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href="#睇埋"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>睇埋</span> </div> </a> <ul id="toc-睇埋-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-文獻" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#文獻"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>文獻</span> </div> </a> <ul id="toc-文獻-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-引用" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#引用"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>引用</span> </div> </a> <ul id="toc-引用-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="目錄" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="內容一覽" > <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="開/收內容一覽" > <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">開/收內容一覽</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">資訊熵</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="去睇另一種語文嘅文章。有45種語言版本。" > <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-45" 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">45種語言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Entropie_(Inligtingsteorie)" title="Entropie (Inligtingsteorie) – 南非荷蘭文" lang="af" hreflang="af" data-title="Entropie (Inligtingsteorie)" data-language-autonym="Afrikaans" data-language-local-name="南非荷蘭文" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D8%B9%D8%AA%D9%84%D8%A7%D8%AC_(%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D9%85%D8%B9%D9%84%D9%88%D9%85%D8%A7%D8%AA)" title="اعتلاج (نظرية المعلومات) – 阿拉伯文" lang="ar" hreflang="ar" data-title="اعتلاج (نظرية المعلومات)" data-language-autonym="العربية" data-language-local-name="阿拉伯文" 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/Entropie_(Informationstheorie)" title="Entropie (Informationstheorie) – 巴伐利亞文" lang="bar" hreflang="bar" data-title="Entropie (Informationstheorie)" data-language-autonym="Boarisch" data-language-local-name="巴伐利亞文" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%BD%D1%82%D1%80%D0%BE%D0%BF%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%A8%D0%B0%D0%BD%D1%8A%D0%BD" title="Ентропия на Шанън – 保加利亞文" lang="bg" hreflang="bg" data-title="Ентропия на Шанън" data-language-autonym="Български" data-language-local-name="保加利亞文" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Entropija_(teorija_informacija)" title="Entropija (teorija informacija) – 波士尼亞文" lang="bs" hreflang="bs" data-title="Entropija (teorija informacija)" data-language-autonym="Bosanski" data-language-local-name="波士尼亞文" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Entropia_de_Shannon" title="Entropia de Shannon – 加泰羅尼亞文" lang="ca" hreflang="ca" data-title="Entropia de Shannon" data-language-autonym="Català" data-language-local-name="加泰羅尼亞文" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A6%D8%A7%D9%86%D8%AA%D8%B1%DB%86%D9%BE%DB%8C%DB%8C_%D8%B2%D8%A7%D9%86%DB%8C%D8%A7%D8%B1%DB%8C" title="ئانترۆپیی زانیاری – 索拉尼庫爾德文" lang="ckb" hreflang="ckb" data-title="ئانترۆپیی زانیاری" data-language-autonym="کوردی" data-language-local-name="索拉尼庫爾德文" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Informa%C4%8Dn%C3%AD_entropie" title="Informační entropie – 捷克文" lang="cs" hreflang="cs" data-title="Informační entropie" data-language-autonym="Čeština" data-language-local-name="捷克文" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Entropi_gwybodaeth" title="Entropi gwybodaeth – 威爾斯文" lang="cy" hreflang="cy" data-title="Entropi gwybodaeth" data-language-autonym="Cymraeg" data-language-local-name="威爾斯文" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Entropi_(informationsteori)" title="Entropi (informationsteori) – 丹麥文" lang="da" hreflang="da" data-title="Entropi (informationsteori)" data-language-autonym="Dansk" data-language-local-name="丹麥文" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Entropie_(Informationstheorie)" title="Entropie (Informationstheorie) – 德文" lang="de" hreflang="de" data-title="Entropie (Informationstheorie)" data-language-autonym="Deutsch" data-language-local-name="德文" 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%95%CE%BD%CF%84%CF%81%CE%BF%CF%80%CE%AF%CE%B1_%CF%80%CE%BB%CE%B7%CF%81%CE%BF%CF%86%CE%BF%CF%81%CE%B9%CF%8E%CE%BD" title="Εντροπία πληροφοριών – 希臘文" lang="el" hreflang="el" data-title="Εντροπία πληροφοριών" data-language-autonym="Ελληνικά" data-language-local-name="希臘文" 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/Entropy_(information_theory)" title="Entropy (information theory) – 英文" lang="en" hreflang="en" data-title="Entropy (information theory)" data-language-autonym="English" data-language-local-name="英文" 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/Entrop%C3%ADa_(informaci%C3%B3n)" title="Entropía (información) – 西班牙文" lang="es" hreflang="es" data-title="Entropía (información)" data-language-autonym="Español" data-language-local-name="西班牙文" 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/Entropia_(informazio-teoria)" title="Entropia (informazio-teoria) – 巴斯克文" lang="eu" hreflang="eu" data-title="Entropia (informazio-teoria)" data-language-autonym="Euskara" data-language-local-name="巴斯克文" 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%A2%D9%86%D8%AA%D8%B1%D9%88%D9%BE%DB%8C_%D8%A7%D8%B7%D9%84%D8%A7%D8%B9%D8%A7%D8%AA" title="آنتروپی اطلاعات – 波斯文" lang="fa" hreflang="fa" data-title="آنتروپی اطلاعات" data-language-autonym="فارسی" data-language-local-name="波斯文" 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/Entropie_de_Shannon" title="Entropie de Shannon – 法文" lang="fr" hreflang="fr" data-title="Entropie de Shannon" data-language-autonym="Français" data-language-local-name="法文" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Entrop%C3%ADa_da_informaci%C3%B3n" title="Entropía da información – 加里西亞文" lang="gl" hreflang="gl" data-title="Entropía da información" data-language-autonym="Galego" data-language-local-name="加里西亞文" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A0%D7%98%D7%A8%D7%95%D7%A4%D7%99%D7%94_(%D7%A1%D7%98%D7%98%D7%99%D7%A1%D7%98%D7%99%D7%A7%D7%94)" title="אנטרופיה (סטטיסטיקה) – 希伯來文" lang="he" hreflang="he" data-title="אנטרופיה (סטטיסטיקה)" data-language-autonym="עברית" data-language-local-name="希伯來文" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Shannon-entr%C3%B3piaf%C3%BCggv%C3%A9ny" title="Shannon-entrópiafüggvény – 