Informazione mutua

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vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Indice</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">nascondi</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inizio</div> </a> </li> <li id="toc-Definizione" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definizione"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definizione</span> </div> </a> <ul id="toc-Definizione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relazione_con_altre_quantità" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relazione_con_altre_quantità"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Relazione con altre quantità</span> </div> </a> <ul id="toc-Relazione_con_altre_quantità-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Varianti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Varianti"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Varianti</span> </div> </a> <button aria-controls="toc-Varianti-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>Attiva/disattiva la sottosezione Varianti</span> </button> <ul id="toc-Varianti-sublist" class="vector-toc-list"> <li id="toc-Metrica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metrica"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Metrica</span> </div> </a> <ul id="toc-Metrica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Informazione_mutua_condizionale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Informazione_mutua_condizionale"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Informazione mutua condizionale</span> </div> </a> <ul id="toc-Informazione_mutua_condizionale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Informazione_mutua_multivariata" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Informazione_mutua_multivariata"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Informazione mutua multivariata</span> </div> </a> <ul id="toc-Informazione_mutua_multivariata-sublist" class="vector-toc-list"> <li id="toc-Applicazioni" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Applicazioni"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>Applicazioni</span> </div> </a> <ul id="toc-Applicazioni-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Varianti_normalizzate" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Varianti_normalizzate"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Varianti normalizzate</span> </div> </a> <ul id="toc-Varianti_normalizzate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Varianti_ponderate" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Varianti_ponderate"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Varianti ponderate</span> </div> </a> <ul id="toc-Varianti_ponderate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Informazione_mutua_assoluta" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Informazione_mutua_assoluta"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Informazione mutua assoluta</span> </div> </a> <ul id="toc-Informazione_mutua_assoluta-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applicazioni_2" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applicazioni_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applicazioni</span> </div> </a> <ul id="toc-Applicazioni_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-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="Indice" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Indice" > <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="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</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">Informazione mutua</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="Vai a una voce in un'altra lingua. Disponibile in 20 lingue" > <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-20" 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">20 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D9%84%D9%88%D9%85%D8%A7%D8%AA_%D9%85%D8%AA%D8%A8%D8%A7%D8%AF%D9%84%D8%A9" title="معلومات متبادلة - arabo" lang="ar" hreflang="ar" data-title="معلومات متبادلة" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Transinformation" title="Transinformation - bavarese" lang="bar" hreflang="bar" data-title="Transinformation" data-language-autonym="Boarisch" data-language-local-name="bavarese" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vz%C3%A1jemn%C3%A1_informace" title="Vzájemná informace - ceco" lang="cs" hreflang="cs" data-title="Vzájemná informace" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Transinformation" title="Transinformation - tedesco" lang="de" hreflang="de" data-title="Transinformation" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BC%CE%BF%CE%B9%CE%B2%CE%B1%CE%AF%CE%B1_%CF%80%CE%BB%CE%B7%CF%81%CE%BF%CF%86%CE%BF%CF%81%CE%AF%CE%B1" title="Αμοιβαία πληροφορία - greco" lang="el" hreflang="el" data-title="Αμοιβαία πληροφορία" data-language-autonym="Ελληνικά" data-language-local-name="greco" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Mutual_information" title="Mutual information - inglese" lang="en" hreflang="en" data-title="Mutual information" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Informaci%C3%B3n_mutua" title="Información mutua - spagnolo" lang="es" hreflang="es" data-title="Información mutua" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Elkarrekiko_informazio" title="Elkarrekiko informazio - basco" lang="eu" hreflang="eu" data-title="Elkarrekiko informazio" data-language-autonym="Euskara" data-language-local-name="basco" 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/%D8%A7%D8%B7%D9%84%D8%A7%D8%B9%D8%A7%D8%AA_%D9%85%D8%AA%D9%82%D8%A7%D8%A8%D9%84" title="اطلاعات متقابل - persiano" lang="fa" hreflang="fa" data-title="اطلاعات متقابل" data-language-autonym="فارسی" data-language-local-name="persiano" 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/Information_mutuelle" title="Information mutuelle - francese" lang="fr" hreflang="fr" data-title="Information mutuelle" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%99%D7%A0%D7%A4%D7%95%D7%A8%D7%9E%D7%A6%D7%99%D7%94_%D7%94%D7%93%D7%93%D7%99%D7%AA" title="אינפורמציה הדדית - ebraico" lang="he" hreflang="he" data-title="אינפורמציה הדדית" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%9B%B8%E4%BA%92%E6%83%85%E5%A0%B1%E9%87%8F" title="相互情報量 - giapponese" lang="ja" hreflang="ja" data-title="相互情報量" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%83%81%ED%98%B8%EC%A0%95%EB%B3%B4" title="상호정보 - coreano" lang="ko" hreflang="ko" data-title="상호정보" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Informacja_wzajemna" title="Informacja wzajemna - polacco" lang="pl" hreflang="pl" data-title="Informacja wzajemna" data-language-autonym="Polski" data-language-local-name="polacco" 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/Informa%C3%A7%C3%A3o_m%C3%BAtua" title="Informação mútua - portoghese" lang="pt" hreflang="pt" data-title="Informação mútua" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%B7%D0%B0%D0%B8%D0%BC%D0%BD%D0%B0%D1%8F_%D0%B8%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%8F" title="Взаимная информация - russo" lang="ru" hreflang="ru" data-title="Взаимная информация" data-language-autonym="Русский" data-language-local-name="russo" 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/Mutual_information" title="Mutual information - Simple English" lang="en-simple" hreflang="en-simple" data-title="Mutual information" 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-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%B7%D0%B0%D1%94%D0%BC%D0%BD%D0%B0_%D1%96%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D1%96%D1%8F" title="Взаємна інформація - ucraino" lang="uk" hreflang="uk" data-title="Взаємна інформація" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BA%92%E4%BF%A1%E6%81%AF" title="互信息 - cinese" lang="zh" hreflang="zh" data-title="互信息" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%9B%B8%E4%BA%92%E8%B3%87%E8%A8%8A" title="相互資訊 - cantonese" lang="yue" hreflang="yue" data-title="相互資訊" data-language-autonym="粵語" data-language-local-name="cantonese" class="interlanguage-link-target"><span>粵語</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/Q252973#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespace"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Informazione_mutua" title="Vedi la voce [c]" accesskey="c"><span>Voce</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discussione:Informazione_mutua" rel="discussion" title="Vedi le discussioni relative a questa pagina [t]" accesskey="t"><span>Discussione</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Cambia versione linguistica" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">italiano</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Visite"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Informazione_mutua"><span>Leggi</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Informazione_mutua&veaction=edit" title="Modifica questa pagina [v]" accesskey="v"><span>Modifica</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Informazione_mutua&action=edit" title="Modifica il wikitesto di questa pagina [e]" accesskey="e"><span>Modifica wikitesto</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Informazione_mutua&action=history" title="Versioni precedenti di questa pagina [h]" accesskey="h"><span>Cronologia</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Strumenti pagine"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Strumenti" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Strumenti</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Strumenti</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">nascondi</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Altre