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Bayes’ Theorem

TagLast edit: 21 Sep 2020 1:26 UTC by Ruby

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.

See also: Bayesian probability, Priors, Likelihood ratio, Belief update, Probability and statistics, Epistemology, Bayesianism

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:

Visualization of Bayes’ Rule

External links

An In­tu­itive Ex­pla­na­tion of Bayes’s Theorem

Eliezer Yudkowsky1 Jan 2003 20:00 UTC
49 points
8 comments25 min readLW link

Rad­i­cal Probabilism

abramdemski18 Aug 2020 21:14 UTC
154 points
47 comments35 min readLW link1 review

A Tech­ni­cal Ex­pla­na­tion of Tech­ni­cal Explanation

Eliezer Yudkowsky1 Jan 2005 8:00 UTC
60 points
7 comments57 min readLW link

The Equa­tion of Knowledge

Lê Nguyên Hoang7 Jul 2020 16:09 UTC
58 points
3 comments7 min readLW link

Bayes’ The­o­rem Illus­trated (My Way)

komponisto3 Jun 2010 4:40 UTC
161 points
195 comments9 min readLW link

It’s Bayes All The Way Up

Scott Alexander12 Sep 2016 13:35 UTC
29 points
0 comments12 min readLW link

a vi­sual ex­pla­na­tion of Bayesian updating

Jan Christian Refsgaard8 May 2021 19:45 UTC
20 points
7 comments3 min readLW link
(www.badprior.com)

Anec­dotes Can Be Strong Ev­i­dence and Bayes The­o­rem Proves It

FCCC13 Mar 2022 4:06 UTC
14 points
3 comments4 min readLW link

The Moun­tain Troll

lsusr11 Jun 2022 9:14 UTC
65 points
14 comments2 min readLW link

Strong Ev­i­dence is Common

Mark Xu13 Mar 2021 22:04 UTC
162 points
42 comments1 min readLW link
(markxu.com)

Towards a Bayesian model for Em­piri­cal Science

Yair Halberstadt7 Oct 2021 5:38 UTC
11 points
7 comments7 min readLW link

A Bayesian Ag­gre­ga­tion Paradox

Jsevillamol22 Nov 2021 10:39 UTC
72 points
21 comments7 min readLW link

Bayes for Schizophren­ics: Rea­son­ing in Delu­sional Disorders

Scott Alexander13 Aug 2012 19:22 UTC
144 points
155 comments11 min readLW link

What is Bayesi­anism?

Kaj_Sotala26 Feb 2010 7:43 UTC
111 points
217 comments4 min readLW link

Learn Bayes Nets!

abramdemski27 Mar 2018 22:00 UTC
46 points
7 comments2 min readLW link

Bayes’ Law is About Mul­ti­ple Hy­poth­e­sis Testing

abramdemski4 May 2018 5:31 UTC
36 points
5 comments5 min readLW link

Bayesian examination

Lê Nguyên Hoang9 Dec 2019 19:50 UTC
86 points
56 comments5 min readLW link

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

Scott Garrabrant10 Jul 2018 23:47 UTC
77 points
22 comments2 min readLW link

What Bayesi­anism taught me

Tyrrell_McAllister12 Aug 2013 6:59 UTC
111 points
204 comments3 min readLW link

Kal­man Filter for Bayesians

SatvikBeri22 Oct 2018 17:06 UTC
50 points
6 comments3 min readLW link

Bayesi­anism for Humans

ChrisHallquist29 Oct 2013 23:54 UTC
91 points
37 comments3 min readLW link

Against strong bayesianism

Richard_Ngo30 Apr 2020 10:48 UTC
57 points
65 comments6 min readLW link

Bayes’ rule =/​= Bayesian inference

neq116 Sep 2010 6:34 UTC
49 points
70 comments2 min readLW link

Bayes Rule Applied

G Gordon Worley III16 Feb 2018 18:30 UTC
4 points
0 comments1 min readLW link
(towardsdatascience.com)