匈牙利文" lang="hu" hreflang="hu" data-title="Shannon-entrópiafüggvény" data-language-autonym="Magyar" data-language-local-name="匈牙利文" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Entropi_(teori_informasi)" title="Entropi (teori informasi) – 印尼文" lang="id" hreflang="id" data-title="Entropi (teori informasi)" data-language-autonym="Bahasa Indonesia" data-language-local-name="印尼文" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Entropia_(teoria_dell%27informazione)" title="Entropia (teoria dell'informazione) – 義大利文" lang="it" hreflang="it" data-title="Entropia (teoria dell'informazione)" data-language-autonym="Italiano" data-language-local-name="義大利文" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%83%85%E5%A0%B1%E9%87%8F" title="情報量 – 日文" lang="ja" hreflang="ja" data-title="情報量" data-language-autonym="日本語" data-language-local-name="日文" 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%A0%95%EB%B3%B4_%EC%97%94%ED%8A%B8%EB%A1%9C%ED%94%BC" title="정보 엔트로피 – 韓文" lang="ko" hreflang="ko" data-title="정보 엔트로피" data-language-autonym="한국어" data-language-local-name="韓文" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Entropija_(informacijos_teorija)" title="Entropija (informacijos teorija) – 立陶宛文" lang="lt" hreflang="lt" data-title="Entropija (informacijos teorija)" data-language-autonym="Lietuvių" data-language-local-name="立陶宛文" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%A8%D0%B5%D0%BD%D0%BD%D0%BE%D0%BD%D1%8B%D0%BD_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B6%D0%BE" title="Шеннонын формулжо – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Шеннонын формулжо" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Entropie_(informatietheorie)" title="Entropie (informatietheorie) – 荷蘭文" lang="nl" hreflang="nl" data-title="Entropie (informatietheorie)" data-language-autonym="Nederlands" data-language-local-name="荷蘭文" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%90%E0%A8%A8%E0%A8%9F%E0%A9%8D%E0%A8%B0%E0%A9%8C%E0%A8%AA%E0%A9%80_(%E0%A8%87%E0%A8%A8%E0%A8%AB%E0%A9%8D%E0%A8%B0%E0%A8%AE%E0%A9%87%E0%A8%B8%E0%A8%BC%E0%A8%A8_%E0%A8%A5%E0%A8%BF%E0%A8%8A%E0%A8%B0%E0%A9%80)" title="ਐਨਟ੍ਰੌਪੀ (ਇਨਫ੍ਰਮੇਸ਼ਨ ਥਿਊਰੀ) – 旁遮普文" lang="pa" hreflang="pa" data-title="ਐਨਟ੍ਰੌਪੀ (ਇਨਫ੍ਰਮੇਸ਼ਨ ਥਿਊਰੀ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="旁遮普文" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Entropia_(teoria_informacji)" title="Entropia (teoria informacji) – 波蘭文" lang="pl" hreflang="pl" data-title="Entropia (teoria informacji)" data-language-autonym="Polski" data-language-local-name="波蘭文" 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/Entropia_da_informa%C3%A7%C3%A3o" title="Entropia da informação – 葡萄牙文" lang="pt" hreflang="pt" data-title="Entropia da informação" data-language-autonym="Português" data-language-local-name="葡萄牙文" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Entropie_informa%C8%9Bional%C4%83" title="Entropie informațională – 羅馬尼亞文" lang="ro" hreflang="ro" data-title="Entropie informațională" data-language-autonym="Română" data-language-local-name="羅馬尼亞文" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%B0%D1%8F_%D1%8D%D0%BD%D1%82%D1%80%D0%BE%D0%BF%D0%B8%D1%8F" title="Информационная энтропия – 俄文" lang="ru" hreflang="ru" data-title="Информационная энтропия" data-language-autonym="Русский" data-language-local-name="俄文" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Information_entropy" title="Information entropy – Simple English" lang="en-simple" hreflang="en-simple" data-title="Information entropy" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Entropia_(te%C3%B3ria_inform%C3%A1ci%C3%AD)" title="Entropia (teória informácií) – 斯洛伐克文" lang="sk" hreflang="sk" data-title="Entropia (teória informácií)" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克文" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Entropija_(informatika)" title="Entropija (informatika) – 斯洛維尼亞文" lang="sl" hreflang="sl" data-title="Entropija (informatika)" data-language-autonym="Slovenščina" data-language-local-name="斯洛維尼亞文" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Entropia_e_informacionit" title="Entropia e informacionit – 阿爾巴尼亞文" lang="sq" hreflang="sq" data-title="Entropia e informacionit" data-language-autonym="Shqip" data-language-local-name="阿爾巴尼亞文" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%BD%D1%82%D1%80%D0%BE%D0%BF%D0%B8%D1%98%D0%B0_(%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%B8%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%98%D0%B0)" title="Ентропија (теорија информација) – 塞爾維亞文" lang="sr" hreflang="sr" data-title="Ентропија (теорија информација)" data-language-autonym="Српски / srpski" data-language-local-name="塞爾維亞文" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/%C3%89ntropi_informasi" title="Éntropi informasi – 巽他文" lang="su" hreflang="su" data-title="Éntropi informasi" data-language-autonym="Sunda" data-language-local-name="巽他文" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Entropi_(informationsteori)" title="Entropi (informationsteori) – 瑞典文" lang="sv" hreflang="sv" data-title="Entropi (informationsteori)" data-language-autonym="Svenska" data-language-local-name="瑞典文" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%AD%E0%B8%99%E0%B9%82%E0%B8%97%E0%B8%A3%E0%B8%9B%E0%B8%B5%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%82%E0%B9%89%E0%B8%AD%E0%B8%A1%E0%B8%B9%E0%B8%A5" title="เอนโทรปีของข้อมูล – 泰文" lang="th" hreflang="th" data-title="เอนโทรปีของข้อมูล" data-language-autonym="ไทย" data-language-local-name="泰文" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Entropi_(bilgi_teorisi)" title="Entropi (bilgi teorisi) – 土耳其文" lang="tr" hreflang="tr" data-title="Entropi (bilgi teorisi)" data-language-autonym="Türkçe" data-language-local-name="土耳其文" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D1%96%D0%B9%D0%BD%D0%B0_%D0%B5%D0%BD%D1%82%D1%80%D0%BE%D0%BF%D1%96%D1%8F" title="Інформаційна ентропія – 烏克蘭文" lang="uk" hreflang="uk" data-title="Інформаційна ентропія" data-language-autonym="Українська" data-language-local-name="烏克蘭文" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AF%D8%B1%D9%85%D8%A7%D8%A6%D9%84%D8%AA_(%D8%A7%D8%B7%D9%84%D8%A7%D8%B9%D8%A7%D8%AA%DB%8C_%D9%86%D8%B8%D8%B1%DB%8C%DB%81)" title="درمائلت (اطلاعاتی نظریہ) – 烏都文" lang="ur" hreflang="ur" data-title="درمائلت (اطلاعاتی نظریہ)" data-language-autonym="اردو" data-language-local-name="烏都文" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Entropy_th%C3%B4ng_tin" title="Entropy 