opzioni" > <div class="vector-menu-heading"> Azioni </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Informazione_mutua"><span>Leggi</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Informazione_mutua&veaction=edit" title="Modifica questa pagina [v]" accesskey="v"><span>Modifica</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Informazione_mutua&action=edit" title="Modifica il wikitesto di questa pagina [e]" accesskey="e"><span>Modifica wikitesto</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Informazione_mutua&action=history"><span>Cronologia</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Generale </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speciale:PuntanoQui/Informazione_mutua" title="Elenco di tutte le pagine che sono collegate a questa [j]" accesskey="j"><span>Puntano qui</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speciale:ModificheCorrelate/Informazione_mutua" rel="nofollow" title="Elenco delle ultime modifiche alle pagine collegate a questa [k]" accesskey="k"><span>Modifiche correlate</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Informazione_mutua&oldid=137834241" title="Collegamento permanente a questa versione di questa pagina"><span>Link permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Informazione_mutua&action=info" title="Ulteriori informazioni su questa pagina"><span>Informazioni pagina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speciale:Cita&page=Informazione_mutua&id=137834241&wpFormIdentifier=titleform" title="Informazioni su come citare questa pagina"><span>Cita questa 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img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint torna-a"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/1leftarrow_blue.svg/18px-1leftarrow_blue.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/1leftarrow_blue.svg/27px-1leftarrow_blue.svg.png 1.5x, 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di pagina" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Question_book_magnify.svg/45px-Question_book_magnify.svg.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Question_book_magnify.svg/68px-Question_book_magnify.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Question_book_magnify.svg/90px-Question_book_magnify.svg.png 2x" data-file-width="60" data-file-height="60" /></a></span></div> <div class="avviso-testo mbox-text"> <div class="mbox-text-div"><b>Questa voce o sezione  sull'argomento Matematica è priva o carente di <a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">note</a> e <a href="/wiki/Aiuto:Uso_delle_fonti#Citazioni_nel_testo_.28citazioni_interne.29_e_alla_fine" title="Aiuto:Uso delle fonti">riferimenti bibliografici puntuali</a></b>. <div class="hide-when-compact"> <div class="noprint"><hr />Sebbene vi siano una <a href="/wiki/Aiuto:Bibliografia" title="Aiuto:Bibliografia">bibliografia</a> e/o dei <a href="/wiki/Wikipedia:Collegamenti_esterni" title="Wikipedia:Collegamenti esterni">collegamenti esterni</a>, manca la contestualizzazione delle fonti con <a href="/wiki/Aiuto:Note" title="Aiuto:Note">note a piè di pagina</a> o altri riferimenti precisi che indichino puntualmente la provenienza delle informazioni. Puoi <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Informazione_mutua&action=edit">migliorare questa voce</a> <a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">citando le fonti</a> più precisamente. Segui i suggerimenti del <a href="/wiki/Progetto:Matematica" title="Progetto:Matematica">progetto di riferimento</a>.</div> </div> </div> </div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r133964453" /><div style="" class="ambox metadata noprint plainlinks avviso avviso-stile"> <div class="avviso-immagine mbox-image noprint"><span class="skin-invert" typeof="mw:File"><a href="/wiki/File:Wikitext.svg" class="mw-file-description" title="Questa voce è da wikificare"><img alt="Questa voce è da wikificare" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Wikitext.svg/50px-Wikitext.svg.png" decoding="async" width="50" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Wikitext.svg/75px-Wikitext.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Wikitext.svg/100px-Wikitext.svg.png 2x" data-file-width="220" data-file-height="78" /></a></span></div> <div class="avviso-testo mbox-text"> <div class="mbox-text-div"><b>Questa voce o sezione  sull'argomento Matematica non è ancora <a href="/wiki/Aiuto:Wikificare" title="Aiuto:Wikificare">formattata</a> secondo gli <a href="/wiki/Aiuto:Manuale_di_stile" title="Aiuto:Manuale di stile">standard</a></b>. <div class="hide-when-compact"><b>Commento</b>: <i>In varie occasioni le fonti sono riportate in linea senza utilizzare le note ed i template appropriati</i> <div class="noprint"><hr /><a class="external text" href="https://it.wikipedia.org/w/index.php?title=Informazione_mutua&action=edit">Contribuisci</a> a migliorarla secondo le <a href="/wiki/Aiuto:Manuale_di_stile" title="Aiuto:Manuale di stile">convenzioni di Wikipedia</a>. Segui i suggerimenti del <a href="/wiki/Progetto:Matematica" title="Progetto:Matematica">progetto di riferimento</a>.</div> </div> </div> </div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Entropy-mutual-information-relative-entropy-relation-diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Entropy-mutual-information-relative-entropy-relation-diagram.svg/170px-Entropy-mutual-information-relative-entropy-relation-diagram.svg.png" decoding="async" width="170" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Entropy-mutual-information-relative-entropy-relation-diagram.svg/255px-Entropy-mutual-information-relative-entropy-relation-diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Entropy-mutual-information-relative-entropy-relation-diagram.svg/340px-Entropy-mutual-information-relative-entropy-relation-diagram.svg.png 2x" data-file-width="744" data-file-height="524" /></a><figcaption>Entropie <a href="#informazione_intrinseca">individuali</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 H(X),H(Y)}"> <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> <mi>H</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(X),H(Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6083199a40f7315b4383af784fbad33a272cce47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.533ex; height:2.843ex;" alt="{\displaystyle H(X),H(Y)}" /></span>, <a href="#entropia_congiunta">congiunte</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 H(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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d87c82092f7817e719251729dc0a55289df0eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.66ex; height:2.843ex;" alt="{\displaystyle H(X,Y)}" /></span>, e <a href="#entropia_condizionale">condizionali</a> per una coppia di sottosistemi correlati <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,Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8705438171d938b7f59cd1bfa5b7d99b6afa5cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}" /></span> con <a href="#informazione_mutua">informazione mutua</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;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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X;Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14b479006c523b8459baf3cf169e3f4a7bcd0771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.768ex; height:2.843ex;" alt="{\displaystyle I(X;Y)}" /></span>.</figcaption></figure> <p>Nella <a href="/wiki/Teoria_della_probabilit%C3%A0" title="Teoria della probabilità">teoria della probabilità</a> e nella <a href="/wiki/Teoria_dell%27informazione" title="Teoria dell'informazione">teoria dell'informazione</a>, l'<b>informazione mutua</b> (in passato detta anche <b>transinformazione</b>) di due <a href="/wiki/Variabile_casuale" title="Variabile casuale">variabili casuali</a> è una quantità che misura la mutua dipendenza delle due variabili. La più comune <a href="/wiki/Unit%C3%A0_di_misura" title="Unità di misura">unità di misura</a> della mutua informazione è il <a href="/wiki/Bit" title="Bit">bit</a>, quando si usano i logaritmi in base 2. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definizione">Definizione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Informazione_mutua&veaction=edit&section=1" title="Modifica la sezione Definizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Informazione_mutua&action=edit&section=1" title="Edit section's source code: Definizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Formalmente l'informazione mutua di due variabili casuali discrete <i>X</i> e <i>Y</i> può essere definita come: </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)=\sum _{y\in Y}\sum _{x\in X}p(x,y)\log {\left({\frac {p(x,y)}{p_{1}(x)\,p_{2}(y)}}\right)},\,\!}"> <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> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mrow> </munder> <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>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <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> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X;Y)=\sum _{y\in Y}\sum _{x\in X}p(x,y)\log {\left({\frac {p(x,y)}{p_{1}(x)\,p_{2}(y)}}\right)},\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22587a68a5861412c495386599bd5103a643fc4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-right: -0.387ex; width:44.906ex; height:7.176ex;" alt="{\displaystyle I(X;Y)=\sum _{y\in Y}\sum _{x\in X}p(x,y)\log {\left({\frac {p(x,y)}{p_{1}(x)\,p_{2}(y)}}\right)},\,\!