# Bayes’ Theorem

TagLast edit: 19 Feb 2024 5:40 UTC by

Bayes’ Theorem (also known as Bayes’ Law) is a law of probability that describes the proper way to incorporate new evidence into prior probabilities to form an updated probability estimate. It is commonly regarded as the foundation of consistent rational reasoning under uncertainty. Bayes Theorem is named after Reverend Thomas Bayes who proved the theorem in 1763.

Bayes’ theorem commonly takes the form:

where A is the proposition of interest, B is the observed evidence, P(A) and P(B) are prior probabilities, and P(A|B) is the posterior probability of A.

With the posterior odds, the prior odds and the likelihood ratio written explicitly, the theorem reads:

# An In­tu­itive Ex­pla­na­tion of Bayes’s Theorem

1 Jan 2003 20:00 UTC
64 points

18 Aug 2020 21:14 UTC
176 points

# A Tech­ni­cal Ex­pla­na­tion of Tech­ni­cal Explanation

1 Jan 2005 8:00 UTC
68 points

# Bayes’ The­o­rem Illus­trated (My Way)

3 Jun 2010 4:40 UTC
171 points

# Anec­dotes Can Be Strong Ev­i­dence and Bayes The­o­rem Proves It

13 Mar 2022 4:06 UTC
16 points

# The Equa­tion of Knowledge

7 Jul 2020 16:09 UTC
58 points

# The Moun­tain Troll

11 Jun 2022 9:14 UTC
95 points

# a vi­sual ex­pla­na­tion of Bayesian updating

8 May 2021 19:45 UTC
20 points

# It’s Bayes All The Way Up

12 Sep 2016 13:35 UTC
32 points

# Strong Ev­i­dence is Common

13 Mar 2021 22:04 UTC
244 points
(markxu.com)

# Towards a Bayesian model for Em­piri­cal Science

7 Oct 2021 5:38 UTC
11 points

# Bayesian Utility: Rep­re­sent­ing Prefer­ence by Prob­a­bil­ity Measures

27 Jul 2009 14:28 UTC
48 points

22 Nov 2021 10:39 UTC
79 points

# E.T. Jaynes Prob­a­bil­ity The­ory: The logic of Science I

27 Dec 2023 23:47 UTC
60 points

# The Bayesian Tyrant

20 Aug 2020 0:08 UTC
140 points

# Why Bayes? A Wise Ruling

25 Feb 2013 15:52 UTC
18 points

# Bayesian Evolv­ing-to-Extinction

14 Feb 2020 23:55 UTC
39 points

# Bayes for Schizophren­ics: Rea­son­ing in Delu­sional Disorders

13 Aug 2012 19:22 UTC
151 points

# What is Bayesi­anism?

26 Feb 2010 7:43 UTC
115 points

# Learn Bayes Nets!

27 Mar 2018 22:00 UTC
50 points

# Bayes’ Law is About Mul­ti­ple Hy­poth­e­sis Testing

4 May 2018 5:31 UTC
36 points

# Non-com­mu­ni­ca­ble Evidence

17 Nov 2015 3:46 UTC
17 points

# Bayesian examination

9 Dec 2019 19:50 UTC
86 points

# What can we learn from Bayes about rea­son­ing?

5 May 2023 15:52 UTC
21 points

# Bayesian Prob­a­bil­ity is for things that are Space-like Separated from You

10 Jul 2018 23:47 UTC
80 points

# What Bayesi­anism taught me

12 Aug 2013 6:59 UTC
115 points

# Bayes’ the­o­rem, plau­si­ble de­ni­a­bil­ity, and smiley faces

11 Apr 2021 20:41 UTC
24 points

# Kal­man Filter for Bayesians

22 Oct 2018 17:06 UTC
51 points

# Bayesi­anism for Humans

29 Oct 2013 23:54 UTC
91 points

# Black ravens and red herrings

27 Jul 2021 17:46 UTC
50 points

# Qual­i­ta­tively Confused

14 Mar 2008 17:01 UTC
59 points

# Against strong bayesianism

30 Apr 2020 10:48 UTC
58 points

# Base Rates and Refer­ence Classes

24 Nov 2021 22:30 UTC
20 points
(bounded-regret.ghost.io)

# Bayes’ The­o­rem in three pictures

21 Jul 2019 7:01 UTC
30 points

# [Question] How do Bayesi­ans tell what does and doesn’t count as ev­i­dence (which, e.g., hy­pothe­ses may ren­der more or less prob­a­ble if true)? Is it pos­si­ble for some­thing to fuzzily-be ev­i­dence?