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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">架撐</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">移去側欄</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">收埋</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="多啲選項" > <div class="vector-menu-heading"> 動作 </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 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rel="nofollow" title="由呢版連出去嘅版嘅最近修改[k]" accesskey="k"><span>連結頁嘅更改</span></a></li><li id="t-upload" class="mw-list-item"><a href="//zh-yue.wikipedia.org/wiki/Project:上載" title="上載檔案[u]" accesskey="u"><span>上載檔案</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E8%B3%87%E8%A8%8A%E7%86%B5&oldid=2239228" title="呢版呢個修訂嘅永久網址"><span>固定連結</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E8%B3%87%E8%A8%8A%E7%86%B5&action=info" title="多啲同呢版有拏褦嘅資料"><span>此版明細</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:%E5%BC%95%E7%94%A8&page=%E8%B3%87%E8%A8%8A%E7%86%B5&id=2239228&wpFormIdentifier=titleform" title="點樣引用呢頁嘅資訊"><span>引用呢篇文</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fzh-yue.wikipedia.org%2Fwiki%2F%25E8%25B3%2587%25E8%25A8%258A%25E7%2586%25B5"><span>攞短網址</span></a></li><li id="t-urlshortener-qrcode" 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class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Entropy_and_information" hreflang="en"><span>維基同享</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q204570" title="連到接住資料儲存庫嘅項目[g]" accesskey="g"><span>維基數據項</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="頁面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="閱讀設定"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">閱讀設定</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移去側欄</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">收埋</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">出自維基百科，自由嘅百科全書</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(由<a href="/w/index.php?title=%E8%A8%8A%E6%81%AF%E7%86%B5&redirect=no" class="mw-redirect" title="訊息熵">訊息熵</a>跳轉過嚟)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="yue" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/File:Binary_entropy_plot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Binary_entropy_plot.svg/300px-Binary_entropy_plot.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Binary_entropy_plot.svg/450px-Binary_entropy_plot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Binary_entropy_plot.svg/600px-Binary_entropy_plot.svg.png 2x" data-file-width="300" data-file-height="300" /></a><figcaption>想像家吓<a href="/wiki/%E6%8E%9F%E9%8A%80%E4%BB%94" title="掟銀仔">掟一個銀仔</a>（睇埋<a href="/wiki/%E4%BC%AF%E5%8A%AA%E5%88%A9%E8%A9%A6%E9%A9%97" title="伯努利試驗">伯努利試驗</a>），幅圖 <a href="/wiki/Y_%E8%BB%B8" class="mw-redirect" title="Y 軸">Y 軸</a>係資訊熵，而 <a href="/wiki/X_%E8%BB%B8" class="mw-redirect" title="X 軸">X 軸</a>係出「公」嘅<a href="/wiki/%E6%A9%9F%E7%8E%87" class="mw-redirect" title="機率">機率</a>。由幅圖睇得出，資訊熵喺個銀仔冇出千（「公」同「字」機率都係 50%）嗰時會最大化。</figcaption></figure> <div role="note" class="hatnote"><span typeof="mw:File"><a href="/wiki/Wikipedia:%E6%90%9E%E6%B8%85%E6%A5%9A" title="Wikipedia:搞清楚"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Confusion_grey.svg/24px-Confusion_grey.svg.png" decoding="async" width="24" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Confusion_grey.svg/36px-Confusion_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Confusion_grey.svg/48px-Confusion_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></a></span>  <b>提示：</b>呢篇文講嘅唔係<b><a href="/wiki/%E7%86%B5" title="熵">熵</a></b>。</div> <p><b>資訊熵</b>（<a href="/wiki/Wikipedia:%E7%B2%B5%E6%96%87%E6%8B%BC%E9%9F%B3" title="Wikipedia:粵文拼音">粵拼</a>：<span class="plainlinks" style="display:inline-block"><a rel="nofollow" class="external text" href="http://yue.forvo.com/word/yue/資訊熵/#yue"><span>zi<span class="Jpingtone">1</span> seon<span class="Jpingtone">3</span> soeng<span class="Jpingtone">1</span></span></a></span> | <a href="/wiki/%E8%8B%B1%E6%96%87" title="英文">英文</a>：<span lang="en">information entropy</span>）係<a href="/wiki/%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="資訊理論">資訊理論</a>（研究<a href="/wiki/%E8%B3%87%E8%A8%8A" title="資訊">資訊</a>嘅數學理論）上嘅核心概念。同<a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%B8" title="物理學">物理學</a>上所講嘅<a href="/wiki/%E7%86%B5" title="熵">熵</a>唔同，資訊論當中嘅「熵」係一個指標，用嚟<a href="/wiki/%E9%87%8F%E5%BA%A6" title="量度">量度</a>一個有隨機性喺入面嘅<a href="/wiki/%E9%9A%A8%E6%A9%9F%E8%AE%8A%E6%95%B8" title="隨機變數">變數</a>或者<a href="/wiki/%E9%9A%A8%E6%A9%9F%E9%81%8E%E7%A8%8B" title="隨機過程">過程</a>當中帶有幾多<a href="/wiki/%E4%B8%8D%E7%A2%BA%E5%AE%9A%E6%80%A7" class="mw-redirect" title="不確定性">不確定性</a>喺入面<sup id="cite_ref-entropy_1-0" class="reference"><a href="#cite_note-entropy-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>。 </p><p>舉個例說明，想像家吓<a href="/wiki/%E6%8E%9F%E9%8A%80%E4%BB%94" title="掟銀仔">掟一個銀仔</a>同擲一粒<a href="/wiki/%E9%AA%B0%E4%BB%94" title="骰仔">骰仔</a>，假設個銀仔同粒骰仔係冇出千嘅，前者有 50% <a href="/wiki/%E6%A9%9F%E6%9C%83%E7%8E%87" class="mw-redirect" title="機會率">機會率</a>係公、50% 機會率係字，而後者有 1/6 機會係擲到「1」、1/6 機會係擲到「2」... 如此類推，即係話<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>註 1<span class="cite-bracket">]</span></a></sup>： </p> <ul><li>掟銀仔（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> 表示掟銀仔結果；0 代表公、1 代表字）： <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 p(i)={\begin{cases}{\frac {1}{2}}&{\text{if }}i=0,\\{\frac {1}{2}}&{\text{if }}i=1.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>1.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(i)={\begin{cases}{\frac {1}{2}}&{\text{if }}i=0,\\{\frac {1}{2}}&{\text{if }}i=1.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/637d58a3cfff16fde45b2b6be16346fce4ab457d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:21.224ex; height:7.509ex;" alt="{\displaystyle p(i)={\begin{cases}{\frac {1}{2}}&{\text{if }}i=0,\\{\frac {1}{2}}&{\text{if }}i=1.\end{cases}}}"></span></dd></dl></li> <li>擲骰仔（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> 表示擲到嘅數字）： <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 p(i)={\begin{cases}{\frac {1}{6}}&{\text{if }}i=1,\\{\frac {1}{6}}&{\text{if }}i=2,\\{\frac {1}{6}}&{\text{if }}i=3,\\...\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(i)={\begin{cases}{\frac {1}{6}}&{\text{if }}i=1,\\{\frac {1}{6}}&{\text{if }}i=2,\\{\frac {1}{6}}&{\text{if }}i=3,\\...