}" /></span></dd></dl> <p>dove <i>p</i>(<i>x</i>,<i>y</i>) è la <a href="/wiki/Distribuzione_congiunta" title="Distribuzione congiunta">funzione di distribuzione di probabilità congiunta</a> di <i>X</i> e <i>Y</i>, e <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_{1}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c912c5209a549292927fe5f6c02f93e43dcd4413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.452ex; height:2.843ex;" alt="{\displaystyle p_{1}(x)}" /></span> e <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_{2}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{2}(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8061cb2aad2d132cb3677639fdfd1c10e908bad0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.278ex; height:2.843ex;" alt="{\displaystyle p_{2}(y)}" /></span> sono le funzioni di distribuzione di <a href="/wiki/Distribuzione_marginale" title="Distribuzione marginale">probabilità marginale</a> rispettivamente di <i>X</i> e <i>Y</i>. </p><p>Nel caso <a href="/wiki/Continuo" title="Continuo">continuo</a>, la <a href="/wiki/Sommatoria" title="Sommatoria">sommatoria</a> è sostituita da un <a href="/wiki/Integrale_multiplo" title="Integrale multiplo">integrale doppio</a> definito: </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)=\int _{Y}\int _{X}p(x,y)\log {\left({\frac {p(x,y)}{p_{1}(x)\,p_{2}(y)}}\right)}\;dx\,dy,}"> <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> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <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"> <mrow> <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> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X;Y)=\int _{Y}\int _{X}p(x,y)\log {\left({\frac {p(x,y)}{p_{1}(x)\,p_{2}(y)}}\right)}\;dx\,dy,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7837a2a988fa06db2a51eae497eb2454a042fa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.38ex; height:6.509ex;" alt="{\displaystyle I(X;Y)=\int _{Y}\int _{X}p(x,y)\log {\left({\frac {p(x,y)}{p_{1}(x)\,p_{2}(y)}}\right)}\;dx\,dy,}" /></span></dd></dl> <p>dove <i>p</i>(<i>x</i>,<i>y</i>) è ora la funzione di "densità" di probabilità congiunta di <i>X</i> e <i>Y</i>, e <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_{1}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c912c5209a549292927fe5f6c02f93e43dcd4413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.452ex; height:2.843ex;" alt="{\displaystyle p_{1}(x)}" /></span> e <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_{2}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{2}(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8061cb2aad2d132cb3677639fdfd1c10e908bad0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.278ex; height:2.843ex;" alt="{\displaystyle p_{2}(y)}" /></span> sono le funzioni di densità di probabilità marginale rispettivamente di <i>X</i> e <i>Y</i>. </p><p>Queste definizioni sono ambigue perché la base della funzione logaritmica non è specificata. Per <a href="/wiki/Disambiguazione" title="Disambiguazione">disambiguare</a>, la funzione <i>I</i> potrebbe essere parametrizzata come <i>I</i>(<i>X</i>,<i>Y</i>,<i>b</i>) dove <i>b</i> è la base. Alternativamente, poiché la più comune unità di misura della mutua informazione è il bit, potrebbe essere specificata una base di 2. </p><p>Intuitivamente, l'informazione mutua misura l'informazione che <i>X</i> e <i>Y</i> condividono: essa misura quanto la conoscenza di una di queste variabili riduce la nostra incertezza riguardo all'altra. Ad esempio, se <i>X</i> e <i>Y</i> sono indipendenti, allora la conoscenza di <i>X</i> non dà alcuna informazione riguardo a <i>Y</i> e viceversa, perciò la loro mutua informazione è zero. All'altro estremo, se <i>X</i> e <i>Y</i> sono identiche allora tutte le informazioni trasmesse da <i>X</i> sono condivise con <i>Y</i>: la conoscenza di <i>X</i> determina il valore di <i>Y</i> e viceversa. Come risultato, nel caso di identità l'informazione mutua è la stessa contenuta in <i>Y</i> (o <i>X</i>) da sola, vale a dire l'<a href="/wiki/Entropia_dell%27informazione" class="mw-redirect" title="Entropia dell'informazione">entropia</a> di <i>Y</i> (o <i>X</i>: chiaramente se <i>X</i> e <i>Y</i> sono identiche, hanno identica entropia). </p><p>L'informazione mutua quantifica la dipendenza tra la <a href="/wiki/Distribuzione_congiunta" title="Distribuzione congiunta">distribuzione congiunta</a> di <i>X</i> e <i>Y</i> e quale sarebbe la distribuzione congiunta se <i>X</i> e <i>Y</i> fossero indipendenti. L'informazione mutua è una misura di dipendenza nel seguente senso: <i>I</i>(<i>X</i>; <i>Y</i>) = 0 <a href="/wiki/Se_e_solo_se" title="Se e solo se">se e solo se</a> <i>X</i> e <i>Y</i> sono variabili casuali indipendenti. Questo è facile da vedere in una sola direzione: se <i>X</i> e <i>Y</i> sono indipendenti, allora <i>p</i>(<i>x</i>,<i>y</i>) = <i>p</i>(<i>x</i>) <i>p</i>(<i>y</i>), e perciò: </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 \log {\left({\frac {p(x,y)}{p(x)\,p(y)}}\right)}=\log 1=0.\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <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"></mspace> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mn>1</mn> <mo>=</mo> <mn>0.</mn> <mspace width="thinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log {\left({\frac {p(x,y)}{p(x)\,p(y)}}\right)}=\log 1=0.\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44cd139e70e0324c1223b20f6091ad6cd1b7e31c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-right: -0.387ex; width:29.36ex; height:6.509ex;" alt="{\displaystyle \log {\left({\frac {p(x,y)}{p(x)\,p(y)}}\right)}=\log 1=0.\,\!}" /></span></dd></dl> <p>Inoltre, la mutua informazione è non negativa (cioè <i>I</i>(<i>X</i>;<i>Y</i>) ≥ 0; vedi sotto) e <a href="/wiki/Funzione_simmetrica" title="Funzione simmetrica">simmetrica</a> (cioè <i>I</i>(<i>X</i>;<i>Y</i>) = <i>I</i>(<i>Y</i>;<i>X</i>)). </p> <div class="mw-heading mw-heading2"><h2 id="Relazione_con_altre_quantità"><span id="Relazione_con_altre_quantit.C3.A0"></span>Relazione con altre quantità</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Informazione_mutua&veaction=edit&section=2" title="Modifica la sezione Relazione con altre quantità" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Informazione_mutua&action=edit&section=2" title="Edit section's source code: Relazione con altre quantità"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'informazione mutua può essere espressa in modo equivalente come </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 {\begin{aligned}I(X;Y)&{}=H(X)-H(X|Y)\\&{}=H(Y)-H(Y|X)\\&{}=H(X)+H(Y)-H(X,Y)\\&{}=H(X,Y)-H(X|Y)-H(Y|X)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <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> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <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>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <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> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <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>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>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I(X;Y)&{}=H(X)-H(X|Y)\\&{}=H(Y)-H(Y|X)\\&{}=H(X)+H(Y)-H(X,Y)\\&{}=H(X,Y)-H(X|Y)-H(Y|X)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c13e97646e808df00481954f411dfcf1da9a4b60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:42.505ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}I(X;Y)&{}=H(X)-H(X|Y)\\&{}=H(Y)-H(Y|X)\\&{}=H(X)+H(Y)-H(X,Y)\\&{}=H(X,Y)-H(X|Y)-H(Y|X)\end{aligned}}}" /></span></dd></dl> <p>dove <i>H</i>(<i>X</i>) e <i>H</i>(<i>Y</i>) sono le <a href="/wiki/Entropia_(teoria_dell%27informazione)" title="Entropia (teoria dell'informazione)">entropie</a> marginali, <i>H</i>(<i>X</i>|<i>Y</i>) e <i>H</i>(<i>Y</i>|<i>X</i>) sono le <a href="/wiki/Entropia_condizionale" title="Entropia condizionale">entropie condizionali</a>, e <i>H</i>(<i>X</i>,<i>Y</i>) è l'<a href="/wiki/Entropia_congiunta" title="Entropia congiunta">entropia congiunta</a> di <i>X</i> e <i>Y</i>. Poiché <i>H</i>(<i>X</i>) ≥ <i>H</i>(<i>X</i>|<i>Y</i>), questa caratterizzazione è coerente con la proprietà di non negatività enunciata sopra. </p><p>Intuitivamente, se l'entropia <i>H</i>(<i>X</i>) è considerata una misura dell'incertezza riguardo a una variabile casuale, allora <i>H</i>(<i>X</i>|<i>Y</i>) è una misura di ciò che <i>Y</i> <i>non</i> dice riguardo a <i>X</i>. Questo è "l'ammontare di incertezza che rimane riguardo a <i>X</i> dopo che <i>Y</i> è noto", e così il lato destro di queste uguaglianze può essere letto come "l'ammontare di incertezza in <i>X</i>, meno l'ammontare di incertezza in <i>X</i> che rimane dopo che <i>Y</i> è noto", che è equivalente "all'ammontare di incertezza in <i>X</i> che è eliminato conoscendo <i>Y</i>". Questo corrobora il significato intuitivo di informazione mutua come l'ammontare di informazione (cioè, la riduzione di incertezza) che la conoscenza di una delle due variabili fornisce riguardo all'altra. </p><p>Si noti che nel caso discreto <i>H</i>(<i>X</i>|<i>X</i>) = 0 e perciò <i>H</i>(<i>X</i>) = <i>I</i>(<i>X</i>;<i>X</i>). Così <i>I</i>(<i>X</i>;<i>X</i>) ≥ <i>I</i>(<i>X</i>;<i>Y</i>), e si può formulare il principio basilare che una variabile contiene almeno altrettanta informazione riguardo a sé stessa di quanta ne può fornire una qualsiasi altra variabile. </p><p>L'informazione mutua può essere espressa anche come una <a href="/wiki/Divergenza_di_Kullback-Leibler" title="Divergenza di Kullback-Leibler">divergenza di Kullback-Leibler</a>, del prodotto <i>p</i>(<i>x</i>) × <i>p</i>(<i>y</i>) della <a href="/wiki/Distribuzione_marginale" title="Distribuzione marginale">distribuzioni marginali</a> delle due variabili casuali <i>X</i> e <i>Y</i>, per <i>p</i>(<i>x</i>,<i>y</i>), la <a href="/wiki/Distribuzione_congiunta" title="Distribuzione congiunta">distribuzione congiunta</a> delle variabili casuali: </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/2972c67fb6bc52c6c15d72c4073f26aa16ccf0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.888ex; height:2.843ex;" alt="{\displaystyle I(X;Y)=D_{\mathrm {KL} }(p(x,y)\|p(x)p(y)).