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

Yoav Ravid11 May 2019 7:29 UTC
7 points
3 comments1 min readLW link

nos­talge­braist—bayes: a kinda-sorta masterpost

Kaj_Sotala4 Sep 2018 11:08 UTC
25 points
19 comments1 min readLW link
(nostalgebraist.tumblr.com)

Fal­la­cies as weak Bayesian evidence

Kaj_Sotala18 Mar 2012 3:53 UTC
85 points
42 comments10 min readLW link

An­thropic rea­son­ing isn’t magic

Stuart_Armstrong1 Nov 2017 8:57 UTC
22 points
3 comments2 min readLW link

Kelly bettors

DanielFilan13 Nov 2018 0:40 UTC
23 points
3 comments10 min readLW link
(danielfilan.com)

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

ChrisHallquist7 Nov 2013 7:45 UTC
68 points
70 comments6 min readLW link

A His­tory of Bayes’ Theorem

lukeprog29 Aug 2011 7:04 UTC
74 points
90 comments15 min readLW link

Rad­i­cal Prob­a­bil­ism [Tran­script]

26 Jun 2020 22:14 UTC
46 points
12 comments6 min readLW link

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

lukeprog1 Feb 2012 19:20 UTC
27 points
34 comments1 min readLW link

Search­ing for Bayes-Structure

Eliezer Yudkowsky28 Feb 2008 22:01 UTC
58 points
48 comments5 min readLW link

How to Mea­sure Anything

lukeprog7 Aug 2013 4:05 UTC
100 points
52 comments22 min readLW link

Refer­ences & Re­sources for LessWrong

XiXiDu10 Oct 2010 14:54 UTC
151 points
106 comments20 min readLW link

Bayes Questions

Bucky7 Nov 2018 16:54 UTC
21 points
13 comments2 min readLW link

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

Bucky27 Jun 2019 7:28 UTC
36 points
12 comments1 min readLW link

Prob­a­bil­ity is in the Mind

Eliezer Yudkowsky12 Mar 2008 4:08 UTC
116 points
194 comments6 min readLW link

Bayesian Evolv­ing-to-Extinction

abramdemski14 Feb 2020 23:55 UTC
38 points
13 comments5 min readLW link

Non-com­mu­ni­ca­ble Evidence

adamzerner17 Nov 2015 3:46 UTC
17 points
49 comments2 min readLW link

Qual­i­ta­tively Confused

Eliezer Yudkowsky14 Mar 2008 17:01 UTC
53 points
82 comments4 min readLW link

Bayes’ The­o­rem in three pictures

Sunny from QAD21 Jul 2019 7:01 UTC
30 points
3 comments4 min readLW link

My Bayesian Enlightenment

Eliezer Yudkowsky5 Oct 2008 16:45 UTC
54 points
65 comments7 min readLW link

Beau­tiful Probability

Eliezer Yudkowsky14 Jan 2008 7:19 UTC
81 points
122 comments6 min readLW link

De­co­her­ence is Falsifi­able and Testable

Eliezer Yudkowsky7 May 2008 7:54 UTC
40 points
41 comments9 min readLW link

Prob­a­bil­ity Space & Au­mann Agreement

Wei_Dai10 Dec 2009 21:57 UTC
52 points
76 comments5 min readLW link

Au­mann’s Agree­ment Revisited

agilecaveman27 Aug 2018 6:21 UTC
4 points
1 comment7 min readLW link

The Me­chan­ics of Disagreement

Eliezer Yudkowsky10 Dec 2008 14:01 UTC
13 points
26 comments4 min readLW link

The Joys of Con­ju­gate Priors

TCB21 May 2011 2:41 UTC
63 points
24 comments5 min readLW link

Bayesi­anism for hu­mans: “prob­a­ble enough”