\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5faeb1257b29a30ca18edd0001c8fb8e6531796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; margin-left: -0.089ex; width:22.473ex; height:13.509ex;" alt="{\displaystyle p(i)={\begin{cases}{\frac {1}{6}}&{\text{if }}i=1,\\{\frac {1}{6}}&{\text{if }}i=2,\\{\frac {1}{6}}&{\text{if }}i=3,\\...\end{cases}}}"></span></dd></dl></li></ul> <p>後者嘅情況有更加高嘅資訊熵－後者當中有更多嘅可能性喺度，所以不確定性亦都更加大。亦都因為噉，「話俾人知掟銀仔嘅結果」所俾嘅資訊少過「話俾人知擲骰仔嘅結果」所俾嘅，因為喺後者情況當中，「提供資訊」所消除嘅資訊熵更加多。就係噉，資訊論做到將「資訊」呢一個概念量化，令到資訊成為一個喺<a href="/wiki/%E7%A7%91%E5%AD%B8" title="科學">科學</a>上可以被研究嘅對象<sup id="cite_ref-entropy_1-1" class="reference"><a href="#cite_note-entropy-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>。 </p><p>以下嘅內容係建基於<a href="/wiki/%E6%A6%82%E7%8E%87%E8%AB%96" title="概率論">概率論</a>嘅基礎概念嘅。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="算式"><span id=".E7.AE.97.E5.BC.8F"></span>算式</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%B3%87%E8%A8%8A%E7%86%B5&action=edit&section=1" title="編輯小節: 算式"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote">睇埋：<a href="/wiki/%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="資訊理論">資訊理論</a></div> <p>俾是但一個變數，佢會有一啲可能數值，例如「某年某月某日某刻掟某一個銀仔」嘅可能數值大致上有兩個－「公」同「字」。每個數值都會有一定嘅機會率出現，而描述每一個可能數值出現嘅機會率嘅就係所謂嘅<a href="/wiki/%E6%A6%82%E7%8E%87%E8%B3%AA%E9%87%8F%E5%87%BD%E6%95%B8" title="概率質量函數">概率質量函數</a>（probability mass function）。知道咗某一件事件嘅概率質量函數之後，件事件所帶有嘅<a class="mw-selflink selflink">資訊熵</a>（數學符號係 <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>，單位係<a href="/wiki/%E5%B1%B1%E8%BE%B2_(%E5%96%AE%E4%BD%8D)" title="山農 (單位)">山農</a>）可以用以下呢條式計算<sup id="cite_ref-entropy_1-2" class="reference"><a href="#cite_note-entropy-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=-\sum _{i}p_{i}\log _{2}(p_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=-\sum _{i}p_{i}\log _{2}(p_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b70ea93a2697dc59081eaf13996fb016546785e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.696ex; height:5.509ex;" alt="{\displaystyle S=-\sum _{i}p_{i}\log _{2}(p_{i})}"></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 [1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83021ecdd7307a04dbb7873affcaac031e7e935a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle [1]}"></span></dd></dl> <p>喺呢條式當中，<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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span> 係指第 i 個可能性發生嘅機會率，而 <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 _{2}(p_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}(p_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2066b8377af39569b9373d3045a7c82cf45bd8fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.804ex; height:2.843ex;" alt="{\displaystyle \log _{2}(p_{i})}"></span> 係 <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span> 以 2 做基數嘅對數。成條式用日常用語講嘅話如下：「考慮一件事件嘅所有可能性，將每一個可能性嘅機會率<a href="/wiki/%E4%B9%98" title="乘">乘</a>嗰個可能性嘅機會率以 2 做基數嘅對數，再將呢啲得出嘅數加埋嗮一齊，再將個數<a href="/wiki/%E8%B2%A0%E6%95%B8" title="負數">負向</a>，就會得出件事件所包含嘅資訊熵」。用呢條式計嘅話，「掟一粒冇出千嘅銀仔」（即係話「公」同「字」嘅機會率都係 50%）呢件事件當中帶嘅資訊熵（<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 S(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9a8ce88a4734545697e3dcbc9ec1026a8ed35e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.289ex; height:2.843ex;" alt="{\displaystyle S(X)}"></span>）係<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(X)=-\left({\frac {1}{2}}\log _{2}\left({\frac {1}{2}}\right)+{\frac {1}{2}}\log _{2}\left({\frac {1}{2}}\right)\right)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(X)=-\left({\frac {1}{2}}\log _{2}\left({\frac {1}{2}}\right)+{\frac {1}{2}}\log _{2}\left({\frac {1}{2}}\right)\right)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738b2caaae4abffb48fb283cc1e5248c6c5eca2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.768ex; height:6.176ex;" alt="{\displaystyle S(X)=-\left({\frac {1}{2}}\log _{2}\left({\frac {1}{2}}\right)+{\frac {1}{2}}\log _{2}\left({\frac {1}{2}}\right)\right)=1}"></span>；</dd></dl> <p>喺直覺上，呢條式係喺度量度緊件事件含有嘅不確定性。科學家之所以會揀用呢條式嚟計資訊熵，係因為喺可能嘅算式當中，得呢條符合佢哋心目中「一條計不確定性嘅算式應有嘅特性」：例如一件肯定嘅事件係理應冇資訊熵嘅－而如果其中一個可能性機會率等如「1」（即係其他可能性機會率冚唪唥等如「0」），<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]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83021ecdd7307a04dbb7873affcaac031e7e935a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle [1]}"></span> 呢條式會俾出「0」；而另一方面，喺每個可能性機會率都一樣（不確定性最大化）嗰陣，<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]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83021ecdd7307a04dbb7873affcaac031e7e935a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle [1]}"></span> 俾嘅數值會最大化<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading2"><h2 id="概論"><span id=".E6.A6.82.E8.AB.96"></span>概論</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%B3%87%E8%A8%8A%E7%86%B5&action=edit&section=2" title="編輯小節: 概論"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>資訊熵嘅<a href="/wiki/%E8%A8%88%E9%87%8F%E5%96%AE%E4%BD%8D" title="計量單位">單位</a>係<a href="/wiki/%E4%BD%8D%E5%85%83" title="位元">位元</a>（bit）。有啲研究者會用 2 以外嘅數做上面條式個對數嘅基數嘅，不過道理都一樣，例如用 2<sup>8</sup> = 256 做基數嘅話，條式就會俾出以<a href="/wiki/%E4%BD%8D%E5%85%83%E7%B5%84" class="mw-redirect" title="位元組">位元組</a>（byte）作為單位嘅資訊熵<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>。</li> <li>資訊熵算式嘅一個特殊情況涉及一個得兩個可能數值嘅隨機變數，例如「掟銀仔嘅結果」就屬於呢種變數。喺呢個情況下，兩個可能性嘅機率加埋實會等如 1（<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}+p_{2}=1}"> <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>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}+p_{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab32d5b6eea69b0c54b2c9268167c36c0746ab89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.638ex; height:2.509ex;" alt="{\displaystyle p_{1}+p_{2}=1}"></span>，當中 <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b58f22283ca46dd5da309cc34303b06a797783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{1}}"></span> 同 <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43f1b08d7d69712872e051c2b33fdfa9f5d42319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{2}}"></span> 係兩個可能情況分別嘅機率），而呢件事件條資訊熵式就係所謂嘅<a href="/wiki/%E4%BA%8C%E5%85%83%E7%86%B5%E5%87%BD%E6%95%B8" class="mw-redirect" title="二元熵函數">二元熵函數</a>。