}" /></span></dd></dl> <p>Inoltre, sia <i>p</i>(<i>x</i>|<i>y</i>) = <i>p</i>(<i>x</i>, <i>y</i>) / <i>p</i>(<i>y</i>). Allora </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 {\begin{aligned}I(X;Y)&{}=\sum _{y}p(y)\sum _{x}p(x|y)\log _{2}{\frac {p(x|y)}{p(x)}}\\&{}=\sum _{y}p(y)\;D_{\mathrm {KL} }(p(x|y)\|p(x))\\&{}=\mathbb {E} _{Y}\{D_{\mathrm {KL} }(p(x|y)\|p(x))\}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <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> </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> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace"></mspace> <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 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> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <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 fence="false" 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 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 fence="false" stretchy="false">}</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I(X;Y)&{}=\sum _{y}p(y)\sum _{x}p(x|y)\log _{2}{\frac {p(x|y)}{p(x)}}\\&{}=\sum _{y}p(y)\;D_{\mathrm {KL} }(p(x|y)\|p(x))\\&{}=\mathbb {E} _{Y}\{D_{\mathrm {KL} }(p(x|y)\|p(x))\}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf0bb5a2dbd5987ddb9bbed45831cd24a07cafa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:41.481ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}I(X;Y)&{}=\sum _{y}p(y)\sum _{x}p(x|y)\log _{2}{\frac {p(x|y)}{p(x)}}\\&{}=\sum _{y}p(y)\;D_{\mathrm {KL} }(p(x|y)\|p(x))\\&{}=\mathbb {E} _{Y}\{D_{\mathrm {KL} }(p(x|y)\|p(x))\}.\end{aligned}}}" /></span></dd></dl> <p>Così l'informazione mutua può essere intesa anche come l'<a href="/wiki/Valore_atteso" title="Valore atteso">aspettativa</a> della divergenza di Kullback-Leibler della distribuzione univariata <i>p</i>(<i>x</i>) di <i>X</i> dalla <a href="/w/index.php?title=Distribuzione_condizionale&action=edit&redlink=1" class="new" title="Distribuzione condizionale (la pagina non esiste)">distribuzione condizionale</a> <i>p</i>(<i>x</i>|<i>y</i>) di <i>X</i> data <i>Y</i>: più le distribuzioni <i>p</i>(<i>x</i>|<i>y</i>) e <i>p</i>(<i>x</i>) sono differenti, maggiore è il <a href="/wiki/Divergenza_di_Kullback-Leibler" title="Divergenza di Kullback-Leibler">guadagno di informazione</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Varianti">Varianti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Informazione_mutua&veaction=edit&section=3" title="Modifica la sezione Varianti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Informazione_mutua&action=edit&section=3" title="Edit section's source code: Varianti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sono state proposte parecchie variazioni sull'informazione mutua per adattarsi a varie necessità. Tra queste vi sono varianti normalizzate e generalizzazioni a più di due variabili. </p> <div class="mw-heading mw-heading3"><h3 id="Metrica">Metrica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Informazione_mutua&veaction=edit&section=4" title="Modifica la sezione Metrica" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Informazione_mutua&action=edit&section=4" title="Edit section's source code: Metrica"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Molte applicazioni richiedono una <a href="/wiki/Distanza_(matematica)" title="Distanza (matematica)">metrica</a>, cioè, una misura di distanza tra punti. La quantità </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(X,Y)=H(X,Y)-I(X;Y)=H(X)+H(Y)-2I(X;Y)=H(X|Y)+H(Y|X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</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>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <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>Y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <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> <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>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(X,Y)=H(X,Y)-I(X;Y)=H(X)+H(Y)-2I(X;Y)=H(X|Y)+H(Y|X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b31edf54b872059f9a8c1e121f2317e0876279a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:81.874ex; height:2.843ex;" alt="{\displaystyle d(X,Y)=H(X,Y)-I(X;Y)=H(X)+H(Y)-2I(X;Y)=H(X|Y)+H(Y|X)}" /></span></dd></dl> <p>soddisfa le proprietà basilari di una metrica; in particolare, la <a href="/wiki/Disuguaglianza_triangolare" title="Disuguaglianza triangolare">disuguaglianza triangolare</a>, ma anche la <a href="/w/index.php?title=Non_negativoe&action=edit&redlink=1" class="new" title="Non negativoe (la pagina non esiste)">non negatività</a>, <a href="/wiki/Principio_degli_indiscernibili" class="mw-redirect" title="Principio degli indiscernibili">indiscernibilità</a> e simmetria. Questa metrica della distanza è nota anche come <a href="/w/index.php?title=Variazione_dell%27informazione&action=edit&redlink=1" class="new" title="Variazione dell'informazione (la pagina non esiste)">variazione dell'informazione</a>. </p><p>Poiché si ha <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(X,Y)\leq H(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</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>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(X,Y)\leq H(X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/984a1abe46cfbeeba13c85de75fbdfb2a593be1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.571ex; height:2.843ex;" alt="{\displaystyle d(X,Y)\leq H(X,Y)}" /></span>, una variante normalizzata naturale è </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(X,Y)=d(X,Y)/H(X,Y)\leq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(X,Y)=d(X,Y)/H(X,Y)\leq 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1006fff593af405c328fbb38d1826b2fcaa95c2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.162ex; height:2.843ex;" alt="{\displaystyle D(X,Y)=d(X,Y)/H(X,Y)\leq 1.}" /></span></dd></dl> <p>La metrica <i>D</i> è una <a href="/w/index.php?title=Metrica_universale&action=edit&redlink=1" class="new" title="Metrica universale (la pagina non esiste)">metrica universale</a>, in quanto se qualsiasi altra misura pone vicino <i>X</i> e <i>Y</i>, allora anche la <i>D</i> le stimerà vicine.<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> </p><p>Un'interpretazione dell'informazione secondo la <a href="/wiki/Teoria_degli_insiemi" title="Teoria degli insiemi">teoria degli insiemi</a> (si veda la figura per l'<a href="/wiki/Entropia_condizionale" title="Entropia condizionale">entropia condizionale</a>) mostra che </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(X,Y)=1-I(X;Y)/H(X,Y)=1-H(X\cap Y)/H(X\cup Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>∩</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>∪</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(X,Y)=1-I(X;Y)/H(X,Y)=1-H(X\cap Y)/H(X\cup Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb0a05b57578220bf8f6903c765bca0d4c7d6df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.894ex; height:2.843ex;" alt="{\displaystyle D(X,Y)=1-I(X;Y)/H(X,Y)=1-H(X\cap Y)/H(X\cup Y)}" /></span></dd></dl> <p>che è effettivamente la <a href="/wiki/Indice_di_Jaccard" title="Indice di Jaccard">distanza di Jaccard</a> tra <i>X</i> e <i>Y</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Informazione_mutua_condizionale">Informazione mutua condizionale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Informazione_mutua&veaction=edit&section=5" title="Modifica la sezione Informazione mutua condizionale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Informazione_mutua&action=edit&section=5" title="Edit section's source code: Informazione mutua condizionale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A volte è utile esprimere l'informazione mutua di due variabili casuali condizionate a una terza. </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|Z)=\mathbb {E} _{Z}{\big (}I(X;Y)|Z{\big )}=\sum _{z\in Z}\sum _{y\in Y}\sum _{x\in X}p_{Z}(z)p_{X,Y|Z}(x,y|z)\log {\frac {p_{X,Y|Z}(x,y|z)}{p_{X|Z}(x|z)p_{Y|Z}(y|z)}},}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Z</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>Z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>∈</mo> <mi>Z</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X;Y|Z)=\mathbb {E} _{Z}{\big (}I(X;Y)|Z{\big )}=\sum _{z\in Z}\sum _{y\in Y}\sum _{x\in X}p_{Z}(z)p_{X,Y|Z}(x,y|z)\log {\frac {p_{X,Y|Z}(x,y|z)}{p_{X|Z}(x|z)p_{Y|Z}(y|z)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e09b4c0f0e2bab626d92012afdf6fa0cfb2b0ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:85.057ex; height:7.343ex;" alt="{\displaystyle I(X;Y|Z)=\mathbb {E} _{Z}{\big (}I(X;Y)|Z{\big )}=\sum _{z\in Z}\sum _{y\in Y}\sum _{x\in X}p_{Z}(z)p_{X,Y|Z}(x,y|z)\log {\frac {p_{X,Y|Z}(x,y|z)}{p_{X|Z}(x|z)p_{Y|Z}(y|z)}},}" /></span></dd></dl> <p>che possono essere semplificate come </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|Z)=\sum _{z\in Z}\sum _{y\in Y}\sum _{x\in X}p_{X,Y,Z}(x,y,z)\log {\frac {p_{Z}(z)p_{X,Y,Z}(x,y,z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)}}.