BT_Uytya2 Sep 2014 21:44 UTC
52 points
7 comments4 min readLW link

Nav­i­gat­ing dis­agree­ment: How to keep your eye on the ev­i­dence

AnnaSalamon24 Apr 2010 22:47 UTC
47 points
73 comments6 min readLW link

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

Scott Alexander3 Sep 2017 20:56 UTC
54 points
0 comments5 min readLW link

Beyond Bayesi­ans and Frequentists

jsteinhardt31 Oct 2012 7:03 UTC
53 points
51 comments11 min readLW link

Ab­solute Authority

Eliezer Yudkowsky8 Jan 2008 3:33 UTC
83 points
76 comments7 min readLW link

Very Short In­tro­duc­tion to Bayesian Model Com­par­i­son

johnswentworth16 Jul 2019 19:48 UTC
30 points
5 comments1 min readLW link

In­ter­ac­tive Bayes The­o­rem Visualization

Allen Kim3 Mar 2018 4:00 UTC
14 points
5 comments1 min readLW link
(allenkim67.github.io)

The Bayesian Tyrant

abramdemski20 Aug 2020 0:08 UTC
131 points
17 comments6 min readLW link1 review

Why Bayes? A Wise Ruling

Vaniver25 Feb 2013 15:52 UTC
18 points
117 comments2 min readLW link

Bayes’ the­o­rem, plau­si­ble de­ni­a­bil­ity, and smiley faces

adamzerner11 Apr 2021 20:41 UTC
24 points
9 comments4 min readLW link

Black ravens and red herrings

Matthew Barnett27 Jul 2021 17:46 UTC
50 points
17 comments1 min readLW link

Good books on Bayesian statistics

blackstampede26 Aug 2021 11:02 UTC
1 point
6 comments1 min readLW link

Base Rates and Refer­ence Classes

jsteinhardt24 Nov 2021 22:30 UTC
19 points
7 comments5 min readLW link
(bounded-regret.ghost.io)

[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?

LVSN30 Nov 2021 12:52 UTC
2 points
10 comments1 min readLW link

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

shminux15 Feb 2022 2:48 UTC
22 points
35 comments1 min readLW link

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

lsusr3 Apr 2022 20:52 UTC
23 points
28 comments6 min readLW link

Re­port like­li­hood ratios

Ege Erdil23 Apr 2022 17:10 UTC
80 points
9 comments7 min readLW link

[Link] New SEP ar­ti­cle on Bayesian Epistemology

Aryeh Englander13 Jun 2022 15:03 UTC
6 points
0 comments1 min readLW link

“Friends do not let friends com­pute p val­ues.”

lukeprog9 Sep 2011 7:10 UTC
15 points
4 comments1 min readLW link

Born as the sev­enth month dies …

Rudi C10 Jul 2020 15:07 UTC
6 points
8 comments2 min readLW link

[Question] Can Bayes the­o­rem rep­re­sent in­finite con­fu­sion?

Yoav Ravid22 Mar 2019 18:02 UTC
4 points
13 comments1 min readLW link

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

HoldenKarnofsky18 Aug 2011 23:34 UTC
121 points
252 comments17 min readLW link

Fre­quen­tist Statis­tics are Fre­quently Subjective

Eliezer Yudkowsky4 Dec 2009 20:22 UTC
82 points
82 comments8 min readLW link

Dou­ble Illu­sion of Transparency

Eliezer Yudkowsky24 Oct 2007 23:06 UTC
105 points
33 comments3 min readLW link

Against NHST

gwern21 Dec 2012 4:45 UTC
92 points
64 comments4 min readLW link

In­ter­pre­ta­tions of “prob­a­bil­ity”

So8res9 May 2019 19:16 UTC
68 points
22 comments5 min readLW link

Ein­stein’s Arrogance

Eliezer Yudkowsky25 Sep 2007 1:29 UTC
119 points
89 comments3 min readLW link

A Fer­vent Defense of Fre­quen­tist Statistics

jsteinhardt18 Feb 2014 20:08 UTC
72 points
129 comments16 min readLW link

In­finite Certainty

Eliezer Yudkowsky9 Jan 2008 6:49 UTC
66 points
128 comments4 min readLW link

Fre­quen­tist Magic vs. Bayesian Magic

Wei_Dai8 Apr 2010 20:34 UTC
58 points
79 comments3 min readLW link

How Much Ev­i­dence Does It Take?

