{"id":569,"date":"2021-04-25T20:39:43","date_gmt":"2021-04-25T20:39:43","guid":{"rendered":"https:\/\/www.allendowney.com\/blog\/?p=569"},"modified":"2021-04-25T20:39:43","modified_gmt":"2021-04-25T20:39:43","slug":"bayesian-and-frequentist-results-are-not-the-same-ever","status":"publish","type":"post","link":"https:\/\/www.allendowney.com\/blog\/2021\/04\/25\/bayesian-and-frequentist-results-are-not-the-same-ever\/","title":{"rendered":"Bayesian and frequentist results are not the same, ever"},"content":{"rendered":"\n<p>I often hear people say that the results from Bayesian methods are the same as the results from frequentist methods, at least under certain conditions. And sometimes it even comes from people who understand Bayesian methods.<\/p>\n\n\n\n<p>Today I saw <a href=\"https:\/\/twitter.com\/dingding_peng\/status\/1385550273758867457\">this tweet from Julia Rohrer<\/a>: &#8220;Running a Bayesian multi-membership multi-level probit model with a custom function to generate average marginal effects only to find that the estimate is <em>precisely the same<\/em> as the one generated by linear regression with dummy-coded group membership.&#8221; [emphasis mine]<\/p>\n\n\n\n<p>Which elicited what I interpret as good-natured teasing, like <a href=\"https:\/\/twitter.com\/lakens\/status\/1385551960728915970\">this tweet from Dani\u00ebl Lakens<\/a>: &#8220;I always love it when people realize that the main difference between a frequentist and Bayesian analysis is that for the latter approach you first need to wait 24 hours for the results.&#8221;<\/p>\n\n\n\n<p>Ok, that&#8217;s funny, but there is a serious point here I want to respond to because both of these comments are based on the premise that we can compare the results from Bayesian and frequentist methods. And that&#8217;s not just wrong, it is an important misunderstanding. <\/p>\n\n\n\n<p>You can&#8217;t compare results from Bayesian and frequentist methods because the results are different kinds of things. Results from frequentist methods are generally a point estimate, a confidence interval, and\/or a p-value. Each of those results is an answer to a different question:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Point estimate: If I have to pick a single value, which one minimizes a particular cost function under a particular set of constraints? For example, which one minimizes mean squared error while being unbiased?<\/li><li>Confidence interval: If my estimated parameters are correct and I run the experiment again, how much would the results vary due to random sampling?<\/li><li>p-value: If my estimated parameters are wrong and the actual effect size is zero, what is the probability I would see an effect as big as the one I saw?<\/li><\/ul>\n\n\n\n<p>In contrast, the result from Bayesian methods is a <strong>posterior distribution<\/strong>, which is a different kind of thing from a point estimate, an interval, or a probability. It doesn&#8217;t make any sense to say that a distribution is &#8220;the same as&#8221; or &#8220;close to&#8221; a point estimate because there is no meaningful way to compute a distance between those things. It makes as much sense as comparing 1 second and 1 meter.<\/p>\n\n\n\n<p>If you have a posterior distribution and someone asks for a point estimate, you can compute one. In fact, you can compute several, depending on what you want to minimize. And if someone asks for an interval, you can compute one of those, too. In fact, you could compute several, depending on what you want the interval to contain. And if someone really insists, you can compute something like a p-value, too.<\/p>\n\n\n\n<p>But you shouldn&#8217;t.<\/p>\n\n\n\n<p>The posterior distribution represents everything you know about the parameters; if you reduce it to a single number, an interval, or a probability, you lose useful information. In fact, you lose exactly the information that makes the posterior distribution useful in the first place.<\/p>\n\n\n\n<p>It&#8217;s like comparing a car and an airplane by driving the airplane on the road. You would conclude that the airplane is complicated, expensive, and not particularly good as a car. But that would be a silly conclusion because it&#8217;s a silly comparison. The whole point of an airplane is that it can fly.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"http:\/\/worldonline.media.clients.ellingtoncms.com\/img\/photos\/2006\/08\/05\/na_color_EMERGENCY_LANDING1_t440.jpg?9e2a24ba44807f8f9b96aad7c4082bf6ded075dc\" alt=\"\"\/><figcaption>https:\/\/slate.com\/human-interest\/2010\/03\/how-to-land-a-plane-on-a-highway.html<\/figcaption><\/figure>\n\n\n\n<p>And the whole point of Bayesian methods is that a posterior distribution is more useful than a point estimate or an interval because you can use it to guide decision-making under uncertainty.<\/p>\n\n\n\n<p>For example, suppose you compare two drugs and you estimate that one is 90% effective and the other is 95% effective. And let&#8217;s suppose that difference is statistically significant with p=0.04. For the next patient that comes along, which drug should you prescribe?<\/p>\n\n\n\n<p>You might be tempted to prescribe the second drug, which seems to have higher efficacy. However:<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>You are not actually sure it has higher efficacy; it&#8217;s still possible that the first drug is better. If you always prescribe the second drug, you&#8217;ll never know.<\/li><li> Also, point estimates and p-values don&#8217;t help much if one of the drugs is more expensive or has more side effects.<\/li><\/ol>\n\n\n\n<p>With a posterior distribution, you can use a method like <a href=\"https:\/\/en.wikipedia.org\/wiki\/Thompson_sampling\">Thompson sampling<\/a> to balance exploration and exploitation, choosing each drug in proportion to the probability that it is the best. And you can make better decisions by maximizing expected benefits, taking into account whatever factors you can model, including things like cost and side effects (which is not to say that it&#8217;s easy, but it&#8217;s possible).<\/p>\n\n\n\n<p>Bayesian methods answer different questions, provide different kinds of answers, and solve different problems. The results are not the same as frequentist methods, ever.