30 Nov 2021 12:52 UTC
2 points

# My Bayesian Enlightenment

5 Oct 2008 16:45 UTC
67 points

# Beau­tiful Probability

14 Jan 2008 7:19 UTC
102 points

# Ep­silon is not a prob­a­bil­ity, it’s a cop-out

15 Feb 2022 2:48 UTC
23 points

# The Case for Fre­quen­tism: Why Bayesian Prob­a­bil­ity is Fun­da­men­tally Un­sound and What Science Does Instead

3 Apr 2022 20:52 UTC
22 points

# De­co­her­ence is Falsifi­able and Testable

7 May 2008 7:54 UTC
44 points

# Re­port like­li­hood ratios

23 Apr 2022 17:10 UTC
80 points

# Un­der­stand­ing Sub­jec­tive Probabilities

10 Dec 2023 6:03 UTC
30 points

# Bayes’ rule =/​= Bayesian inference

16 Sep 2010 6:34 UTC
49 points

# [Link] New SEP ar­ti­cle on Bayesian Epistemology

13 Jun 2022 15:03 UTC
6 points

# Prob­a­bil­ity Space & Au­mann Agreement

10 Dec 2009 21:57 UTC
52 points

# Au­mann’s Agree­ment Revisited

27 Aug 2018 6:21 UTC
4 points

# The Me­chan­ics of Disagreement

10 Dec 2008 14:01 UTC
14 points

# “Friends do not let friends com­pute p val­ues.”

9 Sep 2011 7:10 UTC
15 points

# The Joys of Con­ju­gate Priors

21 May 2011 2:41 UTC
63 points

# Bayesi­anism for hu­mans: “prob­a­ble enough”

2 Sep 2014 21:44 UTC
52 points

# Nav­i­gat­ing dis­agree­ment: How to keep your eye on the ev­i­dence

24 Apr 2010 22:47 UTC
47 points

# When ap­par­ently pos­i­tive ev­i­dence can be nega­tive evidence

20 Oct 2022 21:47 UTC
19 points
(www.ncbi.nlm.nih.gov)

# If It’s Worth Do­ing, It’s Worth Do­ing With Made-Up Statistics

3 Sep 2017 20:56 UTC
72 points

# Creat­ing a database for base rates

12 Dec 2022 10:09 UTC
2 points
(forum.effectivealtruism.org)

# Beyond Bayesi­ans and Frequentists

31 Oct 2012 7:03 UTC
55 points

# The role of Bayesian ML in AI safety—an overview

27 Jan 2023 19:40 UTC
30 points

# Prob­a­bil­ity The­ory: The Logic of Science, Jaynes

16 Feb 2023 21:57 UTC
29 points

# Bayesian Sce­nario: Snipers & Soldiers

26 Feb 2023 21:48 UTC
23 points
(h-b-p.github.io)

# Fun­da­men­tal Uncer­tainty: Chap­ter 3 - Why don’t we agree on what’s right?

25 Jun 2022 17:50 UTC
27 points

# Bayes Rule Applied

16 Feb 2018 18:30 UTC
4 points
(towardsdatascience.com)

# [Question] al­ter­na­tive his­tory: what if Bayes rule had never been dis­cov­ered?

11 May 2019 7:29 UTC
7 points

# nos­talge­braist—bayes: a kinda-sorta masterpost

4 Sep 2018 11:08 UTC
25 points
(nostalgebraist.tumblr.com)

# Fal­la­cies as weak Bayesian evidence

18 Mar 2012 3:53 UTC
87 points

# An­thropic rea­son­ing isn’t magic

1 Nov 2017 8:57 UTC
22 points

# Kelly bettors

13 Nov 2018 0:40 UTC
24 points
(danielfilan.com)

# Ab­solute Authority

8 Jan 2008 3:33 UTC
98 points

# In­ter­ac­tive Bayes The­o­rem Visualization

3 Mar 2018 4:00 UTC
14 points
(allenkim67.github.io)

# Very Short In­tro­duc­tion to Bayesian Model Com­par­i­son

16 Jul 2019 19:48 UTC
32 points

# Yes, Virginia, You Can Be 99.99% (Or More!) Cer­tain That 53 Is Prime

7 Nov 2013 7:45 UTC
95 points

# Are Bayesian meth­ods guaran­teed to overfit?

17 Jun 2023 12:52 UTC
52 points
(www.yulingyao.com)

# A His­tory of Bayes’ Theorem

29 Aug 2011 7:04 UTC
75 points

26 Jun 2020 22:14 UTC
48 points

# Mini-re­view: ‘Prov­ing His­tory: Bayes’ The­o­rem and the Quest for the His­tor­i­cal Je­sus’

1 Feb 2012 19:20 UTC
26 points

# In­tro­duc­ing bayescalc.io

7 Jul 2023 16:11 UTC