Eliezer Yudkowsky24 Sep 2007 4:06 UTC
94 points
32 comments4 min readLW link

Bayes Academy: Devel­op­ment re­port 1

Kaj_Sotala19 Nov 2014 22:35 UTC
68 points
28 comments8 min readLW link

Multiplicitous

Jacob Falkovich18 Dec 2016 16:39 UTC
9 points
0 comments12 min readLW link
(putanumonit.com)

Dreams with Da­m­aged Priors

Eliezer Yudkowsky8 Aug 2009 22:31 UTC
52 points
61 comments3 min readLW link

Hearsay, Dou­ble Hearsay, and Bayesian Updates

Mass_Driver16 Feb 2012 22:31 UTC
68 points
107 comments6 min readLW link

Prob­a­bil­ity is Sub­jec­tively Objective

Eliezer Yudkowsky14 Jul 2008 9:16 UTC
37 points
71 comments11 min readLW link

Pri­ors as Math­e­mat­i­cal Objects

Eliezer Yudkowsky12 Apr 2007 3:24 UTC
50 points
19 comments4 min readLW link

Bayesian Flame

cousin_it26 Jul 2009 16:49 UTC
41 points
163 comments4 min readLW link

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

lukeprog2 Oct 2011 7:40 UTC
41 points
46 comments1 min readLW link

When (Not) To Use Probabilities

Eliezer Yudkowsky23 Jul 2008 10:58 UTC
46 points
44 comments6 min readLW link

The Quick Bayes Table

farsan18 Apr 2012 18:00 UTC
57 points
31 comments2 min readLW link

Bayesian Utility: Rep­re­sent­ing Prefer­ence by Prob­a­bil­ity Measures

Vladimir_Nesov27 Jul 2009 14:28 UTC
44 points
36 comments2 min readLW link

De­mands for Par­tic­u­lar Proof: Appendices

Eliezer Yudkowsky15 Feb 2010 7:58 UTC
34 points
70 comments7 min readLW link

(Sub­jec­tive Bayesi­anism vs. Fre­quen­tism) VS. For­mal­ism

Ronny26 Nov 2011 5:05 UTC
32 points
109 comments5 min readLW link

Laplace Approximation

johnswentworth18 Jul 2019 15:23 UTC
28 points
3 comments3 min readLW link

Knigh­tian un­cer­tainty in a Bayesian framework

So8res24 Jul 2014 14:31 UTC
39 points
2 comments11 min readLW link

In­her­ited Im­prob­a­bil­ities: Trans­fer­ring the Bur­den of Proof

komponisto24 Nov 2010 3:40 UTC
45 points
58 comments8 min readLW link

Knigh­tian un­cer­tainty: a re­jec­tion of the MMEU rule

So8res26 Aug 2014 3:03 UTC
40 points
9 comments15 min readLW link

The prior of a hy­poth­e­sis does not de­pend on its complexity

cousin_it26 Aug 2010 13:20 UTC
34 points
69 comments1 min readLW link

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

ialdabaoth7 Oct 2017 22:04 UTC
6 points
3 comments2 min readLW link

In­tro­duc­tion To The In­fra-Bayesi­anism Sequence

Diffractor27 Aug 2020 8:02 UTC
92 points
64 comments13 min readLW link2 reviews

Fre­quen­tist vs Bayesian break­down: in­ter­pre­ta­tion vs inference

Academian30 Aug 2011 15:58 UTC
36 points
4 comments2 min readLW link

[Question] Why is Bayesi­anism im­por­tant for ra­tio­nal­ity?