條式係噉樣嘅：</li></ul> <dl><dd><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 S_{\mathrm {b} }(p)=-p\log _{2}p-(1-p)\log _{2}(1-p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>p</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>p</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\mathrm {b} }(p)=-p\log _{2}p-(1-p)\log _{2}(1-p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae5e5960db147f6d01898f3d6851cbf46e519e77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.812ex; height:2.843ex;" alt="{\displaystyle S_{\mathrm {b} }(p)=-p\log _{2}p-(1-p)\log _{2}(1-p)}"></span>－係將 <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]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83021ecdd7307a04dbb7873affcaac031e7e935a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle [1]}"></span> 應用落去「得兩個可能性」嘅情況嘅樣。</dd></dl></dd></dl> <ul><li>想像家吓有一個資訊源，佢傳送一連串有 N 個符號嘅資訊，而每個符號都係<a href="/wiki/%E7%8D%A8%E7%AB%8B%E5%90%8C%E5%88%86%E4%BD%88" title="獨立同分佈">獨立同分佈</a>（iid）－指一個符號係乜唔會影響嗰串嘢入面第啲符號係乜，但每個符號嘅<a href="/wiki/%E6%A6%82%E7%8E%87%E5%88%86%E4%BD%88" title="概率分佈">概率分佈</a>相同，例如一連串互不相干，每個都係隨機噉揀嘅<a href="/wiki/%E8%8B%B1%E6%96%87%E5%AD%97%E6%AF%8D" title="英文字母">英文字母</a>－呢串嘢嘅資訊熵係 <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 NS_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NS_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67493adc35ba43ae74b27d43f606885c12bd16d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.288ex; height:2.509ex;" alt="{\displaystyle NS_{i}}"></span>（當中 <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 S_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de6e810a93f67802ecb603ee0e3324005c6e583e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}"></span> 係每一個個別符號嘅資訊熵）咁多位元；而如果呢串嘢係同分佈（每個符號概率分佈相同）但係唔獨立（一個符號係乜可以影響第啲符號係乜嘅機率）－例如係一篇用英文寫嘅文章噉，如果佢某一橛係「information」呢個字嘅話，噉呢橛嘅下一格好大可能係一個空位－喺呢種情況下，串嘢嘅資訊熵會細過 <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 NS_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NS_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67493adc35ba43ae74b27d43f606885c12bd16d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.288ex; height:2.509ex;" alt="{\displaystyle NS_{i}}"></span>。</li> <li>如果有個人傳遞 1,000 位元（0 同 1）長嘅<a href="/wiki/%E6%95%B8%E7%A2%BC%E8%A8%8A%E8%99%9F" title="數碼訊號">數碼訊號</a>，而收訊號嗰個人喺訊號傳出去之前經已完全噉知道嗮每一個位元係「1」定「0」，噉傳訊號嗰個人冇傳達到任何新資訊－因為收訊號嗰個人喺收到訊號前後嘅資訊熵都係「0」，所以呢串訊號嘅傳遞並冇消除任何資訊熵。但如果喺收訊號嗰個人眼中嗰 1,000 個位元每一個都係 50% 機率係「1」、50% 機率係「0」嘅話，噉用 <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]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83021ecdd7307a04dbb7873affcaac031e7e935a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle [1]}"></span> 嚟計，佢透過接收呢串訊號會收到 1,000 位元嘅資訊。</li></ul> <div class="mw-heading mw-heading2"><h2 id="註釋"><span id=".E8.A8.BB.E9.87.8B"></span>註釋</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%B3%87%E8%A8%8A%E7%86%B5&action=edit&section=3" title="編輯小節: 註釋"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r2020813">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><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(i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(i)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c0b097fb5b359b7426bbce87b21175ae5fa710" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:3.871ex; height:2.843ex;" alt="{\displaystyle p(i)}"></span> 意思係「結果係 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> 嘅<a href="/wiki/%E6%A9%9F%E6%9C%83%E7%8E%87" class="mw-redirect" title="機會率">機會率</a>」。</span> </li> </ol></div></div> <div style="clear: both;"></div> <div class="mw-heading mw-heading2"><h2 id="睇埋"><span id=".E7.9D.87.E5.9F.8B"></span>睇埋</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%B3%87%E8%A8%8A%E7%86%B5&action=edit&section=4" title="編輯小節: 睇埋"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E9%9B%BB%E8%85%A6" title="電腦">電腦</a></li> <li><a href="/wiki/%E8%A8%8A%E8%99%9F" title="訊號">訊號</a></li> <li><a href="/wiki/%E5%9B%B0%E6%83%91%E5%BA%A6" title="困惑度">困惑度</a></li> <li><a href="/wiki/%E6%B1%BA%E7%AD%96%E6%A8%B9%E5%AD%B8%E7%BF%92" title="決策樹學習">決策樹學習</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="文獻"><span id=".E6.96.87.E7.8D.BB"></span>文獻</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%B3%87%E8%A8%8A%E7%86%B5&action=edit&section=5" title="編輯小節: 文獻"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Cover, T.M., Thomas, J.A. (2006), <i>Elements of Information Theory - 2nd Ed.</i>, Wiley-Interscience, <a href="/wiki/Special:%E6%9B%B8%E6%9C%AC%E4%BE%86%E6%BA%90/9780471241959" class="internal mw-magiclink-isbn">ISBN 978-0-471-24195-9</a></li> <li>MacKay, D.J.C. (2003), <i>Information Theory, Inference and Learning Algorithms</i>, Cambridge University Press, <a href="/wiki/Special:%E6%9B%B8%E6%9C%AC%E4%BE%86%E6%BA%90/9780521642989" class="internal mw-magiclink-isbn">ISBN 978-0-521-64298-9</a></li> <li>Arndt, C. (2004), <i>Information Measures: Information and its Description in Science and Engineering</i>, Springer, <a href="/wiki/Special:%E6%9B%B8%E6%9C%AC%E4%BE%86%E6%BA%90/9783540408550" class="internal mw-magiclink-isbn">ISBN 978-3-540-40855-0</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="引用"><span id=".E5.BC.95.E7.94.A8"></span>引用</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%B3%87%E8%A8%8A%E7%86%B5&action=edit&section=6" title="編輯小節: 引用"><span>編輯</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r2020813"><div class="reflist reflist-columns references-column-width"> <ol class="references"> <li id="cite_note-entropy-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-entropy_1-0">1.0</a></sup> <sup><a href="#cite_ref-entropy_1-1">1.1</a></sup> <sup><a href="#cite_ref-entropy_1-2">1.2</a></sup></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://towardsdatascience.com/demystifying-entropy-f2c3221e2550">Demystifying Entropy</a> <a href="/wiki/Wayback_Machine" title="Wayback Machine">互聯網檔案館</a>嘅<a rel="nofollow" class="external text" href="https://web.archive.org/web/20190710035002/https://towardsdatascience.com/demystifying-entropy-f2c3221e2550">歸檔</a>，歸檔日期2019年7月10號，.. <i>Towards Data Science</i>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Fazlollah M. Reza (1994) [1961]. <i>An Introduction to Information Theory</i>. Dover Publications, Inc., New York.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Fazlollah M. Reza (1994) [1961]. <i>An Introduction to Information Theory</i>. Dover Publications, Inc., New York.