}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>∈</mo> <mi>Z</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>Z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X;Y|Z)=\sum _{z\in Z}\sum _{y\in Y}\sum _{x\in X}p_{X,Y,Z}(x,y,z)\log {\frac {p_{Z}(z)p_{X,Y,Z}(x,y,z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f66c0efb32521bdaf79f90a0d3cf3733af4d0ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:62.436ex; height:7.176ex;" alt="{\displaystyle I(X;Y|Z)=\sum _{z\in Z}\sum _{y\in Y}\sum _{x\in X}p_{X,Y,Z}(x,y,z)\log {\frac {p_{Z}(z)p_{X,Y,Z}(x,y,z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)}}.}" /></span></dd></dl> <p>Il condizionamento a una terza variabile casuale potrebbe o aumentare o diminuire l'informazione mutua, ma è sempre vero che </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|Z)\geq 0}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X;Y|Z)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef9100c5f7c64445ef735d4e7ada6872550bc92b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.356ex; height:2.843ex;" alt="{\displaystyle I(X;Y|Z)\geq 0}" /></span></dd></dl> <p>per le variabili casuali discrete, distribuite congiuntamente <i>X</i>, <i>Y</i>, <i>Z</i>. Questo risultato è stato utilizzato come mattone basilare per provare altre <a href="/w/index.php?title=Disuguaglianze_nella_teoria_dell%27informazione&action=edit&redlink=1" class="new" title="Disuguaglianze nella teoria dell'informazione (la pagina non esiste)">disuguaglianze nella teoria dell'informazione</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Informazione_mutua_multivariata">Informazione mutua multivariata</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Informazione_mutua&veaction=edit&section=6" title="Modifica la sezione Informazione mutua multivariata" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Informazione_mutua&action=edit&section=6" title="Edit section's source code: Informazione mutua multivariata"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sono state proposte parecchie generalizzazioni della informazione mutua a più di due variabili, come la <a href="/w/index.php?title=Correlazione_totale&action=edit&redlink=1" class="new" title="Correlazione totale (la pagina non esiste)">correlazione totale</a> e l'<a href="/w/index.php?title=Informazione_sulle_interazioni&action=edit&redlink=1" class="new" title="Informazione sulle interazioni (la pagina non esiste)">informazione sulle interazioni</a>. Se Shannon è vista come una <a href="/wiki/Misura_con_segno" title="Misura con segno">misura con segno</a> nel contesto dei <a href="/w/index.php?title=Diagramma_dell%27informazione&action=edit&redlink=1" class="new" title="Diagramma dell'informazione (la pagina non esiste)">diagrammi dell'informazione</a> come spiegato nella voce <i><a href="/w/index.php?title=Teoria_dell%27informazione_e_teoria_della_misura&action=edit&redlink=1" class="new" title="Teoria dell'informazione e teoria della misura (la pagina non esiste)">Teoria dell'informazione e teoria della misura</a></i>, allora l'unica definizione di informazione mutua multivariata <style data-mw-deduplicate="TemplateStyles:r140554517">.mw-parser-output .chiarimento{background:#ffeaea;color:#444444}.mw-parser-output .chiarimento-apice{color:#EE0700}@media screen{html.skin-theme-clientpref-night .mw-parser-output .chiarimento{background:rgba(179,36,36,0.21);color:inherit}html.skin-theme-clientpref-night .mw-parser-output .chiarimento-apice{color:#b32424}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .chiarimento{background:rgba(179,36,36,0.21);color:inherit}html.skin-theme-clientpref-os .mw-parser-output .chiarimento-apice{color:#b32424}}</style><span class="chiarimento" title="Queste informazioni non sono comprovate da fonti attendibili."></span><sup class="noprint chiarimento-apice" title="Queste informazioni non sono comprovate da fonti attendibili.">[<i><a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">senza fonte</a></i>]</sup> è come segue: </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_{1};X_{1})=H(X_{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>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</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 I(X_{1};X_{1})=H(X_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00408deacc6b0113c6928a9795c2f9b4b840b454" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.922ex; height:2.843ex;" alt="{\displaystyle I(X_{1};X_{1})=H(X_{1})}" /></span></dd></dl> <p>e per <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 n>1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/957e317312fd02f34c1d1c8c80bd8484c29fde6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.302ex; height:2.509ex;" alt="{\displaystyle n>1,}" /></span> </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_{1};\,...\,;X_{n})=I(X_{1};\,...\,;X_{n-1})-I(X_{1};\,...\,;X_{n-1}|X_{n}),}"> <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> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>;</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>I</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>;</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>;</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>I</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>;</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>;</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <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 I(X_{1};\,...\,;X_{n})=I(X_{1};\,...\,;X_{n-1})-I(X_{1};\,...\,;X_{n-1}|X_{n}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/324a44d075d2519e9276c38976edc986213a4299" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.715ex; height:2.843ex;" alt="{\displaystyle I(X_{1};\,...\,;X_{n})=I(X_{1};\,...\,;X_{n-1})-I(X_{1};\,...\,;X_{n-1}|X_{n}),}" /></span></dd></dl> <p>dove (come sopra) definiamo </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_{1};\,...\,;X_{n-1}|X_{n})=\mathbb {E} _{X_{n}}{\big (}I(X_{1};\,...\,;X_{n-1})|X_{n}{\big )}.}"> <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> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>;</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>;</mo> <mspace width="thinmathspace"></mspace> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mspace width="thinmathspace"></mspace> <mo>;</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X_{1};\,...\,;X_{n-1}|X_{n})=\mathbb {E} _{X_{n}}{\big (}I(X_{1};\,...\,;X_{n-1})|X_{n}{\big )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70547ef37ee078218dc80dd6c06c90c7d1e50f8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:51.856ex; height:3.176ex;" alt="{\displaystyle I(X_{1};\,...\,;X_{n-1}|X_{n})=\mathbb {E} _{X_{n}}{\big (}I(X_{1};\,...\,;X_{n-1})|X_{n}{\big )}.}" /></span></dd></dl> <p>(Questa definizione dell'informazione mutua multivariata è identica a quella dell'<a href="/w/index.php?title=Informazione_sulle_interazioni&action=edit&redlink=1" class="new" title="Informazione sulle interazioni (la pagina non esiste)">informazione sulle interazioni</a> tranne che per un cambiamento di segno dove il numero delle variabili casuali è dispari.) </p> <div class="mw-heading mw-heading4"><h4 id="Applicazioni">Applicazioni</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Informazione_mutua&veaction=edit&section=7" title="Modifica la sezione Applicazioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Informazione_mutua&action=edit&section=7" title="Edit section's source code: Applicazioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Applicare pedissequamente i diagrammi dell'informazione per derivare la definizione di cui sopra<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r140554517" /><span class="chiarimento" title="Queste informazioni non sono comprovate da fonti attendibili."></span><sup class="noprint chiarimento-apice" title="Queste informazioni non sono comprovate da fonti attendibili.">[<i><a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">senza fonte</a></i>]</sup> è stato criticato, e in effetti ha trovato applicazione pratica piuttosto limitata, poiché è difficile visualizzare o afferrare il significato di questa quantità per un gran numero di variabili casuali. Può essere zero, positivo o negativo per qualsiasi <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 n\geq 3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 3.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df39648b3c5061c0ccc992e8374a69d96d9dc88c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.302ex; height:2.343ex;" alt="{\displaystyle n\geq 3.}" /></span> </p><p>Uno schema di generalizzazione ad alta dimensione che massimizza l'informazione mutua tra la distribuzione congiunta e altre variabili obiettivo si trova essere utile nella <a href="/wiki/Selezione_delle_caratteristiche" title="Selezione delle caratteristiche">selezione delle caratteristiche</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Varianti_normalizzate">Varianti normalizzate</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Informazione_mutua&veaction=edit&section=8" title="Modifica la sezione Varianti normalizzate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Informazione_mutua&action=edit&section=8" title="Edit section's source code: Varianti normalizzate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Varianti normalizzate dell'informazione mutua sono fornite dal <i>coefficiente di vincolo</i> (Coombs, Dawes & Tversky, 1970) o dal <i>coefficiente d'incertezza</i> (Press & Flannery, 1988) </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_{XY}={\frac {I(X;Y)}{H(Y)}}~~~~{\mbox{and}}~~~~C_{YX}={\frac {I(X;Y)}{H(X)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>and</mtext> </mstyle> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>X</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>H</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 C_{XY}={\frac {I(X;Y)}{H(Y)}}~~~~{\mbox{and}}~~~~C_{YX}={\frac {I(X;Y)}{H(X)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dcbf6ab9e5300330aca201035c894ae398d55de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.541ex; height:6.509ex;" alt="{\displaystyle C_{XY}={\frac {I(X;Y)}{H(Y)}}~~~~{\mbox{and}}~~~~C_{YX}={\frac {I(X;Y)}{H(X)}}.