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href="#Decision_theory_approach"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Decision theory approach</span> </div> </a> <ul id="toc-Decision_theory_approach-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Personal_probabilities_and_objective_methods_for_constructing_priors" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Personal_probabilities_and_objective_methods_for_constructing_priors"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Personal probabilities and objective methods for constructing priors</span> </div> </a> <ul id="toc-Personal_probabilities_and_objective_methods_for_constructing_priors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" 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mw-first-heading"><span class="mw-page-title-main">Bayesian probability</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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mw-list-item"><a href="https://ca.wikipedia.org/wiki/Probabilitat_bayesiana" title="Probabilitat bayesiana – Catalan" lang="ca" hreflang="ca" data-title="Probabilitat bayesiana" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%91%D0%B0%D0%B9%D0%B5%D1%81_%D0%BF%D1%83%D0%BB%D0%B0%D1%8F%D1%81%D0%BB%C4%83%D1%85%C4%95" title="Байес пулаяслăхĕ – Chuvash" lang="cv" hreflang="cv" data-title="Байес пулаяслăхĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bayesscher_Wahrscheinlichkeitsbegriff" title="Bayesscher Wahrscheinlichkeitsbegriff – German" lang="de" hreflang="de" data-title="Bayesscher Wahrscheinlichkeitsbegriff" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Bayesiaanlus" title="Bayesiaanlus – Estonian" lang="et" hreflang="et" data-title="Bayesiaanlus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Probabilidad_bayesiana" title="Probabilidad bayesiana – Spanish" lang="es" hreflang="es" data-title="Probabilidad bayesiana" data-language-autonym="Español" data-language-local-name="Spanish" 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/Bayestar_probabilitate" title="Bayestar probabilitate – Basque" lang="eu" hreflang="eu" data-title="Bayestar probabilitate" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84%D8%A7%D8%AA_%D8%A8%DB%8C%D8%B2%DB%8C" title="احتمالات بیزی – Persian" lang="fa" hreflang="fa" data-title="احتمالات بیزی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Probabilit%C3%A9_bay%C3%A9sienne" title="Probabilité bayésienne – French" lang="fr" hreflang="fr" data-title="Probabilité bayésienne" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A0%EC%9D%B4%EC%A6%88_%ED%99%95%EB%A5%A0%EB%A1%A0" title="베이즈 확률론 – Korean" lang="ko" hreflang="ko" data-title="베이즈 확률론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Bayesanizam" title="Bayesanizam – Croatian" lang="hr" hreflang="hr" data-title="Bayesanizam" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Probabilit%C3%A0_bayesiana" title="Probabilità bayesiana – Italian" lang="it" hreflang="it" data-title="Probabilità bayesiana" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%A1%D7%AA%D7%91%D7%A8%D7%95%D7%AA_%D7%91%D7%99%D7%99%D7%A1%D7%99%D7%90%D7%A0%D7%99%D7%AA" title="הסתברות בייסיאנית – Hebrew" lang="he" hreflang="he" data-title="הסתברות בייסיאנית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Bayesiaanse_kans" title="Bayesiaanse kans – Dutch" lang="nl" hreflang="nl" data-title="Bayesiaanse kans" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%99%E3%82%A4%E3%82%BA%E7%A2%BA%E7%8E%87" title="ベイズ確率 – Japanese" lang="ja" hreflang="ja" data-title="ベイズ確率" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Bayes-analyse" title="Bayes-analyse – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Bayes-analyse" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Prawdopodobie%C5%84stwo_subiektywne" title="Prawdopodobieństwo subiektywne – Polish" lang="pl" hreflang="pl" data-title="Prawdopodobieństwo subiektywne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Probabilidade_epistemol%C3%B3gica" title="Probabilidade epistemológica – Portuguese" lang="pt" hreflang="pt" data-title="Probabilidade epistemológica" data-language-autonym="Português" data-language-local-name="Portuguese" 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%91%D0%B0%D0%B9%D0%B5%D1%81%D0%BE%D0%B2%D1%81%D0%BA%D0%B0%D1%8F_%D0%B2%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Байесовская вероятность – Russian" lang="ru" hreflang="ru" data-title="Байесовская вероятность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Bayesian_probability" title="Bayesian probability – Simple English" lang="en-simple" hreflang="en-simple" data-title="Bayesian probability" 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-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%B0%D1%98%D0%B5%D1%81%D0%BE%D0%B2%D0%B0_%D0%B2%D0%B5%D1%80%D0%BE%D0%B2%D0%B0%D1%82%D0%BD%D0%BE%D1%9B%D0%B0" title="Бајесова вероватноћа – Serbian" lang="sr" hreflang="sr" data-title="Бајесова вероватноћа" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Bayesova_vjerovatno%C4%87a" title="Bayesova vjerovatnoća – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Bayesova vjerovatnoća" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%99%E0%B9%88%E0%B8%B2%E0%B8%88%E0%B8%B0%E0%B9%80%E0%B8%9B%E0%B9%87%E0%B8%99%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B9%80%E0%B8%9A%E0%B8%AA%E0%B9%8C" title="ความน่าจะเป็นแบบเบส์ – Thai" lang="th" hreflang="th" data-title="ความน่าจะเป็นแบบเบส์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Bayesci_olas%C4%B1l%C4%B1k" title="Bayesci olasılık – Turkish" lang="tr" hreflang="tr" data-title="Bayesci olasılık" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D0%B0%D1%94%D1%81%D0%BE%D0%B2%D0%B0_%D0%B9%D0%BC%D0%BE%D0%B2%D1%96%D1%80%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Баєсова ймовірність – Ukrainian" lang="uk" hreflang="uk" data-title="Баєсова ймовірність" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%B2%9D%E8%91%89%E6%96%AF%E6%A6%82%E7%8E%87" title="貝葉斯概率 – Cantonese" lang="yue" hreflang="yue" data-title="貝葉斯概率" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%B2%9D%E6%B0%8F%E6%A9%9F%E7%8E%87" title="貝氏機率 – Chinese" lang="zh" hreflang="zh" data-title="貝氏機率" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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.sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Bayes_icon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Bayes_icon.svg/80px-Bayes_icon.svg.png" decoding="async" width="80" height="48" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Bayes_icon.svg/120px-Bayes_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Bayes_icon.svg/160px-Bayes_icon.svg.png 2x" data-file-width="500" data-file-height="300" /></a></span></td></tr><tr><td class="sidebar-content"> <a href="/wiki/Posterior_probability" title="Posterior probability">Posterior</a> = <a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> × <a href="/wiki/Prior_probability" title="Prior probability">Prior</a> ÷ <a href="/wiki/Marginal_likelihood" title="Marginal likelihood">Evidence</a></td> </tr><tr><th class="sidebar-heading"> Background</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></li> <li><a class="mw-selflink selflink">Bayesian probability</a></li> <li><a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a></li> <li><a href="/wiki/Bernstein%E2%80%93von_Mises_theorem" title="Bernstein–von Mises theorem">Bernstein–von Mises theorem</a></li> <li><a href="/wiki/Coherence_(philosophical_gambling_strategy)" class="mw-redirect" title="Coherence (philosophical gambling strategy)">Coherence</a></li> <li><a href="/wiki/Cox%27s_theorem" title="Cox's theorem">Cox's theorem</a></li> <li><a href="/wiki/Cromwell%27s_rule" title="Cromwell's rule">Cromwell's rule</a></li> <li><a href="/wiki/Likelihood_principle" title="Likelihood principle">Likelihood principle</a></li> <li><a href="/wiki/Principle_of_indifference" title="Principle of indifference">Principle of indifference</a></li> <li><a href="/wiki/Principle_of_maximum_entropy" title="Principle of maximum entropy">Principle of maximum entropy</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Model building</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Conjugate_prior" title="Conjugate prior">Conjugate prior</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Linear regression</a></li> <li><a href="/wiki/Empirical_Bayes_method" title="Empirical Bayes method">Empirical Bayes</a></li> <li><a href="/wiki/Bayesian_hierarchical_modeling" title="Bayesian hierarchical modeling">Hierarchical model</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Posterior approximation</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a></li> <li><a href="/wiki/Laplace%27s_approximation" title="Laplace's approximation">Laplace's approximation</a></li> <li><a href="/wiki/Integrated_nested_Laplace_approximations" title="Integrated nested Laplace approximations">Integrated nested Laplace approximations</a></li> <li><a href="/wiki/Variational_Bayesian_methods" title="Variational Bayesian methods">Variational inference</a></li> <li><a href="/wiki/Approximate_Bayesian_computation" title="Approximate Bayesian computation">Approximate Bayesian computation</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Estimators</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Bayesian_estimator" class="mw-redirect" title="Bayesian estimator">Bayesian estimator</a></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum a posteriori estimation</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Evidence approximation</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Evidence_lower_bound" title="Evidence lower bound">Evidence lower bound</a></li> <li><a href="/wiki/Nested_sampling_algorithm" title="Nested sampling algorithm">Nested sampling</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Model evaluation</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a> (<a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">Schwarz criterion</a>)</li> <li><a href="/wiki/Bayesian_model_averaging" class="mw-redirect" title="Bayesian model averaging">Model averaging</a></li> <li><a href="/wiki/Posterior_predictive_distribution" title="Posterior predictive distribution">Posterior predictive</a></li></ul></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Bayesian_statistics" title="Template:Bayesian statistics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Bayesian_statistics" title="Template talk:Bayesian