114 points
(bayescalc.io)

# Search­ing for Bayes-Structure

28 Feb 2008 22:01 UTC
60 points

# How to Mea­sure Anything

7 Aug 2013 4:05 UTC
118 points

# Refer­ences & Re­sources for LessWrong

10 Oct 2010 14:54 UTC
162 points

# Bayes Questions

7 Nov 2018 16:54 UTC
21 points

# [Question] What are prin­ci­pled ways for pe­nal­is­ing com­plex­ity in prac­tice?

27 Jun 2019 7:28 UTC
36 points

# [Linkpost] Prac­ti­cally-A-Book Re­view: Root­claim \$100,000 Lab Leak Debate

28 Mar 2024 16:03 UTC
66 points
(www.astralcodexten.com)

# Prob­a­bil­ity is in the Mind

12 Mar 2008 4:08 UTC
129 points

# Boltz­mann brain’s con­di­tional probability

29 Dec 2023 14:44 UTC
5 points

# Bayesi­ans Com­mit the Gam­bler’s Fallacy

7 Jan 2024 12:54 UTC
46 points
(kevindorst.substack.com)

# How do you ac­tu­ally ob­tain and re­port a like­li­hood func­tion for sci­en­tific re­search?

11 Feb 2024 17:42 UTC
55 points

# Not us­ing a pri­ori in­for­ma­tion for Rus­sian propaganda

24 Apr 2023 1:14 UTC
0 points

# Born as the sev­enth month dies …

10 Jul 2020 15:07 UTC
6 points

# [Question] Can Bayes the­o­rem rep­re­sent in­finite con­fu­sion?

22 Mar 2019 18:02 UTC
4 points

# Why We Can’t Take Ex­pected Value Es­ti­mates Liter­ally (Even When They’re Un­bi­ased)

18 Aug 2011 23:34 UTC
123 points

# Fre­quen­tist Statis­tics are Fre­quently Subjective

4 Dec 2009 20:22 UTC
86 points

# Dou­ble Illu­sion of Transparency

24 Oct 2007 23:06 UTC
113 points

# Against NHST

21 Dec 2012 4:45 UTC
93 points

# In­ter­pre­ta­tions of “prob­a­bil­ity”

9 May 2019 19:16 UTC
68 points

# Ein­stein’s Arrogance

25 Sep 2007 1:29 UTC
153 points

# A Fer­vent Defense of Fre­quen­tist Statistics

18 Feb 2014 20:08 UTC
75 points

# In­finite Certainty

9 Jan 2008 6:49 UTC
77 points

# Fre­quen­tist Magic vs. Bayesian Magic

8 Apr 2010 20:34 UTC
58 points

# How Much Ev­i­dence Does It Take?

24 Sep 2007 4:06 UTC
126 points

# Multiplicitous

18 Dec 2016 16:39 UTC
9 points
(putanumonit.com)

# Bayes Academy: Devel­op­ment re­port 1

19 Nov 2014 22:35 UTC
68 points

# Dreams with Da­m­aged Priors

8 Aug 2009 22:31 UTC
59 points

# [Question] Bayesian up­date from sen­sa­tion­al­is­tic sources

11 May 2023 15:26 UTC
1 point

# Hearsay, Dou­ble Hearsay, and Bayesian Updates

16 Feb 2012 22:31 UTC
68 points

# Prob­a­bil­ity is Sub­jec­tively Objective

14 Jul 2008 9:16 UTC
41 points

# Pri­ors as Math­e­mat­i­cal Objects

12 Apr 2007 3:24 UTC
51 points

# Bayesian Flame

26 Jul 2009 16:49 UTC
41 points

# [Funny] Even Clippy can be blamed on the use of non-Bayesian methods

2 Oct 2011 7:40 UTC
41 points

# When (Not) To Use Probabilities

23 Jul 2008 10:58 UTC
71 points

# The Quick Bayes Table

18 Apr 2012 18:00 UTC
60 points

# In­tro­spec­tive Bayes

27 May 2023 19:35 UTC
−5 points

# De­mands for Par­tic­u­lar Proof: Appendices

15 Feb 2010 7:58 UTC
39 points

# (Sub­jec­tive Bayesi­anism vs. Fre­quen­tism) VS. For­mal­ism

26 Nov 2011 5:05 UTC
32 points

# Laplace Approximation

18 Jul 2019 15:23 UTC
29 points

# Knigh­tian un­cer­tainty in a Bayesian framework

24 Jul 2014 14:31 UTC
54 points

# In­her­ited Im­prob­a­bil­ities: Trans­fer­ring the Bur­den of Proof

24 Nov 2010 3:40 UTC
46 points

# Knigh­tian un­cer­tainty: a re­jec­tion of the MMEU rule

26 Aug 2014 3:03 UTC
41 points