}" /></span></dd></dl> <p>I due coefficienti non sono necessariamente uguali. Una misura d'informazione scalata più utile e simmetrica è la <i>ridondanza</i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r140554517" /><span class="chiarimento" title="Queste informazioni non sono comprovate da fonti attendibili."></span><sup class="noprint chiarimento-apice" title="Queste informazioni non sono comprovate da fonti attendibili.">[<i><a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">senza fonte</a></i>]</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 R={\frac {I(X;Y)}{H(X)+H(Y)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <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> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\frac {I(X;Y)}{H(X)+H(Y)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac9aee7c898e1355d03901e591a9e0cfce43aa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.038ex; height:6.509ex;" alt="{\displaystyle R={\frac {I(X;Y)}{H(X)+H(Y)}}}" /></span></dd></dl> <p>che raggiunge un minimo di zero quando le variabili sono indipendenti e un valore massimo di </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_{\max }={\frac {\min(H(X),H(Y))}{H(X)+H(Y)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</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 stretchy="false">)</mo> </mrow> <mrow> <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> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\max }={\frac {\min(H(X),H(Y))}{H(X)+H(Y)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab645b3bb27ec5c7fe343068a5e098331b6ea31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.207ex; height:6.509ex;" alt="{\displaystyle R_{\max }={\frac {\min(H(X),H(Y))}{H(X)+H(Y)}}}" /></span></dd></dl> <p>quando una variabile diventa completamente ridondante con la conoscenza dell'altra. Si veda anche <i><a href="/wiki/Ridondanza_(teoria_dell%27informazione)" title="Ridondanza (teoria dell'informazione)">Ridondanza (teoria dell'informazione)</a></i>. Un'altra misura simmetrica è l'<i>incertezza simmetrica</i> (Witten & Frank, 2005), data da </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 U(X,Y)=2R=2{\frac {I(X;Y)}{H(X)+H(Y)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>R</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <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> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(X,Y)=2R=2{\frac {I(X;Y)}{H(X)+H(Y)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76011414e90b53f7cee5905b5c6861840dc64c87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.841ex; height:6.509ex;" alt="{\displaystyle U(X,Y)=2R=2{\frac {I(X;Y)}{H(X)+H(Y)}}}" /></span></dd></dl> <p>che rappresenta una media ponderata dei due coefficienti d'incertezza (Press & Flannery, 1988). </p><p>Altre versioni normalizzate sono fornite dalle seguenti espressioni (Yao, 2003; Strehl & Ghosh, 2002). </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 {\frac {I(X;Y)}{\operatorname {min} (H(X),H(Y))}},~~~~~~~{\frac {I(X;Y)}{H(X,Y)}},~~~~~~~{\frac {I(X;Y)}{\sqrt {H(X)H(Y)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>min</mi> <mo>⁡</mo> <mo stretchy="false">(</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 stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {I(X;Y)}{\operatorname {min} (H(X),H(Y))}},~~~~~~~{\frac {I(X;Y)}{H(X,Y)}},~~~~~~~{\frac {I(X;Y)}{\sqrt {H(X)H(Y)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efbf11f0aa2b30128b1ecb5e346bf23a537565ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.406ex; height:7.009ex;" alt="{\displaystyle {\frac {I(X;Y)}{\operatorname {min} (H(X),H(Y))}},~~~~~~~{\frac {I(X;Y)}{H(X,Y)}},~~~~~~~{\frac {I(X;Y)}{\sqrt {H(X)H(Y)}}}}" /></span></dd></dl> <p>Se consideriamo la mutua informazione come un caso speciale della <a href="/w/index.php?title=Correlazione_totale&action=edit&redlink=1" class="new" title="Correlazione totale (la pagina non esiste)">correlazione totale</a>, la normalizzazione è: </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 {\frac {I(X;Y)}{\operatorname {min} (H(X),H(Y))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>min</mi> <mo>⁡</mo> <mo stretchy="false">(</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 stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {I(X;Y)}{\operatorname {min} (H(X),H(Y))}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e48ddf8ad38d160bc2baeb9212e6c7a96ff011c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.054ex; height:6.509ex;" alt="{\displaystyle {\frac {I(X;Y)}{\operatorname {min} (H(X),H(Y))}}}" /></span></dd></dl> <p>La quantità </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^{\prime }(X,Y)=1-{\frac {I(X;Y)}{\operatorname {max} (H(X),H(Y))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>max</mi> <mo>⁡</mo> <mo stretchy="false">(</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 stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{\prime }(X,Y)=1-{\frac {I(X;Y)}{\operatorname {max} (H(X),H(Y))}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86047ec33449a628dea2f14653e604607b3727c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.811ex; height:6.509ex;" alt="{\displaystyle D^{\prime }(X,Y)=1-{\frac {I(X;Y)}{\operatorname {max} (H(X),H(Y))}}}" /></span></dd></dl> <p>è una <a href="/wiki/Distanza_(matematica)" title="Distanza (matematica)">metrica</a>, cioè soddisfa la disuguaglianza triangolare ecc. La metrica <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^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4684c384a685428de6f7949a57a4a9c9acf8475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.609ex; height:2.509ex;" alt="{\displaystyle D^{\prime }}" /></span> è anche una metrica universale.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Varianti_ponderate">Varianti ponderate</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Informazione_mutua&veaction=edit&section=9" title="Modifica la sezione Varianti ponderate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Informazione_mutua&action=edit&section=9" title="Edit section's source code: Varianti ponderate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nella formulazione tradizionale dell'informazione mutua, </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)=\sum _{y\in Y}\sum _{x\in X}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> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mrow> </munder> <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>,</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"></mspace> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X;Y)=\sum _{y\in Y}\sum _{x\in X}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/ba4fc3bc74ebebe70e957b7b87b3d32cf6b032ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.989ex; height:7.176ex;" alt="{\displaystyle I(X;Y)=\sum _{y\in Y}\sum _{x\in X}p(x,y)\log {\frac {p(x,y)}{p(x)\,p(y)}},}" /></span></dd></dl> <p>ciascun <i>evento</i> od <i>oggetto</i> specificato da <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,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" /></span> è ponderato mediante la probabilità corrispondente <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,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/089e91a1824e14cebc8e8d04dc652c61b3008e0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.587ex; height:2.843ex;" alt="{\displaystyle p(x,y)}" /></span>. Questo assume che tutti gli oggetti o eventi siano equivalenti <i>a parte</i> la loro probabilità di occorrenza. Tuttavia, in alcune applicazioni potrebbe accadere che certi oggetti o eventi siano più <i>significativi</i> di altri, o che certi schemi di associazione siano semanticamente più importanti di altri. </p><p>Ad esempio, la mappatura deterministica <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 \{(1,1),(2,2),(3,3)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(1,1),(2,2),(3,3)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b13cd0103f9e43410e877754765a7ad994fe3cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.897ex; height:2.843ex;" alt="{\displaystyle \{(1,1),(2,2),(3,3)\}}" /></span> potrebbe essere considerata più forte della mappatura deterministica <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 \{(1,3),(2,1),(3,2)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(1,3),(2,1),(3,2)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/121771fac14b709e1be2dc4d9c3e4a04a0dd9715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.897ex; height:2.843ex;" alt="{\displaystyle \{(1,3),(2,1),(3,2)\}}" /></span>, sebbene queste relazioni produrrebbero la stessa informazione mutua. Ciò accade perché l'informazione mutua non è affatto sensibile ad alcun ordinamento insito nei valori delle variabili (Cronbach, 1954; Coombs & Dawes, 1970; Lockhead, 1970), e perciò non è affatto sensibile alla <b>forma</b> della relazione tra le variabili associate. Se si deesidera che la precedente relazione — che mostrava accordo su tutti i valori delle variabili — sia stimata più forte della relazione successiva, allora è possibile utilizzate la seguente <i>informazione mutua ponderata</i> (Guiasu, 1977) </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)=\sum _{y\in Y}\sum _{x\in