statistics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Bayesian_statistics" title="Special:EditPage/Template:Bayesian statistics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Interpretation of probability</div><p><b>Bayesian probability</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'b' in 'buy'">b</span><span title="/eɪ/: 'a' in 'face'">eɪ</span><span title="'z' in 'zoom'">z</span><span title="/i/: 'y' in 'happy'">i</span><span title="/ə/: 'a' in 'about'">ə</span><span title="'n' in 'nigh'">n</span></span>/</a></span></span> <a href="/wiki/Help:Pronunciation_respelling_key" title="Help:Pronunciation respelling key"><i title="English pronunciation respelling"><span style="font-size:90%">BAY</span>-zee-ən</i></a> or <span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'b' in 'buy'">b</span><span title="/eɪ/: 'a' in 'face'">eɪ</span><span title="/ʒ/: 's' in 'pleasure'">ʒ</span><span title="/ən/: 'on' in 'button'">ən</span></span>/</a></span></span> <a href="/wiki/Help:Pronunciation_respelling_key" title="Help:Pronunciation respelling key"><i title="English pronunciation respelling"><span style="font-size:90%">BAY</span>-zhən</i></a>)<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> is an <a href="/wiki/Probability_interpretations" title="Probability interpretations">interpretation of the concept of probability</a>, in which, instead of <a href="/wiki/Frequentist_probability" title="Frequentist probability">frequency</a> or <a href="/wiki/Propensity_probability" title="Propensity probability">propensity</a> of some phenomenon, probability is interpreted as reasonable expectation<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> representing a state of knowledge<sup id="cite_ref-ghxaib_3-0" class="reference"><a href="#cite_note-ghxaib-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> or as quantification of a personal belief.<sup id="cite_ref-Finetti,_B._1974_4-0" class="reference"><a href="#cite_note-Finetti,_B._1974-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The Bayesian interpretation of probability can be seen as an extension of <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a> that enables reasoning with <a href="/wiki/Hypothesis" title="Hypothesis">hypotheses</a>;<sup id="cite_ref-Hailperin,_T._1996_5-0" class="reference"><a href="#cite_note-Hailperin,_T._1996-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> that is, with propositions whose <a href="/wiki/Truth_value" title="Truth value">truth or falsity</a> is unknown. In the Bayesian view, a probability is assigned to a hypothesis, whereas under <a href="/wiki/Frequentist_inference" title="Frequentist inference">frequentist inference</a>, a hypothesis is typically tested without being assigned a probability. </p><p>Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies a <a href="/wiki/Prior_probability" title="Prior probability">prior probability</a>. This, in turn, is then updated to a <a href="/wiki/Posterior_probability" title="Posterior probability">posterior probability</a> in the light of new, relevant <a href="/wiki/Data" title="Data">data</a> (evidence).<sup id="cite_ref-paulos_7-0" class="reference"><a href="#cite_note-paulos-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The Bayesian interpretation provides a standard set of procedures and formulae to perform this calculation. </p><p>The term <i>Bayesian</i> derives from the 18th-century mathematician and theologian <a href="/wiki/Thomas_Bayes" title="Thomas Bayes">Thomas Bayes</a>, who provided the first mathematical treatment of a non-trivial problem of statistical <a href="/wiki/Data_analysis" title="Data analysis">data analysis</a> using what is now known as <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a>.<sup id="cite_ref-HOS_8-0" class="reference"><a href="#cite_note-HOS-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 131">: 131 </span></sup> Mathematician <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> pioneered and popularized what is now called Bayesian probability.<sup id="cite_ref-HOS_8-1" class="reference"><a href="#cite_note-HOS-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 97–98">: 97–98 </span></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Bayesian_methodology">Bayesian methodology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_probability&action=edit&section=1" title="Edit section: Bayesian methodology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bayesian methods are characterized by concepts and procedures as follows: </p> <ul><li>The use of <a href="/wiki/Random_variable" title="Random variable">random variables</a>, or more generally unknown quantities,<sup id="cite_ref-rbp_9-0" class="reference"><a href="#cite_note-rbp-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> to model all sources of <a href="/wiki/Uncertainty" title="Uncertainty">uncertainty</a> in statistical models including uncertainty resulting from lack of information (see also <a href="/wiki/Uncertainty_quantification#Aleatoric_and_epistemic_uncertainty" title="Uncertainty quantification">aleatoric and epistemic uncertainty</a>).</li> <li>The need to determine the prior <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> taking into account the available (prior) information.</li> <li>The sequential use of <a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a>: as more data become available, calculate the posterior distribution using Bayes' theorem; subsequently, the posterior distribution becomes the next prior.</li> <li>While for the frequentist, a <a href="/wiki/Null_hypothesis" title="Null hypothesis">hypothesis</a> is a <a href="/wiki/Proposition#Treatment_in_logic" title="Proposition">proposition</a> (which must be <a href="/wiki/Principle_of_bivalence" title="Principle of bivalence">either true or false</a>) so that the frequentist probability of a hypothesis is either 0 or 1, in Bayesian statistics, the probability that can be assigned to a hypothesis can also be in a range from 0 to 1 if the truth value is uncertain.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Objective_and_subjective_Bayesian_probabilities">Objective and subjective Bayesian probabilities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_probability&action=edit&section=2" title="Edit section: Objective and subjective Bayesian probabilities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Broadly speaking, there are two interpretations of Bayesian probability. For objectivists, who interpret probability as an extension of <a href="/wiki/Logic" title="Logic">logic</a>, <i>probability</i> quantifies the reasonable expectation that everyone (even a "robot") who shares the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by <a href="/wiki/Cox%27s_theorem" title="Cox's theorem">Cox's theorem</a>.<sup id="cite_ref-ghxaib_3-1" class="reference"><a href="#cite_note-ghxaib-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-vkdmsn_10-0" class="reference"><a href="#cite_note-vkdmsn-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> For subjectivists, <i>probability</i> corresponds to a personal belief.<sup id="cite_ref-Finetti,_B._1974_4-1" class="reference"><a href="#cite_note-Finetti,_B._1974-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Rationality and coherence allow for substantial variation within the constraints they pose; the constraints are justified by the <a href="/wiki/Dutch_book" class="mw-redirect" title="Dutch book">Dutch book</a> argument or by <a href="/wiki/Decision_theory" title="Decision theory">decision theory</a> and <a href="/wiki/De_Finetti%27s_theorem" title="De Finetti's theorem">de Finetti's theorem</a>.<sup id="cite_ref-Finetti,_B._1974_4-2" class="reference"><a href="#cite_note-Finetti,_B._1974-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The objective and subjective variants of Bayesian probability differ mainly in their interpretation and construction of the prior probability. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_probability&action=edit&section=3" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_statistics#Bayesian_statistics" title="History of statistics">History of statistics § Bayesian statistics</a></div> <p>The term <i>Bayesian</i> derives from <a href="/wiki/Thomas_Bayes" title="Thomas Bayes">Thomas Bayes</a> (1702–1761), who proved a special case of what is now called <a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a> in a paper titled "<a href="/wiki/An_Essay_Towards_Solving_a_Problem_in_the_Doctrine_of_Chances" title="An Essay Towards Solving a Problem in the Doctrine of Chances">An Essay Towards Solving a Problem in the Doctrine of Chances</a>".<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> In that special case, the prior and posterior distributions were <a href="/wiki/Beta_distribution" title="Beta distribution">beta distributions</a> and the data came from <a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trials</a>. It was <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> (1749–1827) who introduced a general version of the theorem and used it to approach problems in <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a>, medical statistics, <a href="/wiki/Reliability_(statistics)" title="Reliability (statistics)">reliability</a>, and <a href="/wiki/Jurisprudence" title="Jurisprudence">jurisprudence</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Early Bayesian inference, which used uniform priors following Laplace's <a href="/wiki/Principle_of_insufficient_reason" class="mw-redirect" title="Principle of insufficient reason">principle of insufficient reason</a>, was called "<a href="/wiki/Inverse_probability" title="Inverse probability">inverse probability</a>" (because it <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">infers</a> backwards from observations to parameters, or from effects to causes).<sup id="cite_ref-Fienberg2006_13-0" class="reference"><a href="#cite_note-Fienberg2006-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> After the 1920s, "inverse probability" was largely supplanted by a collection of methods that came to be called <a href="/wiki/Frequentist_statistics" class="mw-redirect" title="Frequentist statistics">frequentist statistics</a>.