# The prior of a hy­poth­e­sis does not de­pend on its complexity

26 Aug 2010 13:20 UTC
34 points

# RFC on an open prob­lem: how to de­ter­mine prob­a­bil­ities in the face of so­cial distortion

7 Oct 2017 22:04 UTC
6 points

# In­tro­duc­tion To The In­fra-Bayesi­anism Sequence

26 Aug 2020 20:31 UTC
108 points

# Fre­quen­tist vs Bayesian break­down: in­ter­pre­ta­tion vs inference

30 Aug 2011 15:58 UTC
36 points

# [Question] Why is Bayesi­anism im­por­tant for ra­tio­nal­ity?

1 Sep 2020 4:24 UTC
37 points

# Weak Ev­i­dence is Common

16 Jul 2023 23:37 UTC
7 points
(dkl9.net)

# Embed­ding Eth­i­cal Pri­ors into AI Sys­tems: A Bayesian Approach

3 Aug 2023 15:31 UTC
−5 points

# Evolved Bayesi­ans will be biased

20 Aug 2009 14:54 UTC
28 points

# Ap­plied Bayes’ The­o­rem: Read­ing People

30 Jun 2010 17:21 UTC
37 points

# Lev­er­ag­ing Bayes’ The­o­rem to Su­per­charge Me­mory Techniques

9 Oct 2023 3:34 UTC
−15 points

# Dutch Books and De­ci­sion The­ory: An In­tro­duc­tion to a Long Conversation

21 Dec 2010 4:55 UTC
30 points

# LINK: Bayesian statis­tics is so sub­ver­sive, it’s banned in China!

11 Jan 2011 0:50 UTC
4 points

# The med­i­cal test para­dox: Can re­design­ing Bayes rule help?

24 Dec 2020 22:08 UTC
34 points

# Con­ser­va­tion of Ex­pected Evidence

13 Aug 2007 15:55 UTC
241 points

# [Question] What’s the big deal about Bayes’ The­o­rem?

26 Jan 2021 6:08 UTC
21 points

# Bayesian in­fer­ence on 1st or­der logic

3 Feb 2021 1:03 UTC
11 points

# An In­tu­itive Ex­pla­na­tion of Eliezer Yud­kowsky’s In­tu­itive Ex­pla­na­tion of Bayes’ Theorem

18 Dec 2010 13:26 UTC
24 points

# Par­ti­ci­pat­ing in a Covid-19 Vac­cine Trial #3: I Hope I feel Worse Tomorrow

5 Mar 2021 0:02 UTC
8 points

11 Mar 2021 8:11 UTC
18 points

# The Ja­panese Quiz: a Thought Ex­per­i­ment of Statis­ti­cal Epistemology

8 Apr 2021 17:37 UTC
11 points

# Book Re­view of 5 Ap­plied Bayesian Statis­tics Books

21 May 2021 10:23 UTC
66 points

# Dooms­day, Sam­pling As­sump­tions, and Bayes

14 Jun 2021 12:41 UTC
6 points

# In­ter­pret­ing Di­ag­nos­tic Tests

15 Oct 2021 2:58 UTC
6 points

# XKCD—Fre­quen­tist vs. Bayesians

9 Nov 2012 5:25 UTC
24 points

# Find­ing the Cen­tral Limit The­o­rem in Bayes’ rule

27 Nov 2021 5:48 UTC
23 points

# Buck­ets & Bayes

20 Dec 2021 5:54 UTC
34 points

# [Question] How would you ex­plain Bayesian think­ing to a ten year old?

5 Jan 2022 17:25 UTC
7 points

# Ret­ro­spec­tive forecasting

30 Jan 2022 16:38 UTC
22 points

# Com­men­su­rable Scien­tific Paradigms; or, com­putable induction

13 Apr 2022 0:01 UTC
14 points

# RL with KL penalties is bet­ter seen as Bayesian inference

25 May 2022 9:23 UTC
114 points

# How to Vi­su­al­ize Bayesianism

22 Jun 2022 13:57 UTC
9 points

# King David’s %: Estab­lish­ing a new sym­bol for Bayesian prob­a­bil­ity.

26 Jun 2022 19:47 UTC
−11 points
(laulpogan.substack.com)

# Five views of Bayes’ Theorem

2 Jul 2022 2:25 UTC
38 points

# An at­tempt to break cir­cu­lar­ity in science

15 Jul 2022 18:32 UTC
3 points

# Bayesian prob­a­bil­ity the­ory as ex­tended logic—a new result

6 Jul 2017 19:14 UTC
37 points

# Think­ing with­out pri­ors?

2 Aug 2022 9:17 UTC
7 points

# Mak­ing Vaccine

3 Feb 2021 20:24 UTC
574 points

# Log-odds are bet­ter than Probabilities

12 Dec 2022 20:10 UTC
22 points