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Gray, R. M. (2011), <i>Entropy and Information Theory</i>, Springer.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">Norman Abramson (1963), <i>Information theory and coding</i>. McGraw-Hill.</span> </li> </ol></div> <div role="navigation" class="navbox" aria-labelledby="資訊理論" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div class="plainlinks hlist navbar mini"><ul><li class="nv-view"><a href="/wiki/Template:%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="Template:資訊理論"><abbr title="去睇呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">睇</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96&action=edit&redlink=1" class="new" title="Template talk:資訊理論 (無呢版)"><abbr title="傾呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">傾</abbr></a></li><li class="nv-edit"><a class="external text" href="https://zh-yue.wikipedia.org/w/index.php?title=Template:%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96&action=edit"><abbr title="編輯呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">改</abbr></a></li></ul></div><div id="資訊理論" style="font-size:114%;margin:0 4em"><a href="/wiki/%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="資訊理論">資訊理論</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"><b><a href="/wiki/%E8%B3%87%E8%A8%8A" title="資訊">資訊</a></b> <ul><li><a href="/wiki/%E9%80%9A%E8%A8%8A" title="通訊">通訊</a>同<a href="/wiki/%E8%B3%87%E8%A8%8A%E8%99%95%E7%90%86" title="資訊處理">資訊處理</a></li> <li><a href="/wiki/%E8%A8%8A%E8%99%9F" title="訊號">Signal</a>（<a href="/wiki/%E6%95%B8%E7%A2%BC%E8%A8%8A%E8%99%9F" title="數碼訊號">數碼</a>定<a href="/wiki/%E6%A8%A1%E6%93%AC%E8%A8%8A%E8%99%9F" title="模擬訊號">模擬</a>）</li> <li><a href="/wiki/%E9%9B%9C%E9%9F%B3" title="雜音">雜音</a></li> <li><a href="/wiki/%E8%A8%8A%E8%99%9F%E5%B0%8D%E9%9B%9C%E8%A8%8A%E6%AF%94%E4%BE%8B" title="訊號對雜訊比例">訊號對雜訊比例</a></li> <li><a href="/wiki/%E4%BD%8D%E5%85%83" title="位元">Bit</a></li> <li><a href="/wiki/%E7%9B%B8%E4%BC%BC%E5%BA%A6%E9%87%8F%E5%BA%A6" title="相似度量度">相似度量度</a></li></ul> <ul><li><a class="mw-selflink selflink">資訊熵</a>（<a href="/wiki/%E8%81%AF%E5%90%88%E7%86%B5" title="聯合熵">聯合熵</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A2%9D%E4%BB%B6%E7%86%B5" title="條件熵">條件熵</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%9B%B8%E4%BA%92%E8%B3%87%E8%A8%8A" title="相互資訊">相互資訊</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A2%9D%E4%BB%B6%E7%9B%B8%E4%BA%92%E8%B3%87%E8%A8%8A" title="條件相互資訊">條件相互資訊</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%9B%B8%E5%B0%8D%E7%86%B5" title="相對熵">相對熵</a>）</li> <li><a href="/wiki/%E7%86%B5%E7%8E%87" title="熵率">熵率</a></li> <li><a href="/wiki/%E8%B3%87%E8%A8%8A%E6%9C%80%E5%A4%A7%E5%8C%96%E5%8E%9F%E5%89%87" title="資訊最大化原則">資訊最大化原則</a></li></ul></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"><b>第啲概念</b><br /> <li><a href="/wiki/%E6%A6%82%E7%8E%87%E8%AB%96" title="概率論">概率論</a>（<a href="/wiki/%E4%B8%8D%E7%A2%BA%E5%AE%9A%E6%80%A7" class="mw-redirect" title="不確定性">不確定性</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A6%82%E7%8E%87" title="概率">概率</a>）</li> <li><a href="/wiki/%E7%B7%A8%E7%A2%BC%E7%90%86%E8%AB%96" title="編碼理論">編碼理論</a>（<a href="/wiki/%E9%9B%9C%E8%A8%8A%E9%80%9A%E9%81%93%E7%B7%A8%E7%A2%BC%E5%AE%9A%E7%90%86" title="雜訊通道編碼定理">雜訊通道編碼定理</a>）</li> <li><a href="/wiki/%E6%95%B8%E6%93%9A%E5%A3%93%E7%B8%AE" title="數據壓縮">數據壓縮</a>（<a href="/wiki/%E7%8E%87%E5%A4%B1%E7%9C%9F%E7%90%86%E8%AB%96" title="率失真理論">率失真理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%84%A1%E9%9B%9C%E8%A8%8A%E7%B7%A8%E7%A2%BC%E5%AE%9A%E7%90%86" title="無雜訊編碼定理">無雜訊編碼定理</a>）</li> <li><a href="/wiki/%E9%A0%BB%E9%81%93" title="頻道">頻道</a>（<a href="/wiki/%E4%BA%8C%E5%85%83%E5%B0%8D%E7%A8%B1%E9%A0%BB%E9%81%93" title="二元對稱頻道">二元對稱</a>同<a href="/wiki/%E4%BA%8C%E5%85%83%E6%B6%88%E9%99%A4%E9%A0%BB%E9%81%93" title="二元消除頻道">二元消除</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%A0%BB%E9%81%93%E5%AE%B9%E9%87%8F" title="頻道容量">容量</a>）</li></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"><b>相關領域</b><br /> <li><a href="/wiki/%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8" title="電腦科學">電腦科學</a>（<a href="/wiki/%E5%AF%86%E7%A2%BC%E5%AD%B8" title="密碼學">密碼學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%B3%87%E8%A8%8A%E7%B3%BB%E7%B5%B1" title="資訊系統">資訊系統</a>）</li> <li><a href="/wiki/%E9%9B%BB%E5%AD%90%E5%B7%A5%E7%A8%8B" title="電子工程">電子工程</a>同<a href="/wiki/%E9%9B%BB%E8%A8%8A" title="電訊">電訊</a></li> <li><a href="/wiki/%E8%AA%8D%E7%9F%A5%E7%A7%91%E5%AD%B8" title="認知科學">認知科學</a>（<a href="/wiki/%E7%A5%9E%E7%B6%93%E7%A7%91%E5%AD%B8" title="神經科學">神經科學</a>）</li> <li><a href="/wiki/%E8%AA%9E%E8%A8%80%E5%AD%B8" title="語言學">語言學</a></li> <li><a href="/wiki/%E7%B5%B1%E8%A8%88%E5%AD%B8" title="統計學">統計學</a></li> <li><a href="/wiki/%E6%A8%A1%E6%8E%A7%E5%AD%B8" title="模控學">模控學</a></li> <li><a href="/wiki/%E8%B3%87%E8%A8%8A%E7%B6%93%E6%BF%9F%E5%AD%B8" title="資訊經濟學">資訊經濟學</a></li></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b>拉雜相關</b><br /> <li><a href="/wiki/%E8%B2%9D%E8%91%89%E6%96%AF%E6%8E%A8%E8%AB%96" title="貝葉斯推論">貝葉斯推論</a></li> <li><a href="/wiki/%E6%BC%94%E7%AE%97%E6%B3%95%E7%86%B5" title="演算法熵">演算法熵</a></li> <li><a href="/wiki/%E8%B5%A4%E6%B1%A0%E8%B3%87%E8%A8%8A%E9%87%8F%E6%BA%96%E5%89%87" title="赤池資訊量準則">赤池</a>同<a href="/wiki/%E8%B2%9D%E8%91%89%E6%96%AF%E8%B3%87%E8%A8%8A%E9%87%8F%E6%BA%96%E5%89%87" title="貝葉斯資訊量準則">貝葉斯</a>資訊量準則</li> <li><a href="/wiki/%E6%93%AC%E4%BA%82%E6%95%B8%E7%94%A2%E7%94%9F" title="擬亂數產生">擬亂數產生</a>同<a href="/wiki/%E9%9A%A8%E6%A9%9F%E7%A8%AE%E5%AD%90" title="隨機種子">隨機種子</a></li> <li><a href="/wiki/%E6%8E%A7%E5%88%B6%E7%B3%BB%E7%B5%B1" title="控制系統">控制系統</a></li> <li><a href="/wiki/%E8%B3%87%E8%A8%8A%E9%81%8E%E5%A4%9A" class="mw-redirect" title="資訊過多">資訊過多</a></li> <li><a href="/wiki/%E8%AA%8D%E7%9F%A5" title="認知">認知</a>同<a href="/wiki/%E7%A5%9E%E7%B6%93%E7%B7%A8%E7%A2%BC" title="神經編碼">神經編碼</a></li></div></td></tr></tbody></table></div> <div role="navigation" class="navbox" aria-labelledby="電腦科學" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div class="plainlinks hlist navbar mini"><ul><li class="nv-view"><a href="/wiki/Template:%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8" title="Template:電腦科學"><abbr