<sup id="cite_ref-Fienberg2006_13-1" class="reference"><a href="#cite_note-Fienberg2006-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>In the 20th century, the ideas of Laplace developed in two directions, giving rise to <i>objective</i> and <i>subjective</i> currents in Bayesian practice. <a href="/wiki/Harold_Jeffreys" title="Harold Jeffreys">Harold Jeffreys</a>' <i>Theory of Probability</i> (first published in 1939) played an important role in the revival of the Bayesian view of probability, followed by works by <a href="/wiki/Abraham_Wald" title="Abraham Wald">Abraham Wald</a> (1950) and <a href="/wiki/Leonard_J._Savage" class="mw-redirect" title="Leonard J. Savage">Leonard J. Savage</a> (1954). The adjective <i>Bayesian</i> itself dates to the 1950s; the derived <i>Bayesianism</i>, <i>neo-Bayesianism</i> is of 1960s coinage.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> In the objectivist stream, the statistical analysis depends on only the model assumed and the data analysed.<sup id="cite_ref-Bernardo_17-0" class="reference"><a href="#cite_note-Bernardo-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> No subjective decisions need to be involved. In contrast, "subjectivist" statisticians deny the possibility of fully objective analysis for the general case. </p><p>In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of <a href="/wiki/Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a> methods and the consequent removal of many of the computational problems, and to an increasing interest in nonstandard, complex applications.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> While frequentist statistics remains strong (as demonstrated by the fact that much of undergraduate teaching is based on it <sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>), Bayesian methods are widely accepted and used, e.g., in the field of <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>.<sup id="cite_ref-ReferenceA_20-0" class="reference"><a href="#cite_note-ReferenceA-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Justification">Justification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_probability&action=edit&section=4" title="Edit section: Justification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The use of Bayesian probabilities as the basis of <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a> has been supported by several arguments, such as <a href="/wiki/Cox%27s_theorem" title="Cox's theorem">Cox axioms</a>, the <a href="/wiki/Dutch_book" class="mw-redirect" title="Dutch book">Dutch book argument</a>, arguments based on <a href="/wiki/Decision_theory" title="Decision theory">decision theory</a> and <a href="/wiki/De_Finetti%27s_theorem" title="De Finetti's theorem">de Finetti's theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Axiomatic_approach">Axiomatic approach</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_probability&action=edit&section=5" title="Edit section: Axiomatic approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Richard_Threlkeld_Cox" title="Richard Threlkeld Cox">Richard T. Cox</a> showed that Bayesian updating follows from several axioms, including two <a href="/wiki/Functional_equations" class="mw-redirect" title="Functional equations">functional equations</a> and a hypothesis of differentiability.<sup id="cite_ref-vkdmsn_10-1" class="reference"><a href="#cite_note-vkdmsn-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> The assumption of differentiability or even continuity is controversial; Halpern found a counterexample based on his observation that the Boolean algebra of statements may be finite.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Other axiomatizations have been suggested by various authors with the purpose of making the theory more rigorous.<sup id="cite_ref-rbp_9-1" class="reference"><a href="#cite_note-rbp-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Dutch_book_approach">Dutch book approach</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_probability&action=edit&section=6" title="Edit section: Dutch book approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Dutch_book" class="mw-redirect" title="Dutch book">Dutch book</a></div> <p><a href="/wiki/Bruno_de_Finetti" title="Bruno de Finetti">Bruno de Finetti</a> proposed the Dutch book argument based on betting. A clever <a href="/wiki/Bookmaker" title="Bookmaker">bookmaker</a> makes a <a href="/wiki/Dutch_book" class="mw-redirect" title="Dutch book">Dutch book</a> by setting the <a href="/wiki/Odds" title="Odds">odds</a> and bets to ensure that the bookmaker profits—at the expense of the gamblers—regardless of the outcome of the event (a horse race, for example) on which the gamblers bet. It is associated with <a href="/wiki/Probability" title="Probability">probabilities</a> implied by the odds not being <a href="/wiki/Coherence_(philosophical_gambling_strategy)" class="mw-redirect" title="Coherence (philosophical gambling strategy)">coherent</a>. </p><p>However, <a href="/wiki/Ian_Hacking" title="Ian Hacking">Ian Hacking</a> noted that traditional Dutch book arguments did not specify Bayesian updating: they left open the possibility that non-Bayesian updating rules could avoid Dutch books. For example, <a href="/wiki/Ian_Hacking" title="Ian Hacking">Hacking</a> writes<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> "And neither the Dutch book argument, nor any other in the personalist arsenal of proofs of the probability axioms, entails the dynamic assumption. Not one entails Bayesianism. So the personalist requires the dynamic assumption to be Bayesian. It is true that in consistency a personalist could abandon the Bayesian model of learning from experience. Salt could lose its savour." </p><p>In fact, there are non-Bayesian updating rules that also avoid Dutch books (as discussed in the literature on "<a href="/wiki/Probability_kinematics" class="mw-redirect" title="Probability kinematics">probability kinematics</a>"<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> following the publication of <a href="/wiki/Richard_Jeffrey" title="Richard Jeffrey">Richard C. Jeffrey</a>'s rule, which is itself regarded as Bayesian<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>). The additional hypotheses sufficient to (uniquely) specify Bayesian updating are substantial<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> and not universally seen as satisfactory.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Decision_theory_approach">Decision theory approach</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_probability&action=edit&section=7" title="Edit section: Decision theory approach"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Statistical_decision_theory" class="mw-redirect" title="Statistical decision theory">decision-theoretic</a> justification of the use of Bayesian inference (and hence of Bayesian probabilities) was given by <a href="/wiki/Abraham_Wald" title="Abraham Wald">Abraham Wald</a>, who proved that every <a href="/wiki/Admissible_decision_rule" title="Admissible decision rule">admissible</a> statistical procedure is either a Bayesian procedure or a limit of Bayesian procedures.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> Conversely, every Bayesian procedure is <a href="/wiki/Admissible_decision_rule" title="Admissible decision rule">admissible</a>.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Personal_probabilities_and_objective_methods_for_constructing_priors">Personal probabilities and objective methods for constructing priors<span class="anchor" id="subjective"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_probability&action=edit&section=8" title="Edit section: Personal probabilities and objective methods for constructing priors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Following the work on <a href="/wiki/Expected_utility" class="mw-redirect" title="Expected utility">expected utility</a> <a href="/wiki/Optimal_decision" title="Optimal decision">theory</a> of <a href="/wiki/Frank_P._Ramsey" class="mw-redirect" title="Frank P. Ramsey">Ramsey</a> and <a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a>, decision-theorists have accounted for <a href="/wiki/Optimal_decision" title="Optimal decision">rational behavior</a> using a probability distribution for the <a href="/wiki/Agent-based_model" title="Agent-based model">agent</a>. <a href="/wiki/Johann_Pfanzagl" title="Johann Pfanzagl">Johann Pfanzagl</a> completed the <i><a href="/wiki/Theory_of_Games_and_Economic_Behavior" title="Theory of Games and Economic Behavior">Theory of Games and Economic Behavior</a></i> by providing an axiomatization of subjective probability and utility, a task left uncompleted by von Neumann and <a href="/wiki/Oskar_Morgenstern" title="Oskar Morgenstern">Oskar Morgenstern</a>: their original theory supposed that all the agents had the same probability distribution, as a convenience.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Pfanzagl's axiomatization was endorsed by Oskar Morgenstern: "Von Neumann and I have anticipated ... [the question whether probabilities] might, perhaps more typically, be subjective and have stated specifically that in the latter case axioms could be found from which could derive the desired numerical utility together with a number for the probabilities (cf. p. 19 of The Theory of Games and Economic Behavior). We did not carry this out; it was demonstrated by Pfanzagl ... with all the necessary rigor".<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p>Ramsey and <a href="/wiki/Leonard_Jimmie_Savage" title="Leonard Jimmie Savage">Savage</a> noted that the individual agent's probability distribution could be objectively studied in experiments. Procedures for <a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">testing hypotheses</a> about probabilities (using finite samples) are due to <a href="/wiki/Frank_P._Ramsey" class="mw-redirect" title="Frank P. Ramsey">Ramsey</a> (1931) and <a href="/wiki/Bruno_de_Finetti" title="Bruno de Finetti">de Finetti</a> (1931, 1937, 1964, 1970). Both <a href="/wiki/Bruno_de_Finetti" title="Bruno de Finetti">Bruno de Finetti</a><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_34-0" class="reference"><a href="#cite_note-:0-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Frank_P._Ramsey" class="mw-redirect" title="Frank P. Ramsey">Frank P. Ramsey</a><sup id="cite_ref-:0_34-1" class="reference"><a href="#cite_note-:0-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_35-0" class="reference"><a href="#cite_note-:1-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> acknowledge their debts to <a href="/wiki/Pragmatic_philosophy" class="mw-redirect" title="Pragmatic philosophy">pragmatic philosophy</a>, particularly (for Ramsey) to <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles S. Peirce</a>.