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Bayes' theorem

From Wikipedia, the free encyclopedia

Jump to: navigation, search This article is about the theorem. For its application in legal evidence diagnostics and E-

discovery , see Bayesian theory in E-discovery. For its application in marketing, see Bayesian

theory in marketing . For other uses, see Bayes theorem (disambiguation).

The simple statement of Bayes' theorem

Bayesian statistics

Theory

Bayesian probability

Probability interpretations

Bayes' theorem

Bayes' rule · Bayes factor

Bayesian inference

Bayesian netork

Prior · Posterior · !ikelihood

"on#ugate prior Posterior predictive

$yperparameter · $yperprior

Principle of indifference

Principle of ma%imum entropy

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&mpirical Bayes method

"romell's rule

Bernsteinvon (ises theorem

Bayesian information criterion

"redible interval

(a%imum a posteriori estimation

Techniques

Bayesian linear regression

Bayesian estimator

)ppro%imate Bayesian computation

UsesBayesian spam filtering

Binary classification

*aive Bayes classifier

• v

• t

• e

+n probability theory and statistics, Bayes' theorem alternatively Bayes' law or Bayes' rule- is

a theorem ith to distinct interpretations. +n the Bayesian interpretation, it e%presses ho a

sub#ective degree of belief should rationally change to account for evidence. +n the freuentistinterpretation, it relates inverse representations of the probabilities concerning to events. +n the

Bayesian interpretation, Bayes' theorem is fundamental to Bayesian statistics, and has

applications in fields including science, engineering, economics particularly microeconomics-,game theory, medicine and la. The application of Bayes' theorem to update beliefs is called

Bayesian inference.

Bayes' theorem is named for Thomas Bayes /ˈb eɪ z// 01200130-, ho first suggested using the

theorem to update beliefs. $is ork as significantly edited and updated by 4ichard Price before

it as posthumously read at the 4oyal 5ociety. The ideas gained limited e%posure until they ereindependently rediscovered and further developed by !aplace, ho first published the modern

formulation in his 0607 Th!orie analytiue des probabilit!s. 8ntil the second half of the 72th

century, the Bayesian interpretation as largely re#ected by the mathematics community as

unscientific9citation needed . $oever, it is no idely accepted. This may have been due to the

http://en.wikipedia.org/wiki/Empirical_Bayes_methodhttp://en.wikipedia.org/wiki/Cromwell's_rulehttp://en.wikipedia.org/wiki/Bernstein%E2%80%93von_Mises_theoremhttp://en.wikipedia.org/wiki/Bayesian_information_criterionhttp://en.wikipedia.org/wiki/Credible_intervalhttp://en.wikipedia.org/wiki/Maximum_a_posteriori_estimationhttp://en.wikipedia.org/wiki/Bayesian_linear_regressionhttp://en.wikipedia.org/wiki/Bayesian_estimatorhttp://en.wikipedia.org/wiki/Approximate_Bayesian_computationhttp://en.wikipedia.org/wiki/Bayesian_spam_filteringhttp://en.wikipedia.org/wiki/Binary_classificationhttp://en.wikipedia.org/wiki/Naive_Bayes_classifierhttp://en.wikipedia.org/wiki/Template:Bayesian_statisticshttp://en.wikipedia.org/wiki/Template_talk:Bayesian_statisticshttp://en.wikipedia.org/w/index.php?title=Template:Bayesian_statistics&action=edithttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Bayesian_statisticshttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Economicshttp://en.wikipedia.org/wiki/Microeconomicshttp://en.wikipedia.org/wiki/Game_theoryhttp://en.wikipedia.org/wiki/Medicinehttp://en.wikipedia.org/wiki/Lawhttp://en.wikipedia.org/wiki/Bayesian_inferencehttp://en.wikipedia.org/wiki/Thomas_Bayeshttp://en.wikipedia.org/wiki/Wikipedia:IPA_for_Englishhttp://en.wikipedia.org/wiki/Wikipedia:IPA_for_English#Keyhttp://en.wikipedia.org/wiki/Wikipedia:IPA_for_Englishhttp://en.wikipedia.org/wiki/Richard_Pricehttp://en.wikipedia.org/wiki/Royal_Societyhttp://en.wikipedia.org/wiki/Royal_Societyhttp://en.wikipedia.org/wiki/Pierre-Simon_Laplacehttp://en.wikipedia.org/wiki/Pierre-Simon_Laplacehttp://en.wikipedia.org/wiki/Pierre-Simon_Laplace#Analytic_theory_of_probabilitieshttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Empirical_Bayes_methodhttp://en.wikipedia.org/wiki/Cromwell's_rulehttp://en.wikipedia.org/wiki/Bernstein%E2%80%93von_Mises_theoremhttp://en.wikipedia.org/wiki/Bayesian_information_criterionhttp://en.wikipedia.org/wiki/Credible_intervalhttp://en.wikipedia.org/wiki/Maximum_a_posteriori_estimationhttp://en.wikipedia.org/wiki/Bayesian_linear_regressionhttp://en.wikipedia.org/wiki/Bayesian_estimatorhttp://en.wikipedia.org/wiki/Approximate_Bayesian_computationhttp://en.wikipedia.org/wiki/Bayesian_spam_filteringhttp://en.wikipedia.org/wiki/Binary_classificationhttp://en.wikipedia.org/wiki/Naive_Bayes_classifierhttp://en.wikipedia.org/wiki/Template:Bayesian_statisticshttp://en.wikipedia.org/wiki/Template_talk:Bayesian_statisticshttp://en.wikipedia.org/w/index.php?title=Template:Bayesian_statistics&action=edithttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Bayesian_statisticshttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Economicshttp://en.wikipedia.org/wiki/Microeconomicshttp://en.wikipedia.org/wiki/Game_theoryhttp://en.wikipedia.org/wiki/Medicinehttp://en.wikipedia.org/wiki/Lawhttp://en.wikipedia.org/wiki/Bayesian_inferencehttp://en.wikipedia.org/wiki/Thomas_Bayeshttp://en.wikipedia.org/wiki/Wikipedia:IPA_for_Englishhttp://en.wikipedia.org/wiki/Wikipedia:IPA_for_English#Keyhttp://en.wikipedia.org/wiki/Wikipedia:IPA_for_English#Keyhttp://en.wikipedia.org/wiki/Wikipedia:IPA_for_English#Keyhttp://en.wikipedia.org/wiki/Wikipedia:IPA_for_English#Keyhttp://en.wikipedia.org/wiki/Wikipedia:IPA_for_Englishhttp://en.wikipedia.org/wiki/Richard_Pricehttp://en.wikipedia.org/wiki/Royal_Societyhttp://en.wikipedia.org/wiki/Pierre-Simon_Laplacehttp://en.wikipedia.org/wiki/Pierre-Simon_Laplace#Analytic_theory_of_probabilitieshttp://en.wikipedia.org/wiki/Wikipedia:Citation_needed