title="去睇呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">睇</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8&action=edit&redlink=1" class="new" title="Template talk:電腦科學 (無呢版)"><abbr title="傾呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">傾</abbr></a></li><li class="nv-edit"><a class="external text" href="https://zh-yue.wikipedia.org/w/index.php?title=Template:%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8&action=edit"><abbr title="編輯呢個模" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">改</abbr></a></li></ul></div><div id="電腦科學" style="font-size:114%;margin:0 4em"><a href="/wiki/%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8" title="電腦科學">電腦科學</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div id="研究運算嘅科學工程領域">研究<a href="/wiki/%E9%81%8B%E7%AE%97" title="運算">運算</a>嘅<a href="/wiki/%E7%A7%91%E5%AD%B8" title="科學">科學</a><a href="/wiki/%E5%B7%A5%E7%A8%8B" class="mw-redirect" title="工程">工程</a>領域</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">重要概念</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%81%8B%E7%AE%97" title="運算">運算</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/I/O" title="I/O">I/O</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%BB%E8%85%A6" title="電腦">電腦</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%81%8B%E7%AE%97%E9%82%8F%E8%BC%AF" title="運算邏輯">運算邏輯</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E8%AB%96" title="數論">數論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%BA%8C%E9%80%B2%E5%88%B6" title="二進制">二進制</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E5%AD%B8%E9%82%8F%E8%BC%AF" title="數學邏輯">數學邏輯</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%A2%E6%95%A3%E6%95%B8%E5%AD%B8" title="離散數學">離散數學</a>（<a href="/wiki/%E5%9C%96%E8%AB%96" title="圖論">圖論</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%8A%BD%E8%B1%A1%E5%8C%96_(%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8)" title="抽象化 (電腦科學)">抽象化</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8%E8%A9%9E%E5%BD%99%E8%A1%A8" title="電腦科學詞彙表">電腦科學詞彙表</a></div></td><td class="navbox-image" rowspan="8" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/File:Binario_cropped.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Binario_cropped.png/150px-Binario_cropped.png" decoding="async" width="150" height="129" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Binario_cropped.png/225px-Binario_cropped.png 1.5x, //upload.wikimedia.org/wikipedia/commons/1/17/Binario_cropped.png 2x" data-file-width="254" data-file-height="218" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E7%90%86%E8%AB%96%E9%9B%BB%E8%85%A6%E7%A7%91%E5%AD%B8" title="理論電腦科學">電腦理論</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="運算理論" scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E9%81%8B%E7%AE%97%E7%90%86%E8%AB%96" title="運算理論">運算理論</a></div></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E8%87%AA%E5%8B%95%E6%A9%9F%E7%90%86%E8%AB%96" title="自動機理論">自動機理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%8F%AF%E9%81%8B%E7%AE%97%E5%BA%A6%E7%90%86%E8%AB%96" title="可運算度理論">可運算度理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%81%8B%E7%AE%97%E8%A4%87%E9%9B%9C%E5%BA%A6%E7%90%86%E8%AB%96" title="運算複雜度理論">運算複雜度理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%9C%96%E9%9D%88%E6%A9%9F" title="圖靈機">圖靈機</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%87%8F%E5%AD%90%E9%81%8B%E7%AE%97" title="量子運算">量子運算</a></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E8%B3%87%E8%A8%8A%E7%90%86%E8%AB%96" title="資訊理論">資訊理論</a>（<a href="/wiki/%E8%B3%87%E8%A8%8A" title="資訊">資訊</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%A8%8A%E8%99%9F" title="訊號">訊號</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a class="mw-selflink selflink">資訊熵</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B7%A8%E7%A2%BC%E7%90%86%E8%AB%96" title="編碼理論">編碼理論</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%B3%87%E6%96%99%E9%A1%9E%E5%9E%8B" title="資料類型">資料類型‎</a>同<a href="/wiki/%E6%95%B8%E6%93%9A%E7%B5%90%E6%A7%8B" title="數據結構">數據結構</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B3%BB%E7%B5%B1%E8%B3%87%E6%BA%90" title="系統資源">系統資源</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%BC%94%E7%AE%97%E6%B3%95" title="演算法">演算法</a>（<a href="/wiki/%E6%BC%94%E7%AE%97%E6%B3%95%E5%88%86%E6%9E%90" title="演算法分析">分析</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%BD%A2%E5%BC%8F%E5%8C%96%E6%96%B9%E6%B3%95" title="形式化方法">形式化方法</a></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">電腦系統</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%9B%BB%E8%85%A6%E6%9E%B6%E6%A7%8B" title="電腦架構">電腦架構</a>（<a href="/wiki/%E9%82%8F%E8%BC%AF%E9%96%80" title="邏輯門">邏輯門</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%8C%87%E4%BB%A4%E9%9B%86%E6%9E%B6%E6%A7%8B" title="指令集架構">指令集架構</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%BE%AE%E6%9E%B6%E6%A7%8B" title="微架構">微架構</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%A4%9A%E5%85%83%E8%99%95%E7%90%86" title="多元處理">多元處理</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%BD%9C%E6%A5%AD%E7%B3%BB%E7%B5%B1" title="作業系統">作業系統</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%B8%A6%E8%A1%8C%E9%81%8B%E7%AE%97" title="並行運算">並行</a>、<a href="/wiki/%E5%B9%B3%E8%A1%8C%E9%81%8B%E7%AE%97" title="平行運算">平行</a>同<a href="/wiki/%E5%88%86%E6%95%A3%E9%81%8B%E7%AE%97" title="分散運算">分散</a>運算<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%BB%E8%85%A6%E6%80%A7%E8%83%BD" title="電腦性能">性能</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%B6%85%E7%B4%9A%E9%9B%BB%E8%85%A6" title="超級電腦">超級電腦</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">電腦通訊</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E9%9B%BB%E8%85%A6%E7%B6%B2%E7%B5%A1" title="電腦網絡">電腦網絡</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E7%B6%B2%E7%B5%A1%E5%82%B3%E8%BC%B8%E5%8D%94%E8%AD%B0" title="網絡傳輸協議">網絡傳輸協議</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B6%B2%E7%B5%A1%E6%8B%93%E6%A8%B8" title="網絡拓樸">網絡拓樸</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%95%B8%E6%93%9A%E5%82%B3%E8%BC%B8" title="數據傳輸">數據傳輸</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%BA%92%E8%81%AF%E7%B6%B2" title="互聯網">互聯網</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B6%B2%E9%A0%81%E8%A8%AD%E8%A8%88" title="網頁設計">網頁設計</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E9%9B%BB%E8%85%A6%E4%BF%9D%E5%AE%89" title="電腦保安">電腦保安</a></div></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E7%B6%B2%E7%B5%A1%E4%BF%9D%E5%AE%89" title="網絡保安">網絡保安</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%BC%8F%E6%B4%9E" title="漏洞">漏洞</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%85%AC%E5%85%B1%E6%BC%8F%E6%B4%9E%E5%90%8C%E6%9A%B4%E9%9C%B2" title="公共漏洞同暴露">公共漏洞同暴露</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B6%B2%E7%B5%A1%E6%94%BB%E6%93%8A" title="網絡攻擊">網絡攻擊</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%BB%E8%85%A6%E7%97%85%E6%AF%92" title="電腦病毒">電腦病毒</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%AF%86%E7%A2%BC%E5%AD%B8" title="密碼學">密碼學</a></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9B%BB%E8%85%A6%E7%A8%8B%E5%BC%8F%E7%B7%A8%E5%AF%AB" title="電腦程式編寫">程式編寫</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E7%A8%8B%E5%BC%8F%E8%AA%9E%E8%A8%80" title="程式語言">程式語言</a>（<a