<sup id="cite_ref-:0_34-2" class="reference"><a href="#cite_note-:0-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_35-1" class="reference"><a href="#cite_note-:1-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p>The "Ramsey test" for evaluating probability distributions is implementable in theory, and has kept experimental psychologists occupied for a half century.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> This work demonstrates that Bayesian-probability propositions can be <a href="/wiki/Falsifiability" title="Falsifiability">falsified</a>, and so meet an empirical criterion of <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles S. Peirce</a>, whose work inspired Ramsey. (This <a href="/wiki/Falsifiability" title="Falsifiability">falsifiability</a>-criterion was popularized by <a href="/wiki/Karl_Popper" title="Karl Popper">Karl Popper</a>.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup>) </p><p>Modern work on the experimental evaluation of personal probabilities uses the randomization, <a href="/wiki/Double_blind" class="mw-redirect" title="Double blind">blinding</a>, and Boolean-decision procedures of the Peirce-Jastrow experiment.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> Since individuals act according to different probability judgments, these agents' probabilities are "personal" (but amenable to objective study). </p><p>Personal probabilities are problematic for science and for some applications where decision-makers lack the knowledge or time to specify an informed probability-distribution (on which they are prepared to act). To meet the needs of science and of human limitations, Bayesian statisticians have developed "objective" methods for specifying prior probabilities. </p><p>Indeed, some Bayesians have argued the prior state of knowledge defines <i>the</i> (unique) prior probability-distribution for "regular" statistical problems; cf. <a href="/wiki/Well-posed_problem" title="Well-posed problem">well-posed problems</a>. Finding the right method for constructing such "objective" priors (for appropriate classes of regular problems) has been the quest of statistical theorists from Laplace to <a href="/wiki/John_Maynard_Keynes" title="John Maynard Keynes">John Maynard Keynes</a>, <a href="/wiki/Harold_Jeffreys" title="Harold Jeffreys">Harold Jeffreys</a>, and <a href="/wiki/Edwin_Thompson_Jaynes" title="Edwin Thompson Jaynes">Edwin Thompson Jaynes</a>. These theorists and their successors have suggested several methods for constructing "objective" priors (Unfortunately, it is not always clear how to assess the relative "objectivity" of the priors proposed under these methods): </p> <ul><li><a href="/wiki/Principle_of_maximum_entropy" title="Principle of maximum entropy">Maximum entropy</a></li> <li><a href="/wiki/Haar_measure" title="Haar measure">Transformation group analysis</a></li> <li><a href="/wiki/Jos%C3%A9-Miguel_Bernardo" title="José-Miguel Bernardo">Reference analysis</a></li></ul> <p>Each of these methods contributes useful priors for "regular" one-parameter problems, and each prior can handle some challenging <a href="/wiki/Statistical_model" title="Statistical model">statistical models</a> (with "irregularity" or several parameters). Each of these methods has been useful in Bayesian practice. Indeed, methods for constructing "objective" (alternatively, "default" or "ignorance") priors have been developed by avowed subjective (or "personal") Bayesians like <a href="/wiki/James_Berger_(statistician)" class="mw-redirect" title="James Berger (statistician)">James Berger</a> (<a href="/wiki/Duke_University" title="Duke University">Duke University</a>) and <a href="/wiki/Jos%C3%A9-Miguel_Bernardo" title="José-Miguel Bernardo">José-Miguel Bernardo</a> (<a href="/wiki/University_of_Valencia" title="University of Valencia">Universitat de València</a>), simply because such priors are needed for Bayesian practice, particularly in science.<sup id="cite_ref-refa_40-0" class="reference"><a href="#cite_note-refa-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> The quest for "the universal method for constructing priors" continues to attract statistical theorists.<sup id="cite_ref-refa_40-1" class="reference"><a href="#cite_note-refa-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p><p>Thus, the Bayesian statistician needs either to use informed priors (using relevant expertise or previous data) or to choose among the competing methods for constructing "objective" priors. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_probability&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" 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href="/wiki/An_Essay_Towards_Solving_a_Problem_in_the_Doctrine_of_Chances" title="An Essay Towards Solving a Problem in the Doctrine of Chances">An Essay Towards Solving a Problem in the Doctrine of Chances</a></i></li> <li><a href="/wiki/Bayesian_epistemology" title="Bayesian epistemology">Bayesian epistemology</a></li> <li><a href="/wiki/Bertrand_paradox_(probability)" title="Bertrand paradox (probability)">Bertrand paradox</a>—a paradox in classical probability</li> <li><a href="/wiki/Credal_network" title="Credal network">Credal network</a></li> <li><a href="/wiki/Credence_(statistics)" title="Credence (statistics)">Credence (statistics)</a></li> <li><a href="/wiki/De_Finetti%27s_game" class="mw-redirect" title="De Finetti's game">De Finetti's game</a>—a procedure for evaluating someone's subjective probability</li> <li><a href="/wiki/Evidence_under_Bayes%27_theorem" title="Evidence under Bayes' theorem">Evidence under Bayes' theorem</a></li> <li><a href="/wiki/Monty_Hall_problem" title="Monty Hall problem">Monty Hall problem</a></li> <li><a href="/wiki/QBism" class="mw-redirect" title="QBism">QBism</a>—an <a href="/wiki/Interpretation_of_quantum_mechanics" class="mw-redirect" title="Interpretation of quantum mechanics">interpretation of quantum mechanics</a> based on subjective Bayesian probability</li> <li><a href="/wiki/Reference_class_problem" title="Reference class problem">Reference class problem</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_probability&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist 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data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.merriam-webster.com/dictionary/Bayesian">"Bayesian"</a>. <i><a href="/wiki/Merriam-Webster" title="Merriam-Webster">Merriam-Webster.com Dictionary</a></i>. 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"Probability, Frequency, and Reasonable Expectation". <i>American Journal of Physics</i>. <b>14</b> (1): 1–10. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1946AmJPh..14....1C">1946AmJPh..14....1C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.1990764">10.1119/1.1990764</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Probability%2C+Frequency%2C+and+Reasonable+Expectation&rft.volume=14&rft.issue=1&rft.pages=1-10&rft.date=1946&rft_id=info%3Adoi%2F10.1119%2F1.1990764&rft_id=info%3Abibcode%2F1946AmJPh..14....1C&rft.aulast=Cox&rft.aufirst=R.T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-ghxaib-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-ghxaib_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ghxaib_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJaynes,_E.T.1986" class="citation book cs1">Jaynes, E.T. 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Retrieved <span class="nowrap">2011-08-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=New+York+Times&rft.atitle=The+Mathematics+of+Changing+Your+Mind+%5Bby+Sharon+Bertsch+McGrayne%5D&rft.date=2011-08-05&rft.aulast=Paulos&rft.aufirst=John+Allen&rft_id=https%3A%2F%2Fwww.nytimes.com%2F2011%2F08%2F07%2Fbooks%2Freview%2Fthe-theory-that-would-not-die-by-sharon-bertsch-mcgrayne-book-review.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-HOS-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-HOS_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HOS_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStigler1990" class="citation book cs1">Stigler, Stephen M. (March 1990). <i>The history of statistics</i>. Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780674403413" title="Special:BookSources/9780674403413"><bdi>9780674403413</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+history+of+statistics&rft.pub=Harvard+University+Press&rft.date=1990-03&rft.isbn=9780674403413&rft.aulast=Stigler&rft.aufirst=Stephen+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-rbp-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-rbp_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-rbp_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDupré,_Maurice_J.Tipler,_Frank_J.2009" class="citation journal cs1">Dupré, Maurice J.; Tipler, Frank J. (2009). <a rel="nofollow" class="external text" href="http://projecteuclid.org/download/pdf_1/euclid.ba/1340369856">"New axioms for rigorous Bayesian probability"</a>. <i>Bayesian Analysis</i>. <b>4</b> (3): 599–606. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.612.3036">10.1.1.612.3036</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F09-BA422">10.1214/09-BA422</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bayesian+Analysis&rft.atitle=New+axioms+for+rigorous+Bayesian+probability&rft.volume=4&rft.issue=3&rft.pages=599-606&rft.date=2009&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.612.3036%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1214%2F09-BA422&rft.au=Dupr%C3%A9%2C+Maurice+J.&rft.au=Tipler%2C+Frank+J.&rft_id=http%3A%2F%2Fprojecteuclid.org%2Fdownload%2Fpdf_1%2Feuclid.ba%2F1340369856&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-vkdmsn-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-vkdmsn_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-vkdmsn_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCox1961" class="citation book cs1">Cox, Richard T. (1961). <i>The algebra of probable inference</i> (Reprint ed.). Baltimore, MD; London, UK: Johns Hopkins Press; Oxford University Press [distributor]. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780801869822" title="Special:BookSources/9780801869822"><bdi>9780801869822</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+algebra+of+probable+inference&rft.place=Baltimore%2C+MD%3B+London%2C+UK&rft.edition=Reprint&rft.pub=Johns+Hopkins+Press%3B+Oxford+University+Press+%5Bdistributor%5D&rft.date=1961&rft.isbn=9780801869822&rft.aulast=Cox&rft.aufirst=Richard+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcGrayne,_Sharon_Bertsch2011" class="citation book cs1">McGrayne, Sharon Bertsch (2011). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/theorythatwouldn0000mcgr"><i>The Theory that Would not Die</i></a></span>. <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=_Kx5xVGuLRIC&pg=PA">[https://archive.org/details/theorythatwouldn0000mcgr/page/10 10</a>  ]</i>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/theorythatwouldn0000mcgr/page/10">10</a>, at <a href="/wiki/Google_Books" title="Google Books">Google Books</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+that+Would+not+Die&rft.pages=%27%2710+-%27%27%2C+p.+10%2C+at+Google+Books&rft.date=2011&rft.au=McGrayne%2C+Sharon+Bertsch&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheorythatwouldn0000mcgr&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStigler,_Stephen_M.1986" class="citation book cs1">Stigler, Stephen M. (1986). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/historyofstatist00stig">"Chapter 3"</a></span>. <i>The History of Statistics</i>. Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780674403406" title="Special:BookSources/9780674403406"><bdi>9780674403406</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+3&rft.btitle=The+History+of+Statistics&rft.pub=Harvard+University+Press&rft.date=1986&rft.isbn=9780674403406&rft.au=Stigler%2C+Stephen+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofstatist00stig&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-Fienberg2006-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Fienberg2006_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Fienberg2006_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFienberg,_Stephen._E.2006" class="citation journal cs1">Fienberg, Stephen. E. (2006). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140910070556/http://ba.stat.cmu.edu/journal/2006/vol01/issue01/fienberg.pdf">"When did Bayesian Inference become "Bayesian"?"</a> <span class="cs1-format">(PDF)</span>. <i>Bayesian Analysis</i>. <b>1</b> (1): 5, 1–40. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F06-BA101">10.1214/06-BA101</a></span>. Archived from <a rel="nofollow" class="external text" href="http://ba.stat.cmu.edu/journal/2006/vol01/issue01/fienberg.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 10 September 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bayesian+Analysis&rft.atitle=When+did+Bayesian+Inference+become+%22Bayesian%22%3F&rft.volume=1&rft.issue=1&rft.pages=5%2C+1-40&rft.date=2006&rft_id=info%3Adoi%2F10.1214%2F06-BA101&rft.au=Fienberg%2C+Stephen.+E.&rft_id=http%3A%2F%2Fba.stat.cmu.edu%2Fjournal%2F2006%2Fvol01%2Fissue01%2Ffienberg.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarris1959" class="citation journal cs1">Harris, Marshall Dees (1959). "Recent developments of the so-called Bayesian approach to statistics". Agricultural Law Center. <i>Legal-Economic Research</i>. University of Iowa: 125 (fn. #52), 126. <q>The works of <a href="/wiki/Abraham_Wald" title="Abraham Wald">Wald</a>, <i>Statistical Decision Functions</i> (1950) and <a href="/wiki/Leonard_J._Savage" class="mw-redirect" title="Leonard J. Savage">Savage</a>, <i>The Foundation of Statistics</i> (1954) are commonly regarded starting points for current Bayesian approaches</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Legal-Economic+Research&rft.atitle=Recent+developments+of+the+so-called+Bayesian+approach+to+statistics&rft.pages=125+%28fn.+%2352%29%2C+126&rft.date=1959&rft.aulast=Harris&rft.aufirst=Marshall+Dees&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>Annals of the Computation Laboratory of Harvard University</i>. Vol. 31. 1962. p. 180. <q>This revolution, which may or may not succeed, is neo-Bayesianism. Jeffreys tried to introduce this approach, but did not succeed at the time in giving it general appeal.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Annals+of+the+Computation+Laboratory+of+Harvard+University&rft.pages=180&rft.date=1962&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKempthorne1967" class="citation conference cs1">Kempthorne, Oscar (1967). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IC4Ku_7dBFUC&pg=PA235"><i>The Classical Problem of Inference—Goodness of Fit</i></a>. Fifth Berkeley Symposium on Mathematical Statistics and Probability. p. 235. <q>It is curious that even in its activities unrelated to ethics, humanity searches for a religion. At the present time, the religion being 'pushed' the hardest is Bayesianism.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=The+Classical+Problem+of+Inference%E2%80%94Goodness+of+Fit&rft.pages=235&rft.date=1967&rft.aulast=Kempthorne&rft.aufirst=Oscar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIC4Ku_7dBFUC%26pg%3DPA235&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-Bernardo-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bernardo_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernardo2005" class="citation book cs1"><a href="/wiki/Jos%C3%A9-Miguel_Bernardo" title="José-Miguel Bernardo">Bernardo, J.M.</a> (2005). "Reference analysis". <i>Bayesian Thinking - Modeling and Computation</i>. Handbook of Statistics. Vol. 25. Handbook of Statistics. pp. 17–90. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0169-7161%2805%2925002-2">10.1016/S0169-7161(05)25002-2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780444515391" title="Special:BookSources/9780444515391"><bdi>9780444515391</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Reference+analysis&rft.btitle=Bayesian+Thinking+-+Modeling+and+Computation&rft.series=Handbook+of+Statistics&rft.pages=17-90&rft.pub=Handbook+of+Statistics&rft.date=2005&rft_id=info%3Adoi%2F10.1016%2FS0169-7161%2805%2925002-2&rft.isbn=9780444515391&rft.aulast=Bernardo&rft.aufirst=J.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolpert,_R.L.2004" class="citation journal cs1">Wolpert, R.L. (2004). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F088342304000000053">"A conversation with James O. Berger"</a>. <i>Statistical Science</i>. <b>9</b>: 205–218. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F088342304000000053">10.1214/088342304000000053</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Science&rft.atitle=A+conversation+with+James+O.+Berger&rft.volume=9&rft.pages=205-218&rft.date=2004&rft_id=info%3Adoi%2F10.1214%2F088342304000000053&rft.au=Wolpert%2C+R.L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252F088342304000000053&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernardo,_José_M.2006" class="citation conference cs1"><a href="/wiki/Jos%C3%A9-Miguel_Bernardo" title="José-Miguel Bernardo">Bernardo, José M.</a> (2006). <a rel="nofollow" class="external text" href="http://www.ime.usp.br/~abe/ICOTS7/Proceedings/PDFs/InvitedPapers/3I2_BERN.pdf"><i>A Bayesian mathematical statistics primer</i></a> <span class="cs1-format">(PDF)</span>. ICOTS-7. Bern. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.ime.usp.br/~abe/ICOTS7/Proceedings/PDFs/InvitedPapers/3I2_BERN.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=A+Bayesian+mathematical+statistics+primer&rft.place=Bern&rft.date=2006&rft.au=Bernardo%2C+Jos%C3%A9+M.&rft_id=http%3A%2F%2Fwww.ime.usp.br%2F~abe%2FICOTS7%2FProceedings%2FPDFs%2FInvitedPapers%2F3I2_BERN.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-ReferenceA-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-ReferenceA_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBishop,_C.M.2007" class="citation book cs1">Bishop, C.M. (2007). <i>Pattern Recognition and Machine Learning</i>. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pattern+Recognition+and+Machine+Learning&rft.pub=Springer&rft.date=2007&rft.au=Bishop%2C+C.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmithErickson1989" class="citation book cs1">Smith, C. Ray; Erickson, Gary (1989). "From Rationality and Consistency to Bayesian Probability". In Skilling, John (ed.). <i>Maximum Entropy and Bayesian Methods</i>. Dordrecht: Kluwer. pp. 29–44. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-015-7860-8_2">10.1007/978-94-015-7860-8_2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-0224-9" title="Special:BookSources/0-7923-0224-9"><bdi>0-7923-0224-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=From+Rationality+and+Consistency+to+Bayesian+Probability&rft.btitle=Maximum+Entropy+and+Bayesian+Methods&rft.place=Dordrecht&rft.pages=29-44&rft.pub=Kluwer&rft.date=1989&rft_id=info%3Adoi%2F10.1007%2F978-94-015-7860-8_2&rft.isbn=0-7923-0224-9&rft.aulast=Smith&rft.aufirst=C.+Ray&rft.au=Erickson%2C+Gary&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalpern,_J.1999" class="citation journal cs1">Halpern, J. (1999). <a rel="nofollow" class="external text" href="http://www.cs.cornell.edu/info/people/halpern/papers/cox.pdf">"A counterexample to theorems of Cox and Fine"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Artificial Intelligence Research</i>. <b>10</b>: 67–85. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1613%2Fjair.536">10.1613/jair.536</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1538503">1538503</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.cs.cornell.edu/info/people/halpern/papers/cox.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Artificial+Intelligence+Research&rft.atitle=A+counterexample+to+theorems+of+Cox+and+Fine&rft.volume=10&rft.pages=67-85&rft.date=1999&rft_id=info%3Adoi%2F10.1613%2Fjair.536&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1538503%23id-name%3DS2CID&rft.au=Halpern%2C+J.&rft_id=http%3A%2F%2Fwww.cs.cornell.edu%2Finfo%2Fpeople%2Fhalpern%2Fpapers%2Fcox.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Hacking (1967), Section 3, page 316</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">Hacking (1988, page 124)</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSkyrms1987" class="citation journal cs1">Skyrms, Brian (1 January 1987). "Dynamic Coherence and Probability Kinematics". <i>Philosophy of Science</i>. <b>54</b> (1): 1–20. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.395.5723">10.1.1.395.5723</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F289350">10.1086/289350</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/187470">187470</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120881078">120881078</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophy+of+Science&rft.atitle=Dynamic+Coherence+and+Probability+Kinematics&rft.volume=54&rft.issue=1&rft.pages=1-20&rft.date=1987-01-01&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.395.5723%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120881078%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F187470%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1086%2F289350&rft.aulast=Skyrms&rft.aufirst=Brian&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoyce2003" class="citation encyclopaedia cs1">Joyce, James (30 September 2003). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/bayes-theorem/">"Bayes' Theorem"</a>. <i>The Stanford Encyclopedia of Philosophy</i>. stanford.edu.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bayes%27+Theorem&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.pub=stanford.edu&rft.date=2003-09-30&rft.aulast=Joyce&rft.aufirst=James&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Fbayes-theorem%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFuchsSchack2012" class="citation book cs1">Fuchs, Christopher A.; Schack, Rüdiger (1 January 2012). "Bayesian Conditioning, the Reflection Principle, and Quantum Decoherence". In Ben-Menahem, Yemima; Hemmo, Meir (eds.). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/probabilityphysi00benm"><i>Probability in Physics</i></a></span>. The Frontiers Collection. Springer Berlin Heidelberg. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/probabilityphysi00benm/page/n245">233</a>–247. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1103.5950">1103.5950</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-21329-8_15">10.1007/978-3-642-21329-8_15</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783642213281" title="Special:BookSources/9783642213281"><bdi>9783642213281</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119215115">119215115</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bayesian+Conditioning%2C+the+Reflection+Principle%2C+and+Quantum+Decoherence&rft.btitle=Probability+in+Physics&rft.series=The+Frontiers+Collection&rft.pages=233-247&rft.pub=Springer+Berlin+Heidelberg&rft.date=2012-01-01&rft_id=info%3Aarxiv%2F1103.5950&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119215115%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-642-21329-8_15&rft.isbn=9783642213281&rft.aulast=Fuchs&rft.aufirst=Christopher+A.&rft.au=Schack%2C+R%C3%BCdiger&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprobabilityphysi00benm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_Frassen1989" class="citation book cs1"><a href="/wiki/Bas_van_Fraassen" title="Bas van Fraassen">van Frassen, Bas</a> (1989). <i>Laws and Symmetry</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-824860-1" title="Special:BookSources/0-19-824860-1"><bdi>0-19-824860-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Laws+and+Symmetry&rft.pub=Oxford+University+Press&rft.date=1989&rft.isbn=0-19-824860-1&rft.aulast=van+Frassen&rft.aufirst=Bas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWald,_Abraham1950" class="citation book cs1">Wald, Abraham (1950). <i>Statistical Decision Functions</i>. Wiley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Decision+Functions&rft.pub=Wiley&rft.date=1950&rft.au=Wald%2C+Abraham&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernardo,_José_M.Smith,_Adrian_F.M.1994" class="citation book cs1">Bernardo, José M.; Smith, Adrian F.M. (1994). <i>Bayesian Theory</i>. John Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-92416-4" title="Special:BookSources/0-471-92416-4"><bdi>0-471-92416-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bayesian+Theory&rft.pub=John+Wiley&rft.date=1994&rft.isbn=0-471-92416-4&rft.au=Bernardo%2C+Jos%C3%A9+M.&rft.au=Smith%2C+Adrian+F.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Pfanzagl (1967, 1968)</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Morgenstern (1976, page 65)</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGalavotti1989" class="citation journal cs1"><a href="/wiki/Maria_Carla_Galavotti" title="Maria Carla Galavotti">Galavotti, Maria Carla</a> (1 January 1989). "Anti-Realism in the Philosophy of Probability: Bruno de Finetti's Subjectivism". <i>Erkenntnis</i>. <b>31</b> (2/3): 239–261. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01236565">10.1007/bf01236565</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/20012239">20012239</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:170802937">170802937</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Erkenntnis&rft.atitle=Anti-Realism+in+the+Philosophy+of+Probability%3A+Bruno+de+Finetti%27s+Subjectivism&rft.volume=31&rft.issue=2%2F3&rft.pages=239-261&rft.date=1989-01-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A170802937%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F20012239%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1007%2Fbf01236565&rft.aulast=Galavotti&rft.aufirst=Maria+Carla&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-:0-34"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_34-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_34-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_34-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGalavotti1991" class="citation journal cs1"><a href="/wiki/Maria_Carla_Galavotti" title="Maria Carla Galavotti">Galavotti, Maria Carla</a> (1 December 1991). "The notion of subjective probability in the work of Ramsey and de Finetti". <i>Theoria</i>. <b>57</b> (3): 239–259. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1755-2567.1991.tb00839.x">10.1111/j.1755-2567.1991.tb00839.x</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1755-2567">1755-2567</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoria&rft.atitle=The+notion+of+subjective+probability+in+the+work+of+Ramsey+and+de+Finetti&rft.volume=57&rft.issue=3&rft.pages=239-259&rft.date=1991-12-01&rft_id=info%3Adoi%2F10.1111%2Fj.1755-2567.1991.tb00839.x&rft.issn=1755-2567&rft.aulast=Galavotti&rft.aufirst=Maria+Carla&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-:1-35"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_35-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_35-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDokicEngel2003" class="citation book cs1">Dokic, Jérôme; Engel, Pascal (2003). <i>Frank Ramsey: Truth and Success</i>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781134445936" title="Special:BookSources/9781134445936"><bdi>9781134445936</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Frank+Ramsey%3A+Truth+and+Success&rft.pub=Routledge&rft.date=2003&rft.isbn=9781134445936&rft.aulast=Dokic&rft.aufirst=J%C3%A9r%C3%B4me&rft.au=Engel%2C+Pascal&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text">Davidson et al. (1957)</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThornton2018" class="citation encyclopaedia cs1">Thornton, Stephen (7 August 2018). "Karl Popper". <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/popper/#ProDem"><i>Stanford Encyclopedia of Philosophy</i></a>. Metaphysics Research Lab, Stanford University.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Karl+Popper&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2018-08-07&rft.aulast=Thornton&rft.aufirst=Stephen&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Fpopper%2F%23ProDem&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPopper,_Karl2002" class="citation book cs1">Popper, Karl (2002) [1959]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=T76Zd20IYlgC&q=logic+of+scientific+discovery"><i>The Logic of Scientific Discovery</i></a> (2nd ed.). Routledge. p. 57. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-415-27843-0" title="Special:BookSources/0-415-27843-0"><bdi>0-415-27843-0</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Logic+of+Scientific+Discovery&rft.pages=57&rft.edition=2nd&rft.pub=Routledge&rft.date=2002&rft.isbn=0-415-27843-0&rft.au=Popper%2C+Karl&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DT76Zd20IYlgC%26q%3Dlogic%2Bof%2Bscientific%2Bdiscovery&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span> (translation of 1935 original, in German).</span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text">Peirce & Jastrow (1885)</span> </li> <li id="cite_note-refa-40"><span class="mw-cite-backlink">^ <a href="#cite_ref-refa_40-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-refa_40-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernardo,_J._M.2005" class="citation book cs1">Bernardo, J. M. (2005). "Reference Analysis". In Dey, D.K.; <a href="/wiki/C._R._Rao" title="C. R. Rao">Rao, C. R.</a> (eds.). <a rel="nofollow" class="external text" href="http://www.uv.es/~bernardo/RefAna.pdf"><i>Handbook of Statistics</i></a> <span class="cs1-format">(PDF)</span>. Vol. 25. Amsterdam: Elsevier. pp. 17–90. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.uv.es/~bernardo/RefAna.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Reference+Analysis&rft.btitle=Handbook+of+Statistics&rft.place=Amsterdam&rft.pages=17-90&rft.pub=Elsevier&rft.date=2005&rft.au=Bernardo%2C+J.+M.&rft_id=http%3A%2F%2Fwww.uv.es%2F~bernardo%2FRefAna.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bayesian_probability&action=edit&section=11" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerger,_James_O.1985" class="citation book cs1"><a href="/wiki/James_Berger_(statistician)" class="mw-redirect" title="James Berger (statistician)">Berger, James O.</a> (1985). <i>Statistical Decision Theory and Bayesian Analysis</i>. Springer Series in Statistics (Second ed.). Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96098-2" title="Special:BookSources/978-0-387-96098-2"><bdi>978-0-387-96098-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Decision+Theory+and+Bayesian+Analysis&rft.series=Springer+Series+in+Statistics&rft.edition=Second&rft.pub=Springer-Verlag&rft.date=1985&rft.isbn=978-0-387-96098-2&rft.au=Berger%2C+James+O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBessièreMazerAhuacatzinMekhnacha2013" class="citation book cs1">Bessière, Pierre; Mazer, E.; Ahuacatzin, J.-M.; Mekhnacha, K. (2013). <i>Bayesian Programming</i>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781439880326" title="Special:BookSources/9781439880326"><bdi>9781439880326</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bayesian+Programming&rft.pub=CRC+Press&rft.date=2013&rft.isbn=9781439880326&rft.aulast=Bessi%C3%A8re&rft.aufirst=Pierre&rft.au=Mazer%2C+E.