<\/p>\n\n\n\n<p>Conciliatory postscript: If you don&#8217;t need a posterior distribution &#8212; if you just want a point estimate or an interval &#8212; and you conclude that you don&#8217;t need Bayesian methods, that&#8217;s fine. But it&#8217;s not because the results are the same.<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I often hear people say that the results from Bayesian methods are the same as the results from frequentist methods, at least under certain conditions. And sometimes it even comes from people who understand Bayesian methods. Today I saw this tweet from Julia Rohrer: &#8220;Running a Bayesian multi-membership multi-level probit model with a custom function to generate average marginal effects only to find that the estimate is precisely the same as the one generated by linear regression with dummy-coded group&#8230;<\/p>\n<p class=\"read-more\"><a class=\"btn btn-default\" href=\"https:\/\/www.allendowney.com\/blog\/2021\/04\/25\/bayesian-and-frequentist-results-are-not-the-same-ever\/\"> Read More<span class=\"screen-reader-text\">  Read More<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[1],"tags":[9,83],"class_list":["post-569","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-bayesian","tag-frequentist"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Bayesian and frequentist results are not the same, ever - Probably Overthinking It<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.allendowney.com\/blog\/2021\/04\/25\/bayesian-and-frequentist-results-are-not-the-same-ever\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Bayesian and frequentist results are not the same, ever - Probably Overthinking It\" \/>\n<meta property=\"og:description\" content=\"I often hear people say that the results from Bayesian methods are the same as the results from frequentist methods, at least under certain conditions. 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Eric Ma served as moderator, introducing me and joining me to answer questions at the end. The example I presented is an updated version of\u2026","rel":"","context":"In \"bayesian\"","block_context":{"text":"bayesian","link":"https:\/\/www.allendowney.com\/blog\/tag\/bayesian\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":883,"url":"https:\/\/www.allendowney.com\/blog\/2023\/03\/20\/the-bayesian-killer-app\/","url_meta":{"origin":569,"position":1},"title":"The Bayesian Killer App","author":"AllenDowney","date":"March 20, 2023","format":false,"excerpt":"It's been a while since anyone said \"killer app\" without irony, so let me remind you that a killer app is software \"so necessary or desirable that it proves the core value of some larger technology,\" quoth Wikipedia. For example, most people didn't have much use for the internet until\u2026","rel":"","context":"Similar post","block_context":{"text":"Similar post","link":""},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/img.youtube.com\/vi\/fsdbneHgi58\/0.jpg?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":1551,"url":"https:\/\/www.allendowney.com\/blog\/2025\/05\/22\/my-very-busy-week\/","url_meta":{"origin":569,"position":2},"title":"My very busy week","author":"AllenDowney","date":"May 22, 2025","format":false,"excerpt":"I'm not sure who scheduled ODSC and PyConUS during the same week, but I am unhappy with their decisions. Last Tuesday I presented a talk and co-presented a workshop at ODSC, and on Thursday I presented a tutorial at PyCon. If you would like to follow along with my very\u2026","rel":"","context":"In \"bayesian statistics\"","block_context":{"text":"bayesian statistics","link":"https:\/\/www.allendowney.com\/blog\/tag\/bayesian-statistics\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/img.youtube.com\/vi\/foMbacbuAQk\/0.jpg?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":609,"url":"https:\/\/www.allendowney.com\/blog\/2021\/05\/07\/founded-upon-an-error\/","url_meta":{"origin":569,"position":3},"title":"Founded Upon an Error","author":"AllenDowney","date":"May 7, 2021","format":false,"excerpt":"A recent post on Reddit asks, \"Why was Bayes' Theory not accepted\/popular historically until the late 20th century?\" Great question! As always, there are many answers to a question like this, and the good people of Reddit provide several. 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If you would like to learn more, it\u2026","rel":"","context":"In \"bayesian statistics\"","block_context":{"text":"bayesian statistics","link":"https:\/\/www.allendowney.com\/blog\/tag\/bayesian-statistics\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/img.youtube.com\/vi\/PLGVZCDnMOq0qmerwB1eITnr5AfYRGm0DF\/0.jpg?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":1684,"url":"https:\/\/www.allendowney.com\/blog\/2025\/12\/16\/sat-math-scores-gender-difference-or-selection-bias\/","url_meta":{"origin":569,"position":5},"title":"SAT math scores: gender difference or selection bias?","author":"AllenDowney","date":"December 16, 2025","format":false,"excerpt":"The video from my PyData Boston talk is up now: https:\/\/www.youtube.com\/watch?v=6pwtbNVgyzg Resources The slides are here Run the first notebook (Poincar\u00e9 problem) on Colab Run the second notebook (analysis of SAT data) on Colab If you want to learn to do this kind of analysis, you can sign up for\u2026","rel":"","context":"In \"bayesian statistics\"","block_context":{"text":"bayesian statistics","link":"https:\/\/www.allendowney.com\/blog\/tag\/bayesian-statistics\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/img.youtube.com\/vi\/6pwtbNVgyzg\/0.jpg?resize=350%2C200","width":350,"height":200},"classes":[]}],"_links":{"self":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts\/569","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/comments?post=569"}],"version-history":[{"count":2,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts\/569\/revisions"}],"predecessor-version":[{"id":571,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts\/569\/revisions\/571"}],"wp:attachment":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/media?parent=569"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/categories?post=569"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/tags?post=569"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}