href="/wiki/%E7%A8%8B%E5%BC%8F%E8%AA%9E%E8%A8%80%E7%90%86%E8%AB%96" title="程式語言理論">程式語言理論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%BD%8E%E7%B4%9A%E7%A8%8B%E5%BC%8F%E8%AA%9E%E8%A8%80" title="低級程式語言">低級</a>同<a href="/wiki/%E9%AB%98%E7%B4%9A%E7%A8%8B%E5%BC%8F%E8%AA%9E%E8%A8%80" title="高級程式語言">高級</a>程式語言<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%8E%A7%E5%88%B6%E6%B5%81%E7%A8%8B" title="控制流程">控制流程</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%A8%8B%E5%BC%8F%E5%9F%B7%E8%A1%8C" title="程式執行">程式執行</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%B7%A8%E8%AD%AF%E5%99%A8" title="編譯器">編譯器</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%A8%8B%E5%BC%8F%E7%B7%A8%E5%AF%AB%E7%AF%84%E5%BC%8F" title="程式編寫範式">程式編寫範式</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">電腦應用</th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E6%95%B8%E7%A2%BC%E8%97%9D%E8%A1%93" title="數碼藝術">數碼藝術</a></div></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%9B%BB%E8%85%A6%E5%9C%96%E5%83%8F" title="電腦圖像">電腦圖像</a>（<a href="/wiki/%E4%BA%8C%E7%B6%AD%E9%9B%BB%E8%85%A6%E5%9C%96%E5%83%8F" class="mw-redirect" title="二維電腦圖像">2D</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E4%B8%89%E7%B6%AD%E9%9B%BB%E8%85%A6%E5%9C%96%E5%83%8F" class="mw-redirect" title="三維電腦圖像">3D</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%BB%E8%85%A6%E5%8B%95%E7%95%AB" title="電腦動畫">電腦動畫</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%B3%87%E8%A8%8A%E5%8F%AF%E8%A6%96%E5%8C%96" title="資訊可視化">資訊可視化</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%99%9B%E6%93%AC%E5%AF%A6%E5%A2%83" title="虛擬實境">VR</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%93%B4%E5%BC%B5%E5%AF%A6%E5%A2%83" title="擴張實境">AR</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%BB%E8%85%A6%E5%9C%96%E5%83%8F%E5%AD%B8" title="電腦圖像學">電腦圖像學</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%BB%E8%85%A6%E9%9F%B3%E6%A8%82" title="電腦音樂">電腦音樂</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E8%BB%9F%E4%BB%B6%E5%B7%A5%E7%A8%8B" title="軟件工程">軟件工程</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E9%9B%BB%E8%85%A6%E8%BB%9F%E4%BB%B6" title="電腦軟件">電腦軟件</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%BB%9F%E4%BB%B6%E9%96%8B%E7%99%BC" title="軟件開發">軟件開發</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%BB%9F%E4%BB%B6%E9%9C%80%E6%B1%82" title="軟件需求">軟件需求</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%BB%9F%E4%BB%B6%E9%A0%85%E7%9B%AE%E7%AE%A1%E7%90%86" title="軟件項目管理">軟件項目管理</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%BB%E5%AD%90%E9%81%8A%E6%88%B2%E8%A3%BD%E4%BD%9C" title="電子遊戲製作">電子遊戲製作</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;"><a href="/wiki/%E4%BA%BA%E6%A9%9F%E4%BA%92%E5%8B%95" title="人機互動">HCI</a></div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E7%94%A8%E5%AE%B6" title="用家">用家</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%98%93%E7%94%A8%E5%BA%A6" title="易用度">易用度</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%94%A8%E5%AE%B6%E4%BB%8B%E9%9D%A2" title="用家介面">用家介面</a>（<a href="/wiki/%E7%94%A8%E5%AE%B6%E4%BB%8B%E9%9D%A2%E8%A8%AD%E8%A8%88" title="用家介面設計">設計</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%9C%96%E5%83%8F%E7%94%A8%E5%AE%B6%E4%BB%8B%E9%9D%A2" title="圖像用家介面">圖像</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%91%BD%E4%BB%A4%E8%A1%8C%E4%BB%8B%E9%9D%A2" title="命令行介面">命令行</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%85%A6%E6%A9%9F%E4%BB%8B%E9%9D%A2" title="腦機介面">腦機</a>）</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;padding-left:0em;padding-right:0em;"><div style="padding:0em 0.75em;">其他應用</div></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E8%B3%87%E8%A8%8A%E7%B3%BB%E7%B5%B1" title="資訊系統">資訊系統</a>（<a href="/wiki/%E6%95%B8%E6%93%9A%E5%BA%AB" title="數據庫">數據庫</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%A7%91%E5%AD%B8%E9%81%8B%E7%AE%97" title="科學運算">科學運算</a>（<a href="/wiki/%E9%9B%BB%E8%85%A6%E6%A8%A1%E6%93%AC" title="電腦模擬">電腦模擬</a>）<span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%81%8B%E7%AE%97%E6%95%B8%E8%AB%96" title="運算數論">運算數論</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%BB%E8%85%A6%E4%BB%A3%E6%95%B8" title="電腦代數">電腦代數</a></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD" title="人工智能">AI</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD%E5%93%B2%E5%AD%B8" title="人工智能哲學">AI 哲學</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%9F%A5%E8%AD%98%E8%A1%A8%E7%A4%BA" title="知識表示">知識表示</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%87%AA%E5%8B%95%E8%A8%88%E5%8A%83" title="自動計劃">自動計劃</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9B%BB%E8%85%A6%E8%A6%96%E8%A6%BA" title="電腦視覺">電腦視覺</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%87%AA%E7%84%B6%E8%AA%9E%E8%A8%80%E8%99%95%E7%90%86" title="自然語言處理">NLP</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E6%A9%9F%E6%A2%B0%E5%AD%B8%E7%BF%92" title="機械學習">機械學習</a>（<a href="/wiki/%E4%BA%BA%E5%B7%A5%E7%A5%9E%E7%B6%93%E7%B6%B2%E7%B5%A1" title="人工神經網絡">ANN</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%9B%A3%E7%9D%A3%E5%BC%8F%E5%AD%B8%E7%BF%92" title="監督式學習">監督式學習</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E9%9D%9E%E7%9B%A3%E7%9D%A3%E5%BC%8F%E5%AD%B8%E7%BF%92" title="非監督式學習">非監督式學習</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%BC%B7%E5%8C%96%E5%AD%B8%E7%BF%92" title="強化學習">強化學習</a>）</div></td></tr><tr><th scope="row" class="navbox-group" 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