&rft.au=Ahuacatzin%2C+J.-M.&rft.au=Mekhnacha%2C+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernardoSmith1994" class="citation book cs1"><a href="/wiki/Jos%C3%A9-Miguel_Bernardo" title="José-Miguel Bernardo">Bernardo, José M.</a>; <a href="/wiki/Adrian_Smith_(statistician)" title="Adrian Smith (statistician)">Smith, Adrian F.M.</a> (1994). <i>Bayesian Theory</i>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-49464-5" title="Special:BookSources/978-0-471-49464-5"><bdi>978-0-471-49464-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bayesian+Theory&rft.pub=Wiley&rft.date=1994&rft.isbn=978-0-471-49464-5&rft.aulast=Bernardo&rft.aufirst=Jos%C3%A9+M.&rft.au=Smith%2C+Adrian+F.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBickelDoksum2001" class="citation book cs1"><a href="/wiki/Peter_J._Bickel" title="Peter J. Bickel">Bickel, Peter J.</a>; Doksum, Kjell A. (2001) [1976]. <i>Basic and selected topics</i>. Mathematical Statistics. Vol. 1 (Second ed.). Pearson Prentice–Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-850363-5" title="Special:BookSources/978-0-13-850363-5"><bdi>978-0-13-850363-5</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0443141">0443141</a>. <q>(updated printing, 2007, of Holden-Day, 1976)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+and+selected+topics&rft.series=Mathematical+Statistics&rft.edition=Second&rft.pub=Pearson+Prentice%E2%80%93Hall&rft.date=2001&rft.isbn=978-0-13-850363-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D443141%23id-name%3DMR&rft.aulast=Bickel&rft.aufirst=Peter+J.&rft.au=Doksum%2C+Kjell+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavidsonSuppesSiegel1957" class="citation book cs1"><a href="/wiki/Donald_Davidson_(philosopher)" title="Donald Davidson (philosopher)">Davidson, Donald</a>; <a href="/wiki/Patrick_Suppes" title="Patrick Suppes">Suppes, Patrick</a>; <a href="/wiki/Sidney_Siegel" title="Sidney Siegel">Siegel, Sidney</a> (1957). <i>Decision-Making: an Experimental Approach</i>. <a href="/wiki/Stanford_University_Press" title="Stanford University Press">Stanford University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Decision-Making%3A+an+Experimental+Approach&rft.pub=Stanford+University+Press&rft.date=1957&rft.aulast=Davidson&rft.aufirst=Donald&rft.au=Suppes%2C+Patrick&rft.au=Siegel%2C+Sidney&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_Finetti,_Bruno1937" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Bruno_de_Finetti" title="Bruno de Finetti">de Finetti, Bruno</a> (1937). <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=AIHP_1937__7_1_1_0">"La Prévision: ses lois logiques, ses sources subjectives"</a> [Foresight: Its logical laws, its subjective sources]. <i>Annales de l'Institut Henri Poincaré</i> (in French). <b>7</b> (1): 1–68.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+de+l%27Institut+Henri+Poincar%C3%A9&rft.atitle=La+Pr%C3%A9vision%3A+ses+lois+logiques%2C+ses+sources+subjectives&rft.volume=7&rft.issue=1&rft.pages=1-68&rft.date=1937&rft.au=de+Finetti%2C+Bruno&rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%3Fid%3DAIHP_1937__7_1_1_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_Finetti,_Bruno1989" class="citation journal cs1">de Finetti, Bruno (September 1989) [1931]. "Probabilism: A critical essay on the theory of probability and on the value of science". <i>Erkenntnis</i>. <b>31</b>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Erkenntnis&rft.atitle=Probabilism%3A+A+critical+essay+on+the+theory+of+probability+and+on+the+value+of+science&rft.volume=31&rft.date=1989-09&rft.au=de+Finetti%2C+Bruno&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span> (translation of de Finetti, 1931)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_Finetti,_Bruno1964" class="citation book cs1">de Finetti, Bruno (1964) [1937]. "Foresight: Its logical laws, its subjective sources". In Kyburg, H.E.; Smokler, H.E. (eds.). <i>Studies in Subjective Probability</i>. New York, NY: Wiley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Foresight%3A+Its+logical+laws%2C+its+subjective+sources&rft.btitle=Studies+in+Subjective+Probability&rft.place=New+York%2C+NY&rft.pub=Wiley&rft.date=1964&rft.au=de+Finetti%2C+Bruno&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span> (translation of de Finetti, 1937, above)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_Finetti1974–1975" class="citation book cs1">de Finetti, Bruno (1974–1975) [1970]. <i>Theory of Probability: A critical introductory treatment</i>. Translated by Machi, A.; <a href="/wiki/AFM_Smith" class="mw-redirect" title="AFM Smith">Smith, AFM</a>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-20141-3" title="Special:BookSources/0-471-20141-3"><bdi>0-471-20141-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Probability%3A+A+critical+introductory+treatment&rft.pub=Wiley&rft.date=1974%2F1975&rft.isbn=0-471-20141-3&rft.aulast=de+Finetti&rft.aufirst=Bruno&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-20142-1" title="Special:BookSources/0-471-20142-1">0-471-20142-1</a>, two volumes.</li> <li>Goertz, Gary and James Mahoney. 2012. <i>A Tale of Two Cultures: Qualitative and Quantitative Research in the Social Sciences</i>. Princeton University Press.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeGroot,_Morris2004" class="citation book cs1"><a href="/wiki/Morris_DeGroot" class="mw-redirect" title="Morris DeGroot">DeGroot, Morris</a> (2004) [1970]. <i>Optimal Statistical Decisions</i>. Wiley Classics Library. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-68029-X" title="Special:BookSources/0-471-68029-X"><bdi>0-471-68029-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optimal+Statistical+Decisions&rft.series=Wiley+Classics+Library&rft.pub=Wiley&rft.date=2004&rft.isbn=0-471-68029-X&rft.au=DeGroot%2C+Morris&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHacking1967" class="citation journal cs1"><a href="/wiki/Ian_Hacking" title="Ian Hacking">Hacking, Ian</a> (December 1967). "Slightly more realistic personal probability". <i>Philosophy of Science</i>. <b>34</b> (4): 311–325. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F288169">10.1086/288169</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/186120">186120</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14344339">14344339</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophy+of+Science&rft.atitle=Slightly+more+realistic+personal+probability&rft.volume=34&rft.issue=4&rft.pages=311-325&rft.date=1967-12&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14344339%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F186120%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1086%2F288169&rft.aulast=Hacking&rft.aufirst=Ian&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li></ul> <dl><dd>(Partly reprinted in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGärdenfors,_PeterSahlin,_Nils-Eric1988" class="citation book cs1"><a href="/wiki/Peter_G%C3%A4rdenfors" title="Peter Gärdenfors">Gärdenfors, Peter</a>; Sahlin, Nils-Eric (1988). <i>Decision, Probability, and Utility: Selected Readings</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-33658-9" title="Special:BookSources/0-521-33658-9"><bdi>0-521-33658-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Decision%2C+Probability%2C+and+Utility%3A+Selected+Readings&rft.pub=Cambridge+University+Press&rft.date=1988&rft.isbn=0-521-33658-9&rft.au=G%C3%A4rdenfors%2C+Peter&rft.au=Sahlin%2C+Nils-Eric&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span>)</dd></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHajek,_A.Hartmann,_S.2010" class="citation book cs1">Hajek, A.; Hartmann, S. (2010) [2001]. "Bayesian Epistemology". In Dancy, J.; Sosa, E.; Steup, M. (eds.). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110728055439/http://stephanhartmann.org/HajekHartmann_BayesEpist.pdf"><i>A Companion to Epistemology</i></a> <span class="cs1-format">(PDF)</span>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4051-3900-7" title="Special:BookSources/978-1-4051-3900-7"><bdi>978-1-4051-3900-7</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://stephanhartmann.org/HajekHartmann_BayesEpist.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2011-07-28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bayesian+Epistemology&rft.btitle=A+Companion+to+Epistemology&rft.pub=Wiley&rft.date=2010&rft.isbn=978-1-4051-3900-7&rft.au=Hajek%2C+A.&rft.au=Hartmann%2C+S.&rft_id=http%3A%2F%2Fstephanhartmann.org%2FHajekHartmann_BayesEpist.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHald1998" class="citation book cs1"><a href="/wiki/Anders_Hald" title="Anders Hald">Hald, Anders</a> (1998). <i>A History of Mathematical Statistics from 1750 to 1930</i>. New York: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-17912-2" title="Special:BookSources/978-0-471-17912-2"><bdi>978-0-471-17912-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematical+Statistics+from+1750+to+1930&rft.place=New+York&rft.pub=Wiley&rft.date=1998&rft.isbn=978-0-471-17912-2&rft.aulast=Hald&rft.aufirst=Anders&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartmann,_S.Sprenger,_J.2011" class="citation book cs1">Hartmann, S.; Sprenger, J. (2011). "Bayesian Epistemology". In Bernecker, S.; Pritchard, D. (eds.). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110728055519/http://stephanhartmann.org/HartmannSprenger_BayesEpis.pdf"><i>Routledge Companion to Epistemology</i></a> <span class="cs1-format">(PDF)</span>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-415-96219-3" title="Special:BookSources/978-0-415-96219-3"><bdi>978-0-415-96219-3</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://stephanhartmann.org/HartmannSprenger_BayesEpis.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2011-07-28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bayesian+Epistemology&rft.btitle=Routledge+Companion+to+Epistemology&rft.pub=Routledge&rft.date=2011&rft.isbn=978-0-415-96219-3&rft.au=Hartmann%2C+S.&rft.au=Sprenger%2C+J.&rft_id=http%3A%2F%2Fstephanhartmann.org%2FHartmannSprenger_BayesEpis.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Bayesian_approach_to_statistical_problems">"Bayesian approach to statistical problems"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bayesian+approach+to+statistical+problems&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DBayesian_approach_to_statistical_problems&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHowsonUrbach2005" class="citation book cs1"><a href="/wiki/Colin_Howson" title="Colin Howson">Howson, C.