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development of computing, hich enabled the successful application of Bayesianism to many

comple% problems.90

Contents

9hide

• 0 +ntroductory e%ample

• 7 5tatement and interpretation

o 7.0 Bayesian interpretation

o 7.7 Fre;uentist interpretation

• < Forms

o


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o ?.7 >rug testing

• 3 $istory

• 1 *otes

o 1.0 Further reading

• 6 &%ternal links

[edit] Introductory example

5uppose someone told you they had a nice conversation ith someone on the train. *ot knoing

anything else about this conversation, the probability that they ere speaking to a oman is?2@. *o suppose they also told you that this person had long hair. +t is no more likely that

they ere speaking to a oman, since most longAhaired people are omen. Bayes' theorem can be used to calculate the probability that the person is a oman.

To see ho this is done, let

represent the event that the conversation as held ith a oman, and

denote the event that the conversation as held ith a longAhaired person.

+t can be assumed that omen constitute half the population for this e%ample. 5o, not knoing

anything else, the probability that occurs is

.

5uppose it is also knon that 1?@ of omen have long hair, hich e denote as

read: the probability of event given event is 2.1?-.

!ikeise, suppose it is knon that


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ur goal is to calculate the probability that the conversation as held ith a oman, given the

fact that the person had long hair, or, in our notation, . 8sing the formula for Bayes'theorem, e have:

here e have used the la of total probability. The numeric anser can be obtained bysubstituting the above values into this formula. This yields

,

i.e., the probability that the conversation as held ith a oman, given that the person had long

hair, is about 10@.

[edit] Statement and interpretation

(athematically, Bayes' theorem gives the relationship beteen the probabilities of and ,

and , and the conditional probabilities of given and given , and

. +n its most common form, it is:

The meaning of this statement depends on the interpretation of probability ascribed to the terms:

[edit] Bayesian interpretation

"ain article# Bayesian probability

+n the Bayesian or epistemological- interpretation, probability measures a degree of belief .Bayes' theorem then links the degree of belief in a proposition before and after accounting for

evidence. For e%ample, suppose somebody proposes that a biased coin is tice as likely to land

heads than tails. >egree of belief in this might initially be ?2@. The coin is then flipped anumber of times to collect evidence. Belief may rise to 12@ if the evidence supports the

proposition.

For proposition and evidence ,

• , the prior , is the initial degree of belief in .

