PPM</a></li> <li><a href="/wiki/Run-length_encoding" title="Run-length encoding">RLE</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Hybrid</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li>LZ77 + Huffman <ul><li><a href="/wiki/Deflate" title="Deflate">Deflate</a></li> <li><a href="/wiki/LZX" title="LZX">LZX</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Stac" title="Lempel–Ziv–Stac">LZS</a></li></ul></li> <li>LZ77 + ANS <ul><li><a href="/wiki/LZFSE" title="LZFSE">LZFSE</a></li></ul></li> <li>LZ77 + Huffman + ANS <ul><li><a href="/wiki/Zstd" title="Zstd">Zstandard</a></li></ul></li> <li>LZ77 + Huffman + context <ul><li><a href="/wiki/Brotli" title="Brotli">Brotli</a></li></ul></li> <li>LZSS + Huffman <ul><li><a href="/wiki/LHA_(file_format)" title="LHA (file format)">LHA/LZH</a></li></ul></li> <li>LZ77 + Range <ul><li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Markov_chain_algorithm" title="Lempel–Ziv–Markov chain algorithm">LZMA</a></li> <li>LZHAM</li></ul></li> <li>RLE + BWT + MTF + Huffman <ul><li><a href="/wiki/Bzip2" title="Bzip2">bzip2</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lossy_compression" title="Lossy compression">Lossy</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Transform_coding" title="Transform coding">Transform type</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">Discrete cosine transform</a> <ul><li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">DCT</a></li> <li><a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">MDCT</a></li></ul></li> <li><a href="/wiki/Discrete_sine_transform" title="Discrete sine transform">DST</a></li> <li><a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a></li> <li><a href="/wiki/Wavelet_transform" title="Wavelet transform">Wavelet</a> <ul><li><a href="/wiki/Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li> <li><a href="/wiki/Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Predictive type</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a> <ul><li><a href="/wiki/Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li></ul></li> <li><a href="/wiki/Linear_predictive_coding" title="Linear predictive coding">LPC</a> <ul><li><a href="/wiki/Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li> <li><a href="/wiki/Code-excited_linear_prediction" title="Code-excited linear prediction">CELP</a></li> <li><a href="/wiki/Log_area_ratio" title="Log area ratio">LAR</a></li> <li><a href="/wiki/Line_spectral_pairs" title="Line spectral pairs">LSP</a></li> <li><a href="/wiki/Warped_linear_predictive_coding" title="Warped linear predictive coding">WLPC</a></li></ul></li> <li>Motion <ul><li><a href="/wiki/Motion_compensation" title="Motion compensation">Compensation</a></li> <li><a href="/wiki/Motion_estimation" title="Motion estimation">Estimation</a></li> <li><a href="/wiki/Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li> <li><a href="/wiki/Psychoacoustics" title="Psychoacoustics">Psychoacoustic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Data_compression#Audio" title="Data compression">Audio</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit_rate" title="Bit rate">Bit rate</a> <ul><li><a href="/wiki/Average_bitrate" title="Average bitrate">ABR</a></li> <li><a href="/wiki/Constant_bitrate" title="Constant bitrate">CBR</a></li> <li><a href="/wiki/Variable_bitrate" title="Variable bitrate">VBR</a></li></ul></li> <li><a href="/wiki/Companding" title="Companding">Companding</a></li> <li><a href="/wiki/Convolution" title="Convolution">Convolution</a></li> <li><a href="/wiki/Dynamic_range" title="Dynamic range">Dynamic range</a></li> <li><a href="/wiki/Latency_(audio)" title="Latency (audio)">Latency</a></li> <li><a href="/wiki/Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon theorem</a></li> <li><a href="/wiki/Sampling_(signal_processing)" title="Sampling (signal processing)">Sampling</a></li> <li><a href="/wiki/Silence_compression" title="Silence compression">Silence compression</a></li> <li><a href="/wiki/Sound_quality" title="Sound quality">Sound quality</a></li> <li><a href="/wiki/Speech_coding" title="Speech coding">Speech coding</a></li> <li><a href="/wiki/Sub-band_coding" title="Sub-band coding">Sub-band coding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Audio_codec" title="Audio codec">Codec</a> parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/A-law_algorithm" title="A-law algorithm">A-law</a></li> <li><a href="/wiki/%CE%9C-law_algorithm" title="Μ-law algorithm">μ-law</a></li> <li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a> <ul><li><a href="/wiki/Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li> <li><a href="/wiki/Delta_modulation" title="Delta modulation">DM</a></li></ul></li> <li><a href="/wiki/Fourier_transform" title="Fourier transform">FT</a> <ul><li><a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a></li></ul></li> <li><a href="/wiki/Linear_predictive_coding" title="Linear predictive coding">LPC</a> <ul><li><a href="/wiki/Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li> <li><a href="/wiki/Code-excited_linear_prediction" title="Code-excited