</a>; Urbach, P. (2005). <i>Scientific Reasoning: The Bayesian approach</i> (3rd ed.). <a href="/wiki/Open_Court_Publishing_Company" title="Open Court Publishing Company">Open Court Publishing Company</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8126-9578-6" title="Special:BookSources/978-0-8126-9578-6"><bdi>978-0-8126-9578-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Scientific+Reasoning%3A+The+Bayesian+approach&rft.edition=3rd&rft.pub=Open+Court+Publishing+Company&rft.date=2005&rft.isbn=978-0-8126-9578-6&rft.aulast=Howson&rft.aufirst=C.&rft.au=Urbach%2C+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJaynes,_E.T.2003" class="citation book cs1"><a href="/wiki/Edwin_Thompson_Jaynes" title="Edwin Thompson Jaynes">Jaynes, E.T.</a> (2003). <i>Probability Theory: The logic of science</i>. C. University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-59271-0" title="Special:BookSources/978-0-521-59271-0"><bdi>978-0-521-59271-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+Theory%3A+The+logic+of+science&rft.pub=C.+University+Press&rft.date=2003&rft.isbn=978-0-521-59271-0&rft.au=Jaynes%2C+E.T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www-biba.inrialpes.fr/Jaynes/prob.html">"Link to fragmentary edition of March 1996"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Link+to+fragmentary+edition+of+March+1996&rft_id=http%3A%2F%2Fwww-biba.inrialpes.fr%2FJaynes%2Fprob.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcGrayne,_S.B.2011" class="citation book cs1">McGrayne, S.B. (2011). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/theorythatwouldn0000mcgr"><i>The Theory that would not Die: How Bayes' rule cracked the Enigma code, hunted down Russian submarines, and emerged triumphant from two centuries of controversy</i></a></span>. New Haven, CT: Yale University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780300169690" title="Special:BookSources/9780300169690"><bdi>9780300169690</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/670481486">670481486</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+that+would+not+Die%3A+How+Bayes%27+rule+cracked+the+Enigma+code%2C+hunted+down+Russian+submarines%2C+and+emerged+triumphant+from+two+centuries+of+controversy&rft.place=New+Haven%2C+CT&rft.pub=Yale+University+Press&rft.date=2011&rft_id=info%3Aoclcnum%2F670481486&rft.isbn=9780300169690&rft.au=McGrayne%2C+S.B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheorythatwouldn0000mcgr&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorgenstern,_Oskar1978" class="citation book cs1"><a href="/wiki/Oskar_Morgenstern" title="Oskar Morgenstern">Morgenstern, Oskar</a> (1978). "Some Reflections on <a href="/wiki/Expected_utility" class="mw-redirect" title="Expected utility">Utility</a>". In Schotter, Andrew (ed.). <i>Selected Economic Writings of Oskar Morgenstern</i>. New York University Press. pp. 65–70. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8147-7771-8" title="Special:BookSources/978-0-8147-7771-8"><bdi>978-0-8147-7771-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Some+Reflections+on+Utility&rft.btitle=Selected+Economic+Writings+of+Oskar+Morgenstern&rft.pages=65-70&rft.pub=New+York+University+Press&rft.date=1978&rft.isbn=978-0-8147-7771-8&rft.au=Morgenstern%2C+Oskar&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeirce,_C.S.Jastrow_J.1885" class="citation journal cs1"><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, C.S.</a> & <a href="/wiki/Joseph_Jastrow" title="Joseph Jastrow">Jastrow J.</a> (1885). <a rel="nofollow" class="external text" href="http://psychclassics.yorku.ca/Peirce/small-diffs.htm">"On Small Differences in Sensation"</a>. <i>Memoirs of the National Academy of Sciences</i>. <b>3</b>: 73–83.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Memoirs+of+the+National+Academy+of+Sciences&rft.atitle=On+Small+Differences+in+Sensation&rft.volume=3&rft.pages=73-83&rft.date=1885&rft.au=Peirce%2C+C.S.&rft.au=Jastrow+J.&rft_id=http%3A%2F%2Fpsychclassics.yorku.ca%2FPeirce%2Fsmall-diffs.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPfanzagl,_J1967" class="citation book cs1">Pfanzagl, J (1967). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/essaysinmathemat0000shub">"Subjective Probability Derived from the Morgenstern-von Neumann Utility Theory"</a></span>. In <a href="/wiki/Martin_Shubik" title="Martin Shubik">Martin Shubik</a> (ed.). <i>Essays in Mathematical Economics In Honor of Oskar Morgenstern</i>. Princeton University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/essaysinmathemat0000shub/page/237">237–251</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Subjective+Probability+Derived+from+the+Morgenstern-von+Neumann+Utility+Theory&rft.btitle=Essays+in+Mathematical+Economics+In+Honor+of+Oskar+Morgenstern&rft.pages=237-251&rft.pub=Princeton+University+Press&rft.date=1967&rft.au=Pfanzagl%2C+J&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fessaysinmathemat0000shub&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPfanzagl,_J.Baumann,_V.Huber,_H.1968" class="citation book cs1">Pfanzagl, J.; Baumann, V. & Huber, H. (1968). "Events, Utility and Subjective Probability". <i>Theory of Measurement</i>. Wiley. pp. 195–220.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Events%2C+Utility+and+Subjective+Probability&rft.btitle=Theory+of+Measurement&rft.pages=195-220&rft.pub=Wiley&rft.date=1968&rft.au=Pfanzagl%2C+J.&rft.au=Baumann%2C+V.&rft.au=Huber%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamsey,_Frank_Plumpton2001" class="citation book cs1"><a href="/wiki/Frank_P._Ramsey" class="mw-redirect" title="Frank P. Ramsey">Ramsey, Frank Plumpton</a> (2001) [1931]. "Chapter VII: Truth and Probability". <i>The Foundations of Mathematics and other Logical Essays</i>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-415-22546-9" title="Special:BookSources/0-415-22546-9"><bdi>0-415-22546-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+VII%3A+Truth+and+Probability&rft.btitle=The+Foundations+of+Mathematics+and+other+Logical+Essays&rft.pub=Routledge&rft.date=2001&rft.isbn=0-415-22546-9&rft.au=Ramsey%2C+Frank+Plumpton&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080227205205/http://cepa.newschool.edu/het//texts/ramsey/ramsess.pdf">"Chapter VII: Truth and Probability"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://cepa.newschool.edu/het//texts/ramsey/ramsess.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2008-02-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Chapter+VII%3A+Truth+and+Probability&rft_id=http%3A%2F%2Fcepa.newschool.edu%2Fhet%2F%2Ftexts%2Framsey%2Framsess.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStigler,_S.M.1990" class="citation book cs1"><a href="/wiki/Stephen_Stigler" title="Stephen Stigler">Stigler, S.M.</a> (1990). <i>The History of Statistics: The Measurement of Uncertainty before 1900</i>. Belknap Press; Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-40341-3" title="Special:BookSources/978-0-674-40341-3"><bdi>978-0-674-40341-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+History+of+Statistics%3A+The+Measurement+of+Uncertainty+before+1900&rft.pub=Belknap+Press%3B+Harvard+University+Press&rft.date=1990&rft.isbn=978-0-674-40341-3&rft.au=Stigler%2C+S.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStigler,_S.M.1999" class="citation book cs1">Stigler, S.M. (1999). <i>Statistics on the Table: The history of statistical concepts and methods</i>. Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-674-83601-4" title="Special:BookSources/0-674-83601-4"><bdi>0-674-83601-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistics+on+the+Table%3A+The+history+of+statistical+concepts+and+methods&rft.pub=Harvard+University+Press&rft.date=1999&rft.isbn=0-674-83601-4&rft.au=Stigler%2C+S.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStone,_J.V.2013" class="citation book cs1">Stone, J.V. (2013). <i>Bayes' Rule: A tutorial introduction to Bayesian analysis</i>. England: Sebtel Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bayes%27+Rule%3A+A+tutorial+introduction+to+Bayesian+analysis&rft.place=England&rft.pub=Sebtel+Press&rft.date=2013&rft.au=Stone%2C+J.V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://jim-stone.staff.shef.ac.uk/BookBayes2012/BayesRuleBookMain.html">"Chapter 1 of <i>Bayes' Rule</i>"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Chapter+1+of+Bayes%27+Rule.&rft_id=http%3A%2F%2Fjim-stone.staff.shef.ac.uk%2FBookBayes2012%2FBayesRuleBookMain.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWinkler,_R.L.2003" class="citation book cs1">Winkler, R.L. (2003). <i>Introduction to Bayesian Inference and Decision</i> (2nd ed.). Probabilistic. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9647938-4-2" title="Special:BookSources/978-0-9647938-4-2"><bdi>978-0-9647938-4-2</bdi></a>. <q>Updated classic textbook. Bayesian theory clearly presented</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Bayesian+Inference+and+Decision&rft.edition=2nd&rft.pub=Probabilistic&rft.date=2003&rft.isbn=978-0-9647938-4-2&rft.au=Winkler%2C+R.L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABayesian+probability" class="Z3988"></span></li></ul> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Bayesian_probability&oldid=1252927037">https://en.wikipedia.org/w/index.php?title=Bayesian_probability&oldid=1252927037</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Bayesian_statistics" title="Category:Bayesian statistics">Bayesian statistics</a></li><li><a href="/wiki/Category:Justification_(epistemology)" title="Category:Justification (epistemology)">Justification (epistemology)</a></li><li><a href="/wiki/Category:Probability_interpretations" title="Category:Probability interpretations">Probability interpretations</a></li><li><a href="/wiki/Category:Philosophy_of_mathematics" title="Category:Philosophy of mathematics">Philosophy of mathematics</a></li><li><a href="/wiki/Category:Philosophy_of_science" title="Category:Philosophy of science">Philosophy of science</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:CS1_French-language_sources_(fr)" title="Category:CS1 French-language sources (fr)">CS1 French-language sources (fr)</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 23 October 2024, at 15:32<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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