Chris_Leong1 Sep 2020 4:24 UTC
37 points
24 comments1 min readLW link

Evolved Bayesi­ans will be biased

taw20 Aug 2009 14:54 UTC
28 points
14 comments2 min readLW link

Ap­plied Bayes’ The­o­rem: Read­ing People

Kaj_Sotala30 Jun 2010 17:21 UTC
36 points
26 comments8 min readLW link

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

Jack21 Dec 2010 4:55 UTC
29 points
102 comments7 min readLW link

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

lukeprog11 Jan 2011 0:50 UTC
4 points
9 comments1 min readLW link

The med­i­cal test para­dox: Can re­design­ing Bayes rule help?

habryka24 Dec 2020 22:08 UTC
34 points
2 comments1 min readLW link
(www.youtube.com)

Con­ser­va­tion of Ex­pected Evidence

Eliezer Yudkowsky13 Aug 2007 15:55 UTC
181 points
80 comments2 min readLW link

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

AVoropaev26 Jan 2021 6:08 UTC
21 points
17 comments3 min readLW link

Bayesian in­fer­ence on 1st or­der logic

Daniel Abolafia3 Feb 2021 1:03 UTC
11 points
6 comments18 min readLW link

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

Alexandros18 Dec 2010 13:26 UTC
24 points
6 comments1 min readLW link

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

ejacob5 Mar 2021 0:02 UTC
8 points
2 comments5 min readLW link

Are dogs bad?

qbolec11 Mar 2021 8:11 UTC
9 points
1 comment1 min readLW link

The Ja­panese Quiz: a Thought Ex­per­i­ment of Statis­ti­cal Epistemology

DanB8 Apr 2021 17:37 UTC
11 points
0 comments9 min readLW link

Book Re­view of 5 Ap­plied Bayesian Statis­tics Books

Jan Christian Refsgaard21 May 2021 10:23 UTC
62 points
6 comments4 min readLW link

Dooms­day, Sam­pling As­sump­tions, and Bayes

Yair Halberstadt14 Jun 2021 12:41 UTC
6 points
2 comments11 min readLW link

In­ter­pret­ing Di­ag­nos­tic Tests

Tornus15 Oct 2021 2:58 UTC
6 points
2 comments3 min readLW link

XKCD—Fre­quen­tist vs. Bayesians

brilee9 Nov 2012 5:25 UTC
24 points
91 comments1 min readLW link

Find­ing the Cen­tral Limit The­o­rem in Bayes’ rule

Maxwell Peterson27 Nov 2021 5:48 UTC
21 points
9 comments3 min readLW link

Buck­ets & Bayes

Duncan_Sabien20 Dec 2021 5:54 UTC
28 points
0 comments18 min readLW link

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

lumenwrites5 Jan 2022 17:25 UTC
7 points
3 comments1 min readLW link

Ret­ro­spec­tive forecasting

Ege Erdil30 Jan 2022 16:38 UTC
21 points
6 comments5 min readLW link

Com­men­su­rable Scien­tific Paradigms; or, com­putable induction

samshap13 Apr 2022 0:01 UTC
14 points
0 comments5 min readLW link

RL with KL penalties is bet­ter seen as Bayesian inference

25 May 2022 9:23 UTC
69 points
12 comments12 min readLW link

How to Vi­su­al­ize Bayesianism

David Udell22 Jun 2022 13:57 UTC
8 points
2 comments3 min readLW link

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

Paul Logan26 Jun 2022 19:47 UTC
−4 points
1 comment5 min readLW link
(laulpogan.substack.com)

Five views of Bayes’ Theorem

Adam Scherlis2 Jul 2022 2:25 UTC
34 points
4 comments1 min readLW link