linear prediction">CELP</a></li> <li><a href="/wiki/Log_area_ratio" title="Log area ratio">LAR</a></li> <li><a href="/wiki/Line_spectral_pairs" title="Line spectral pairs">LSP</a></li> <li><a href="/wiki/Warped_linear_predictive_coding" title="Warped linear predictive coding">WLPC</a></li></ul></li> <li><a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">MDCT</a></li> <li><a href="/wiki/Psychoacoustics" title="Psychoacoustics">Psychoacoustic model</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Image_compression" title="Image compression">Image</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chroma_subsampling" title="Chroma subsampling">Chroma subsampling</a></li> <li><a href="/wiki/Coding_tree_unit" title="Coding tree unit">Coding tree unit</a></li> <li><a href="/wiki/Color_space" title="Color space">Color space</a></li> <li><a href="/wiki/Compression_artifact" title="Compression artifact">Compression artifact</a></li> <li><a href="/wiki/Image_resolution" title="Image resolution">Image resolution</a></li> <li><a href="/wiki/Macroblock" title="Macroblock">Macroblock</a></li> <li><a href="/wiki/Pixel" title="Pixel">Pixel</a></li> <li><a href="/wiki/Peak_signal-to-noise_ratio" title="Peak signal-to-noise ratio">PSNR</a></li> <li><a href="/wiki/Quantization_(image_processing)" title="Quantization (image processing)">Quantization</a></li> <li><a href="/wiki/Standard_test_image" title="Standard test image">Standard test image</a></li> <li><a href="/wiki/Texture_compression" title="Texture compression">Texture compression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Methods</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chain_code" title="Chain code">Chain code</a></li> <li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">DCT</a></li> <li><a href="/wiki/Deflate" title="Deflate">Deflate</a></li> <li><a href="/wiki/Fractal_compression" title="Fractal compression">Fractal</a></li> <li><a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">KLT</a></li> <li><a href="/wiki/Pyramid_(image_processing)" title="Pyramid (image processing)">LP</a></li> <li><a href="/wiki/Run-length_encoding" title="Run-length encoding">RLE</a></li> <li><a href="/wiki/Wavelet_transform" title="Wavelet transform">Wavelet</a> <ul><li><a href="/wiki/Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li> <li><a href="/wiki/Embedded_zerotrees_of_wavelet_transforms" title="Embedded zerotrees of wavelet transforms">EZW</a></li> <li><a href="/wiki/Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Data_compression#Video" title="Data compression">Video</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit_rate" title="Bit rate">Bit rate</a> <ul><li><a href="/wiki/Average_bitrate" title="Average bitrate">ABR</a></li> <li><a href="/wiki/Constant_bitrate" title="Constant bitrate">CBR</a></li> <li><a href="/wiki/Variable_bitrate" title="Variable bitrate">VBR</a></li></ul></li> <li><a href="/wiki/Display_resolution" title="Display resolution">Display resolution</a></li> <li><a href="/wiki/Film_frame" title="Film frame">Frame</a></li> <li><a href="/wiki/Frame_rate" title="Frame rate">Frame rate</a></li> <li><a href="/wiki/Video_compression_picture_types" title="Video compression picture types">Frame types</a></li> <li><a href="/wiki/Interlaced_video" title="Interlaced video">Interlace</a></li> <li><a href="/wiki/Video#Characteristics_of_video_streams" title="Video">Video characteristics</a></li> <li><a href="/wiki/Video_quality" title="Video quality">Video quality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Video_codec" title="Video codec">Codec</a> parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">DCT</a></li> <li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a></li> <li><a href="/wiki/Deblocking_filter" title="Deblocking filter">Deblocking filter</a></li> <li><a href="/wiki/Lapped_transform" title="Lapped transform">Lapped transform</a></li> <li>Motion <ul><li><a href="/wiki/Motion_compensation" title="Motion compensation">Compensation</a></li> <li><a href="/wiki/Motion_estimation" title="Motion estimation">Estimation</a></li> <li><a href="/wiki/Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li> <li><a href="/wiki/Wavelet_transform" title="Wavelet transform">Wavelet</a> <ul><li><a href="/wiki/Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Compressed_data_structure" title="Compressed data structure">Compressed data structures</a> <ul><li><a href="/wiki/Compressed_suffix_array" title="Compressed suffix array">Compressed suffix array</a></li> <li><a href="/wiki/FM-index" title="FM-index">FM-index</a></li></ul></li> <li><a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">Entropy</a></li> <li><a class="mw-selflink selflink">Information theory</a> <ul><li><a href="/wiki/Timeline_of_information_theory" title="Timeline of information theory">Timeline</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Prefix_code" title="Prefix code">Prefix code</a></li> <li><a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a></li> <li><a href="/wiki/Rate%E2%80%93distortion_theory" title="Rate–distortion