X}w(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> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </munder> <mi>w</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <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"></mspace> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(X;Y)=\sum _{y\in Y}\sum _{x\in X}w(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/8d878c6622420225b626259fa56c9e4427c1af1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:45.981ex; height:7.176ex;" alt="{\displaystyle I(X;Y)=\sum _{y\in Y}\sum _{x\in X}w(x,y)p(x,y)\log {\frac {p(x,y)}{p(x)\,p(y)}},}" /></span></dd></dl> <p>che pone un peso <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 w(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c832b255202f489718326a29d60b7c2be485b15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.993ex; height:2.843ex;" alt="{\displaystyle w(x,y)}" /></span> sulla probabilità di ogni co-occorrenza dei valori delle variabili, <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,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/089e91a1824e14cebc8e8d04dc652c61b3008e0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.587ex; height:2.843ex;" alt="{\displaystyle p(x,y)}" /></span>. Questo consente che certe probabilità possano portate più o meno significato di altre, con ciò permettendo la quantificazione dei relativi fattori <i>olistici</i> o <i>pregnanti</i>. Nell'esempio di sopra, utilizzare pesi relativi più grandi per <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 w(1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w(1,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e01c54cc04db0d799c7e7f2407adcee0d6e5b6e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.832ex; height:2.843ex;" alt="{\displaystyle w(1,1)}" /></span>, <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 w(2,2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w(2,2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c748432c52030777e6e130bcf2caa62d3efcab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.832ex; height:2.843ex;" alt="{\displaystyle w(2,2)}" /></span> e <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 w(3,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w(3,3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80b9ac172050e00d5cd80e25baceb427b05413d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.832ex; height:2.843ex;" alt="{\displaystyle w(3,3)}" /></span> avrebbe l'effetto di valutare maggiore <i>informatività</i> per la relazione <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 \{(1,1),(2,2),(3,3)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(1,1),(2,2),(3,3)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b13cd0103f9e43410e877754765a7ad994fe3cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.897ex; height:2.843ex;" alt="{\displaystyle \{(1,1),(2,2),(3,3)\}}" /></span> che per la relazione <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 \{(1,3),(2,1),(3,2)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(1,3),(2,1),(3,2)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/121771fac14b709e1be2dc4d9c3e4a04a0dd9715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.897ex; height:2.843ex;" alt="{\displaystyle \{(1,3),(2,1),(3,2)\}}" /></span>, il che potrebbe essere desiderabile in alcuni casi di riconoscimento degli schemi, e simili. Tuttavia, sono stati realizzati pochi studi matematici sull'informazione mutua ponderata. </p> <div class="mw-heading mw-heading3"><h3 id="Informazione_mutua_assoluta">Informazione mutua assoluta</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Informazione_mutua&veaction=edit&section=10" title="Modifica la sezione Informazione mutua assoluta" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Informazione_mutua&action=edit&section=10" title="Edit section's source code: Informazione mutua assoluta"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Usando i concetti della <a href="/wiki/Complessit%C3%A0_di_Kolmogorov" title="Complessità di Kolmogorov">complessità di Kolmogorov</a>, si può considerare l'informazione mutua di due sequenze indipendente da qualsiasi distribuzione di probabilità: </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_{K}(X;Y)=K(X)-K(X|Y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>K</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 I_{K}(X;Y)=K(X)-K(X|Y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e51aa116a8d55d2bdbeced6a8956c20a7dbdc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.029ex; height:2.843ex;" alt="{\displaystyle I_{K}(X;Y)=K(X)-K(X|Y).}" /></span></dd></dl> <p>Stabilire che questa quantità è simmetrica fino ad un fattore logaritmico (<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_{K}(X;Y)\approx I_{K}(Y;X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>;</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{K}(X;Y)\approx I_{K}(Y;X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6615950d65439cb8bdd1f020a66af2ac348c7c4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.724ex; height:2.843ex;" alt="{\displaystyle I_{K}(X;Y)\approx I_{K}(Y;X)}" /></span>) richiede la <a href="/w/index.php?title=Regola_della_catena_per_la_complessit%C3%A0_di_Kolmogorov&action=edit&redlink=1" class="new" title="Regola della catena per la complessità di Kolmogorov (la pagina non esiste)">regola della catena per la complessità di Kolmogorov</a> (Li, 1997). Si possono usare approssimazioni di questa quantità attraverso la <a href="/wiki/Compressione_dei_dati" title="Compressione dei dati">compressione</a> per definire una <a href="/wiki/Distanza_(matematica)" title="Distanza (matematica)">misura di distanza</a> per eseguire un <a href="/wiki/Clustering_gerarchico" title="Clustering gerarchico">clustering gerarchico</a> di sequenze senza avere alcuna conoscenza del dominio delle sequenze stesse (Cilibrasi, 2005). </p> <div class="mw-heading mw-heading2"><h2 id="Applicazioni_2">Applicazioni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Informazione_mutua&veaction=edit&section=11" title="Modifica la sezione Applicazioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Informazione_mutua&action=edit&section=11" title="Edit section's source code: Applicazioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In molte applicazioni, si vuole massimizzare la mutua informazione (accrescendo così le dipendenze), il che è spesso equivalente a minimizzare l'<a href="/wiki/Entropia_condizionale" title="Entropia condizionale">entropia condizionale</a>. Gli esempi comprendono: </p> <ul><li>La <a href="/wiki/Capacit%C3%A0_di_canale_(teoria_dell%27informazione)" class="mw-redirect" title="Capacità di canale (teoria dell'informazione)">capacità di canale</a> è uguale all'informazione mutua, massimizzata su tutte le distribuzioni di input.</li> <li>Sono state proposte procedure di <a href="/w/index.php?title=Addestramento_discriminatorio&action=edit&redlink=1" class="new" title="Addestramento discriminatorio (la pagina non esiste)">addestramento discriminatorio</a> per i <a href="/wiki/Modello_di_Markov_nascosto" title="Modello di Markov nascosto">modelli di Markov nascosti</a> basate sul criterio della <a href="/w/index.php?title=Massima_informazione_mutua&action=edit&redlink=1" class="new" title="Massima informazione mutua (la pagina non esiste)">massima informazione mutua</a> (MIM).</li> <li>La previsione della [<a href="/wiki/Struttura_secondaria" title="Struttura secondaria">struttura secondaria</a> dell'<a href="/wiki/RNA" title="RNA">RNA</a> da un <a href="/w/index.php?title=Allineamento_multiplo_di_sequenze&action=edit&redlink=1" class="new" title="Allineamento multiplo di sequenze (la pagina non esiste)">allineamento multiplo di sequenze</a>.</li> <li>La previsione del <a href="/wiki/Filogenesi" title="Filogenesi">profilo filogenetico</a> dall'attuale a coppie e dalla scomparsa dei <a href="/wiki/Gene" title="Gene">geni</a> funzionalmente collegati.</li> <li>L'informazione mutua è stata usata come criterio per la <a href="/wiki/Selezione_delle_caratteristiche" title="Selezione delle caratteristiche">selezione delle caratteristiche</a> e per le trasformazioni delle caratteristiche nell'<a href="/wiki/Apprendimento_automatico" title="Apprendimento automatico">apprendimento automatico</a>. Può essere utilizzata per caratterizzare sia la rilevanza che la ridondanza delle variabili, come la <a href="/w/index.php?title=Selezione_delle_caratteristiche_con_la_minima_ridondanza&action=edit&redlink=1" class="new" title="Selezione delle caratteristiche con la minima ridondanza (la pagina non esiste)">selezione delle caratteristiche con la minima ridondanza</a>.</li> <li>L'informazione mutua è spesso usata come funzione significativa per la computazione delle <a href="/wiki/Collocazione_(linguistica)" title="Collocazione (linguistica)">collocazioni</a> nel <a href="/wiki/Corpus#Linguistica_dei_Corpora" title="Corpus">linguistica dei <i>corpora</i></a>.</li> <li>L'informazione mutua è usata nell'<a href="/wiki/Imaging_biomedico" class="mw-redirect" title="Imaging biomedico">imaging biomedico</a> per la <a href="/w/index.php?title=Registrazione_delle_immagini&action=edit&redlink=1" class="new" title="Registrazione delle immagini (la pagina non esiste)">registrazione delle immagini</a>. Data un'immagine di riferimento (ad esempio, una scansione cerebrale), e una seconda immagine che si deve mettere nello stesso <a href="/wiki/Sistema_di_coordinate" title="Sistema di coordinate">sistema di coordinate</a>, come l'immagine di riferimento, questa immagine è deformata finché l'informazione mutua tra di essa e l'immagine di riferimento non è massimizzata.