• , the posterior , is the degree of belief having accounted for .

http://en.wikipedia.org/wiki/Law_of_total_probabilityhttp://en.wikipedia.org/wiki/Law_of_total_probabilityhttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=2http://en.wikipedia.org/wiki/Marginal_probabilityhttp://en.wikipedia.org/wiki/Marginal_probabilityhttp://en.wikipedia.org/wiki/Conditional_probabilityhttp://en.wikipedia.org/wiki/Probability_interpretationshttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=3http://en.wikipedia.org/wiki/Bayesian_probabilityhttp://en.wikipedia.org/wiki/Bayesian_probabilityhttp://en.wikipedia.org/wiki/Bayesian_probabilityhttp://en.wikipedia.org/wiki/Law_of_total_probabilityhttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=2http://en.wikipedia.org/wiki/Marginal_probabilityhttp://en.wikipedia.org/wiki/Conditional_probabilityhttp://en.wikipedia.org/wiki/Probability_interpretationshttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=3http://en.wikipedia.org/wiki/Bayesian_probabilityhttp://en.wikipedia.org/wiki/Bayesian_probability


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• represents the support provides for .

For more on the application of Bayes' theorem under the Bayesian interpretation of probability,see Bayesian inference.

[edit] Frequentist interpretation

+llustration of fre;uentist interpretation ith tree diagrams. Bayes' theorem connects conditional probabilities to their inverses.

+n the fre;uentist interpretation, probability measures a proportion of outcomes. Bayes' theorem

under this interpretation is most easily visualiCed using tree diagrams, as shon to the right. The

to diagrams partition the same population in different ays. The upper diagram partitions the population into to groups, ith and ithout property , and then partitions each group into

to further groups, ith and ithout property . The loer diagram proceeds in reverse, first

partitioning according to and then according to . For e%ample, might be having a risk

factor for a medical condition, and might be having the condition.-

+f e pick every member of a population ith property , and ask Dhat proportion of these

have property ED, this gives the probability . "onversely, if e pick every member of

the same population ith property and ask Dhat proportion of these have property ED this

gives the probability . is the overall proportion ith , and the overall proportion ith . Bayes' theorem links these probabilities, hich are in general different.

http://en.wikipedia.org/wiki/Bayesian_inferencehttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=4http://en.wikipedia.org/wiki/Tree_structurehttp://en.wikipedia.org/wiki/Frequentist_interpretation_of_probabilityhttp://en.wikipedia.org/wiki/Bayesian_inferencehttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=4http://en.wikipedia.org/wiki/Tree_structurehttp://en.wikipedia.org/wiki/Frequentist_interpretation_of_probability


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[edit] Forms

[edit] For eents

[edit] Simple !orm

For events and , provided that .

+n a Bayesian inference step, the probability of evidence is constant for all models . The

posterior may then be e%pressed as proportional to the numerator:

[edit] "xtended !orm

ften, for some partition of the event space, the event space is given or conceptualiCed in

terms of and . +t is then useful to eliminate using the la of total

probability:

+n the special case of a binary partition,

[edit] Three or more eents

&%tensions to Bayes' theorem may be found for three or more events. For e%ample, for threeevents, to possible tree diagrams branch in the order B") and )B". By repeatedly applyingthe definition of conditional probability:

http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=5http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=6http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=7http://en.wikipedia.org/wiki/Bayesian_inferencehttp://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=8http://en.wikipedia.org/wiki/Partition_of_a_sethttp://en.wikipedia.org/wiki/Partition_of_a_sethttp://en.wikipedia.org/wiki/Event_spacehttp://en.wikipedia.org/wiki/Law_of_total_probabilityhttp://en.wikipedia.org/wiki/Law_of_total_probabilityhttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=9http://en.wikipedia.org/wiki/Conditional_probabilityhttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=5http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=6http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=7http://en.wikipedia.org/wiki/Bayesian_inferencehttp://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=8http://en.wikipedia.org/wiki/Partition_of_a_sethttp://en.wikipedia.org/wiki/Event_spacehttp://en.wikipedia.org/wiki/Law_of_total_probabilityhttp://en.wikipedia.org/wiki/Law_of_total_probabilityhttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=9http://en.wikipedia.org/wiki/Conditional_probability


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)s previously, the la of total probability may be used to substitute for unknon marginal

probabilities.

[edit] For random aria#les

>iagram illustrating the meaning of Bayes' theorem as applied to an event space generated bycontinuous random variables and . *ote that there e%ists an instance of Bayes' theorem for

each point in the domain. +n practise, these instances might be parametrised by riting the

specified probability densities as a function of and .

"onsider a sample space generated by to random variables and . +n principle, Bayes'

theorem applies to the events and . $oever, terms become 2

at points here either variable has finite probability density. To remain useful, Bayes' theorem

may be formulated in terms of the relevant densities see >erivation-.