theory">Rate–distortion</a></li> <li><a href="/wiki/Redundancy_(information_theory)" title="Redundancy (information theory)">Redundancy</a></li> <li><a href="/wiki/Data_compression_symmetry" title="Data compression symmetry">Symmetry</a></li> <li><a href="/wiki/Smallest_grammar_problem" title="Smallest grammar problem">Smallest grammar problem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Community</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hutter_Prize" title="Hutter Prize">Hutter Prize</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mark_Adler" title="Mark Adler">Mark Adler</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Major_mathematics_areas1051" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Areas_of_mathematics" title="Template:Areas of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas1051" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a class="mw-selflink selflink">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Systems_science" title="Systems science">Systems science</a> <ul><li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_science" title="Computer science">Computer science</a></li> <li><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a> <ul><li><a href="/wiki/List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li> <li><a href="/wiki/Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li> <li><a href="/wiki/List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li> <li><a href="/wiki/Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a href="/wiki/Category:Fields_of_mathematics" title="Category:Fields of mathematics">Category</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <b><a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw" title="commons:Category:Mathematics">Commons</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="WikiProject"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/16px-People_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/24px-People_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/32px-People_icon.svg.png 2x" data-file-width="100" data-file-height="100" /></span></span> <b><a href="/wiki/Wikipedia:WikiProject_Mathematics" title="Wikipedia:WikiProject Mathematics">WikiProject</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Computer_science1028" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Computer_science" title="Template:Computer science"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Computer_science" title="Template talk:Computer science"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Computer_science" title="Special:EditPage/Template:Computer science"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Computer_science1028" style="font-size:114%;margin:0 4em"><a href="/wiki/Computer_science" title="Computer science">Computer science</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Note: This template roughly follows the 2012 <a href="/wiki/ACM_Computing_Classification_System" title="ACM Computing Classification System">ACM Computing Classification System</a>.</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_hardware" title="Computer hardware">Hardware</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Printed_circuit_board" title="Printed circuit board">Printed circuit board</a></li> <li><a href="/wiki/Peripheral" title="Peripheral">Peripheral</a></li> <li><a href="/wiki/Integrated_circuit" title="Integrated circuit">Integrated circuit</a></li> <li><a href="/wiki/Very_Large_Scale_Integration" class="mw-redirect" title="Very Large Scale Integration">Very Large Scale Integration</a></li> <li><a href="/wiki/System_on_a_chip" title="System on a chip">Systems on Chip (SoCs)</a></li> <li><a href="/wiki/Green_computing" title="Green computing">Energy consumption (Green computing)</a></li> <li><a href="/wiki/Electronic_design_automation" title="Electronic design automation">Electronic design automation</a></li> <li><a href="/wiki/Hardware_acceleration" title="Hardware acceleration">Hardware acceleration</a></li> <li><a href="/wiki/Processor_(computing)" title="Processor (computing)">Processor</a></li> <li><a href="/wiki/List_of_computer_size_categories" title="List of computer size categories">Size</a> / <a href="/wiki/Form_factor_(design)" title="Form factor (design)">Form</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Computer systems organization</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_architecture" title="Computer architecture">Computer architecture</a></li> <li><a href="/wiki/Computational_complexity" title="Computational complexity">Computational complexity</a></li> <li><a href="/wiki/Dependability" title="Dependability">Dependability</a></li> <li><a href="/wiki/Embedded_system" title="Embedded system">Embedded system</a></li> <li><a href="/wiki/Real-time_computing" title="Real-time computing">Real-time computing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_network" title="Computer network">Networks</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Network_architecture" title="Network architecture">Network