</li> <li>La rilevazione della <a href="/w/index.php?title=Sincronizzazione_delle_fasi&action=edit&redlink=1" class="new" title="Sincronizzazione delle fasi (la pagina non esiste)">sincronizzazione delle fasi</a> nell'analisi delle <a href="/w/index.php?title=Serie_temporali&action=edit&redlink=1" class="new" title="Serie temporali (la pagina non esiste)">serie temporali</a>.</li> <li>Nel metodo <a href="/w/index.php?title=Infomax&action=edit&redlink=1" class="new" title="Infomax (la pagina non esiste)">infomax</a> per l'apprendimento delle reti neurali e altre forme di apprendimento automatico, compreso l'algoritmo per l'<a href="/wiki/Analisi_delle_componenti_indipendenti" title="Analisi delle componenti indipendenti">analisi delle componenti indipendenti</a>.</li> <li>L'informazione mutua media nel <a href="/w/index.php?title=Teorema_di_Takens&action=edit&redlink=1" class="new" title="Teorema di Takens (la pagina non esiste)">teorema dell'incorporazione dei ritardi</a> si usa per determinare il parametro del <i>ritardo di incorporazione</i>.</li> <li>L'informazione mutua tra i <a href="/wiki/Gene" title="Gene">geni</a> nei dati delle <a href="/w/index.php?title=Microdisposizione&action=edit&redlink=1" class="new" title="Microdisposizione (la pagina non esiste)">microdisposizioni delle espressioni</a> è usata dall'algoritmo <a href="/w/index.php?title=ARACNE&action=edit&redlink=1" class="new" title="ARACNE (la pagina non esiste)">ARACNE</a> per la ricostruzione delle <a href="/w/index.php?title=Rete_di_regolazione_genica&action=edit&redlink=1" class="new" title="Rete di regolazione genica (la pagina non esiste)">reti geniche</a>.</li> <li>La mutua informazione è usata come misura di confronto dei clustering (Vinh <i>et al.</i>, 2009), fornendo alcuni vantaggi su altre misure classiche come l'<a href="/w/index.php?title=Indice_di_Rand&action=edit&redlink=1" class="new" title="Indice di Rand (la pagina non esiste)">indice di Rand</a> e l'<a href="/w/index.php?title=Indice_di_Rand_aggiustato&action=edit&redlink=1" class="new" title="Indice di Rand aggiustato (la pagina non esiste)">indice di Rand aggiustato</a>.</li> <li>La versione dell'informazione mutua aggiustata per il caso è l'<a href="/w/index.php?title=Informazione_mutua_aggiustata&action=edit&redlink=1" class="new" title="Informazione mutua aggiustata (la pagina non esiste)">informazione mutua aggiustata</a> (IMA) (Vinh et al., 2009). È usata per confrontare i clustering. Corregge l'effetto dell'accordo tra i clustering dovuto unicamente al caso, simile al modo in cui l'<a href="/w/index.php?title=Indice_di_Rand_aggiustato&action=edit&redlink=1" class="new" title="Indice di Rand aggiustato (la pagina non esiste)">indice di Rand aggiustato</a> corregge l'<a href="/w/index.php?title=Indice_di_Rand&action=edit&redlink=1" class="new" title="Indice di Rand (la pagina non esiste)">indice di Rand</a>. Un programma <a href="/wiki/MATLAB" title="MATLAB">MATLAB</a> per calcolare l'IMA può essere trovato da <a rel="nofollow" class="external free" href="https://web.archive.org/web/20110531024715/http://ee.unsw.edu.au/~nguyenv/Software.htm">https://web.archive.org/web/20110531024715/http://ee.unsw.edu.au/~nguyenv/Software.htm</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Informazione_mutua&veaction=edit&section=12" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Informazione_mutua&action=edit&section=12" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text">Alexander Kraskov, Harald Stögbauer, Ralph G. Andrzejak, and Peter Grassberger, "Hierarchical Clustering Based on Mutual Information", (2003) <i><a rel="nofollow" class="external text" href="https://arxiv.org/abs/q-bio/0311039">ArXiv q-bio/0311039</a></i></span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Christopher D. Manning, Prabhakar Raghavan, Hinrich Schütze, <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoin0000mann_b6m0"><span style="font-style:italic;">An Introduction to Information Retrieval</span></a>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, 2008, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-521-86571-9" title="Speciale:RicercaISBN/0-521-86571-9">0-521-86571-9</a>.</cite></span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text">Kraskov, <i>et al.</i> <i>ibid.</i></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Informazione_mutua&veaction=edit&section=13" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Informazione_mutua&action=edit&section=13" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation pubblicazione" style="font-style:normal"> R. Cilibrasi, Paul Vitányi, <a rel="nofollow" class="external text" href="https://www.cwi.nl/~paulv/papers/cluster.pdf"><span style="font-style:italic;">Clustering by compression</span></a> (<span style="font-weight: bolder; font-size:80%"><abbr title="documento in formato PDF">PDF</abbr></span>) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r140554517" /><span class="chiarimento" title="A volte può capitare che un link presente su Wikipedia non sia più raggiungibile. Se possibile ritrova il link e inserisci il collegamento corretto, comunque non rimuovere il collegamento e inserisci il template {{Collegamento interrotto}}"></span><sup class="noprint chiarimento-apice" title="A volte può capitare che un link presente su Wikipedia non sia più raggiungibile. Se possibile ritrova il link e inserisci il collegamento corretto, comunque non rimuovere il collegamento e inserisci il template {{Collegamento interrotto}}">[<i><a href="/wiki/Aiuto:Collegamenti_interrotti" title="Aiuto:Collegamenti interrotti">collegamento interrotto</a></i>]</sup>, in <span style="font-style:italic;">IEEE Transactions on Information Theory</span>, vol. 51, n. 4, 2005, pp. 1523–1545, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1109%2FTIT.2005.844059">10.1109/TIT.2005.844059</a>.</cite></li> <li>Coombs, C. H., Dawes, R. M. & Tversky, A. (1970), <i>Mathematical Psychology: An Elementary Introduction</i>, Prentice-Hall, Englewood Cliffs, NJ.</li> <li>Cronbach L. J. (1954). On the non-rational application of information measures in psychology, in H Quastler, ed., <i>Information Theory in Psychology: Problems and Methods</i>, Free Press, Glencoe, Illinois, pp. 14–30.</li> <li>Kenneth Ward Church and Patrick Hanks. Word association norms, mutual information, and lexicography, <i>Proceedings of the 27th Annual Meeting of the Association for Computational Linguistics</i>, 1989.</li> <li>Guiasu, Silviu (1977), <i>Information Theory with Applications</i>, McGraw-Hill, New York.</li> <li><cite class="citation libro" style="font-style:normal"> Ming Li, Paul Vitányi, <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoko00limi"><span style="font-style:italic;">An introduction to Kolmogorov complexity and its applications</span></a>, New York, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, 1997, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-387-94868-6" title="Speciale:RicercaISBN/0-387-94868-6">0-387-94868-6</a>.</cite></li> <li>Lockhead G. R. (1970). Identification and the form of multidimensional discrimination space, <i>Journal of Experimental Psychology</i> <b>85</b>(1), 1-10.</li> <li><a href="/w/index.php?title=Athanasios_Papoulis&action=edit&redlink=1" class="new" title="Athanasios Papoulis (la pagina non esiste)">Athanasios Papoulis</a>. <i>Probability, Random Variables, and Stochastic Processes</i>, second edition. New York: McGraw-Hill, 1984. <i>(See Chapter 15.)</i></li> <li>Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. (1988), <i>Numerical Recipes in C: The Art of Scientific Computing</i>, Cambridge University Press, Cambridge, p. 634</li> <li><cite class="citation pubblicazione" style="font-style:normal"> Alexander <a href="/w/index.php?title=Alexander_Strehl&action=edit&redlink=1" class="new" title="Alexander Strehl (la pagina non esiste)">Strehl</a>, Joydeep Ghosh, <a rel="nofollow" class="external text" href="http://strehl.com/download/strehl-jmlr02.pdf"><span style="font-style:italic;">Cluster ensembles -- a knowledge reuse framework for combining multiple partitions</span></a> (<span style="font-weight: bolder; font-size:80%"><abbr title="documento in formato PDF">PDF</abbr></span>), in <span style="font-style:italic;">Journal of Machine Learning Research</span>, vol. 3, 2002, pp. 583–617, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1162%2F153244303321897735">10.1162/153244303321897735</a>.</cite></li> <li>Witten, Ian H. & Frank, Eibe (2005), <i>Data Mining: Practical Machine Learning Tools and Techniques</i>, Morgan Kaufmann, Amsterdam.</li> <li>Yao, Y. Y. (2003) Information-theoretic measures for knowledge discovery and data mining, in <i>Entropy Measures, Maximum Entropy Principle and Emerging Applications</i> , Karmeshu (ed.), Springer, pp. 115–136.</li> <li>Peng, H.C., Long, F., and Ding, C., "Feature selection based on mutual information: criteria of max-dependency, max-relevance, and min-redundancy," IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 27, No. 8, pp. 1226–1238, 2005. <a rel="nofollow" class="external text" href="http://research.janelia.org/peng/proj/mRMR/index.htm">Program</a></li> <li>Andre S. Ribeiro, Stuart A. Kauffman, Jason Lloyd-Price, Bjorn Samuelsson, and Joshua Socolar, (2008) "Mutual Information in Random Boolean models of regulatory networks", Physical Review E, Vol.77, No.1. arXiv:0707.3642.</li> <li>N.X. Vinh, Epps, J. and Bailey, J., 'Information Theoretic Measures for Clusterings Comparison: Is a Correction for Chance Necessary?', in Proc. the 26th International Conference on Machine Learning (ICML'09), <a rel="nofollow" class="external text" href="http://www.cs.mcgill.ca/~icml2009/papers/10.pdf">PDF</a>.</li> <li><cite class="citation pubblicazione" style="font-style:normal"> W.M. 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