architecture</a></li> <li><a href="/wiki/Network_protocol" class="mw-redirect" title="Network protocol">Network protocol</a></li> <li><a href="/wiki/Networking_hardware" title="Networking hardware">Network components</a></li> <li><a href="/wiki/Network_scheduler" title="Network scheduler">Network scheduler</a></li> <li><a href="/wiki/Network_performance" title="Network performance">Network performance evaluation</a></li> <li><a href="/wiki/Network_service" title="Network service">Network service</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Software organization</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpreter_(computing)" title="Interpreter (computing)">Interpreter</a></li> <li><a href="/wiki/Middleware" title="Middleware">Middleware</a></li> <li><a href="/wiki/Virtual_machine" title="Virtual machine">Virtual machine</a></li> <li><a href="/wiki/Operating_system" title="Operating system">Operating system</a></li> <li><a href="/wiki/Software_quality" title="Software quality">Software quality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Programming_language_theory" title="Programming language theory">Software notations</a> and <a href="/wiki/Programming_tool" title="Programming tool">tools</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Programming_paradigm" title="Programming paradigm">Programming paradigm</a></li> <li><a href="/wiki/Programming_language" title="Programming language">Programming language</a></li> <li><a href="/wiki/Compiler_construction" class="mw-redirect" title="Compiler construction">Compiler</a></li> <li><a href="/wiki/Domain-specific_language" title="Domain-specific language">Domain-specific language</a></li> <li><a href="/wiki/Modeling_language" title="Modeling language">Modeling language</a></li> <li><a href="/wiki/Software_framework" title="Software framework">Software framework</a></li> <li><a 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development process</a></li> <li><a href="/wiki/Requirements_analysis" title="Requirements analysis">Requirements analysis</a></li> <li><a href="/wiki/Software_design" title="Software design">Software design</a></li> <li><a href="/wiki/Software_construction" title="Software construction">Software construction</a></li> <li><a href="/wiki/Software_deployment" title="Software deployment">Software deployment</a></li> <li><a href="/wiki/Software_engineering" title="Software engineering">Software engineering</a></li> <li><a href="/wiki/Software_maintenance" title="Software maintenance">Software maintenance</a></li> <li><a href="/wiki/Programming_team" title="Programming team">Programming team</a></li> <li><a href="/wiki/Open-source_software" title="Open-source software">Open-source model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></th><td 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style="width:1%"><a href="/wiki/Algorithm" title="Algorithm">Algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algorithm_design" class="mw-redirect" title="Algorithm design">Algorithm design</a></li> <li><a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">Analysis of algorithms</a></li> <li><a href="/wiki/Algorithmic_efficiency" title="Algorithmic efficiency">Algorithmic efficiency</a></li> <li><a href="/wiki/Randomized_algorithm" title="Randomized algorithm">Randomized algorithm</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematics of <a href="/wiki/Computing" title="Computing">computing</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Mathematical_software" title="Mathematical software">Mathematical software</a></li> <li><a class="mw-selflink selflink">Information theory</a></li> <li><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">Theoretical computer science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Information_system" title="Information system">Information systems</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Database" title="Database">Database management system</a></li> <li><a href="/wiki/Computer_data_storage" title="Computer data storage">Information storage systems</a></li> <li><a href="/wiki/Enterprise_information_system" title="Enterprise information system">Enterprise information system</a></li> <li><a href="/wiki/Social_software" title="Social software">Social information systems</a></li> <li><a href="/wiki/Geographic_information_system" title="Geographic information system">Geographic information system</a></li> <li><a href="/wiki/Decision_support_system" title="Decision support system">Decision support system</a></li> <li><a href="/wiki/Process_control" class="mw-redirect" title="Process control">Process control system</a></li> <li><a href="/wiki/Multimedia_database" title="Multimedia database">Multimedia information system</a></li> <li><a href="/wiki/Data_mining" title="Data mining">Data mining</a></li> <li><a href="/wiki/Digital_library" title="Digital library">Digital library</a></li> <li><a href="/wiki/Computing_platform" title="Computing platform">Computing platform</a></li> <li><a href="/wiki/Digital_marketing" title="Digital marketing">Digital marketing</a></li> <li><a href="/wiki/World_Wide_Web" title="World Wide Web">World Wide Web</a></li> <li><a href="/wiki/Information_retrieval" title="Information retrieval">Information retrieval</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_security" title="Computer security">Security</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a></li> <li><a href="/wiki/Formal_methods" title="Formal methods">Formal methods</a></li> <li><a href="/wiki/Security_hacker" title="Security hacker">Security hacker</a></li> <li><a href="/wiki/Security_service_(telecommunication)" title="Security service (telecommunication)">Security services</a></li> <li><a href="/wiki/Intrusion_detection_system" title="Intrusion detection system">Intrusion detection system</a></li> <li><a href="/wiki/Hardware_security" title="Hardware security">Hardware security</a></li> <li><a href="/wiki/Network_security" title="Network security">Network security</a></li> <li><a href="/wiki/Information_security" title="Information security">Information security</a></li> <li><a href="/wiki/Application_security" title="Application security">Application security</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Human%E2%80%93computer_interaction" title="Human–computer interaction">Human–computer interaction</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interaction_design" title="Interaction design">Interaction design</a></li> <li><a href="/wiki/Social_computing" title="Social computing">Social computing</a></li> <li><a href="/wiki/Ubiquitous_computing" title="Ubiquitous computing">Ubiquitous computing</a></li> <li><a href="/wiki/Visualization_(graphics)" title="Visualization (graphics)">Visualization</a></li> <li><a href="/wiki/Computer_accessibility" title="Computer accessibility">Accessibility</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Concurrency_(computer_science)" title="Concurrency (computer science)">Concurrency</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Concurrent_computing" title="Concurrent computing">Concurrent computing</a></li> <li><a href="/wiki/Parallel_computing" title="Parallel computing">Parallel computing</a></li> <li><a href="/wiki/Distributed_computing" title="Distributed computing">Distributed computing</a></li> <li><a href="/wiki/Multithreading_(computer_architecture)" title="Multithreading (computer architecture)">Multithreading</a></li> <li><a href="/wiki/Multiprocessing" title="Multiprocessing">Multiprocessing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Natural_language_processing" title="Natural language processing">Natural language processing</a></li> <li><a href="/wiki/Knowledge_representation_and_reasoning" title="Knowledge representation and reasoning">Knowledge representation and reasoning</a></li> <li><a href="/wiki/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Automated_planning_and_scheduling" title="Automated planning and scheduling">Automated planning and scheduling</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Search methodology</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control method</a></li> <li><a href="/wiki/Philosophy_of_artificial_intelligence" title="Philosophy of artificial intelligence">Philosophy of artificial intelligence</a></li> <li><a href="/wiki/Distributed_artificial_intelligence" title="Distributed artificial intelligence">Distributed artificial intelligence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a></li> <li><a href="/wiki/Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li> <li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li> <li><a href="/wiki/Multi-task_learning" title="Multi-task learning">Multi-task learning</a></li> <li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross-validation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_graphics" title="Computer graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_animation" title="Computer animation">Animation</a></li> <li><a href="/wiki/Extended_reality" title="Extended reality">Extended reality</a> <ul><li><a href="/wiki/Augmented_reality" title="Augmented reality">Augmented</a></li> <li><a href="/wiki/Mixed_reality" title="Mixed reality">Mixed</a></li> <li><a href="/wiki/Virtual_reality" title="Virtual reality">Virtual</a></li></ul></li> <li><a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">Rendering</a></li> <li><a href="/wiki/Photograph_manipulation" title="Photograph manipulation">Photograph manipulation</a></li> <li><a href="/wiki/Graphics_processing_unit" title="Graphics processing unit">Graphics processing unit</a></li> <li><a href="/wiki/Image_compression" title="Image compression">Image compression</a></li> <li><a href="/wiki/Solid_modeling" title="Solid modeling">Solid modeling</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applied computing</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Computing" class="mw-redirect" title="Quantum Computing">Quantum Computing</a></li> <li><a href="/wiki/E-commerce" title="E-commerce">E-commerce</a></li> <li><a href="/wiki/Enterprise_software" 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