[edit] Simple !orm

+f is continuous and is discrete,

+f is discrete and is continuous,

+f both and are continuous,

[edit] "xtended !orm

http://en.wikipedia.org/wiki/Law_of_total_probabilityhttp://en.wikipedia.org/wiki/Law_of_total_probabilityhttp://en.wikipedia.org/wiki/Marginal_probabilitieshttp://en.wikipedia.org/wiki/Marginal_probabilitieshttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=10http://en.wikipedia.org/wiki/Domain_of_a_functionhttp://en.wikipedia.org/wiki/Function_(Mathematics)http://en.wikipedia.org/wiki/Function_(Mathematics)http://en.wikipedia.org/wiki/Sample_spacehttp://en.wikipedia.org/wiki/Sample_spacehttp://en.wikipedia.org/wiki/Random_variableshttp://en.wikipedia.org/wiki/Random_variableshttp://en.wikipedia.org/wiki/Probability_density_functionhttp://en.wikipedia.org/wiki/Probability_density_functionhttp://en.wikipedia.org/wiki/Bayes'_theorem#Derivationhttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=12http://en.wikipedia.org/wiki/Law_of_total_probabilityhttp://en.wikipedia.org/wiki/Marginal_probabilitieshttp://en.wikipedia.org/wiki/Marginal_probabilitieshttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=10http://en.wikipedia.org/wiki/Domain_of_a_functionhttp://en.wikipedia.org/wiki/Function_(Mathematics)http://en.wikipedia.org/wiki/Sample_spacehttp://en.wikipedia.org/wiki/Random_variableshttp://en.wikipedia.org/wiki/Probability_density_functionhttp://en.wikipedia.org/wiki/Bayes'_theorem#Derivationhttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=12


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>iagram illustrating ho an event space generated by continuous random variables and G is

often conceptualiCed.

) continuous event space is often conceptualiCed in terms of the numerator terms. +t is then

useful to eliminate the denominator using the la of total probability. For , this becomes

an integral:

[edit] Bayes' rule

"ain article# Bayes$ rule

8nder the Bayesian interpretation of probability, Bayes' rule may be thought of as Bayes'

theorem in odds form.

here

.

5o the rule says that the posterior odds are the prior odds times the Bayes factor .

[edit] $eriation

[edit] For %eneral eents

Bayes' theorem may be derived from the definition of conditional probability:

http://en.wikipedia.org/wiki/Law_of_total_probabilityhttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=13http://en.wikipedia.org/wiki/Bayes'_rulehttp://en.wikipedia.org/wiki/Bayesian_probabilityhttp://en.wikipedia.org/wiki/Bayes'_rulehttp://en.wikipedia.org/wiki/Oddshttp://en.wikipedia.org/wiki/Bayes_factorhttp://en.wikipedia.org/wiki/Bayes_factorhttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=14http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=15http://en.wikipedia.org/wiki/Conditional_probabilityhttp://en.wikipedia.org/wiki/Law_of_total_probabilityhttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=13http://en.wikipedia.org/wiki/Bayes'_rulehttp://en.wikipedia.org/wiki/Bayesian_probabilityhttp://en.wikipedia.org/wiki/Bayes'_rulehttp://en.wikipedia.org/wiki/Oddshttp://en.wikipedia.org/wiki/Bayes_factorhttp://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=14http://en.wikipedia.org/w/index.php?title=Bayes%27_theorem&action=edit&section=15http://en.wikipedia.org/wiki/Conditional_probability


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[edit] $ru% testin%

Tree diagram illustrating drug testing e%ample. 8, 8 bar, DID and DD are the events representing

user, nonAuser, positive result and negative result. Percentages in parentheses are calculated.

5uppose a drug test is HH@ sensitive and HH@ specific. That is, the test ill produce HH@ true

positive results for drug users and HH@ true negative results for nonAdrug users. 5uppose that2.?@ of people are users of the drug. +f a randomly selected individual tests positive, hat is the

probability he or she is a userE

>espite the apparent accuracy of the test, if an individual tests positive, it is more likely that they

do not use the drug than that they do.

This surprising result arises because the number of nonAusers is very large compared to thenumber of users, such that the number of false positives 2.HH?@- outeighs the number of true

positives 2.=H?@-. To use concrete numbers, if 0222 individuals are tested, there are e%pected to

be HH? nonAusers and ? users. From the HH? nonAusers, false positives are

e%pected. From the ? users, true positives are e%pected. ut of 0? positive

results, only ?, about


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[edit] &istory

Bayes' theorem as named after the 4everend Thomas Bayes 012730-, ho studied ho to

compute a distribution for the probability parameter of a binomial distribution in modernterminology-. $is friend 4ichard Price edited and presented this ork in 013


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• BayesO rule: ) Tutorial +ntroduction by J 5tone.

[edit] "xternal lin+s

• &arliest Qnon 8ses of 5ome of the Words of (athematics B-. "ontains origins of

DBayesianD, DBayes' TheoremD, DBayes &stimateL4iskL5olutionD, D&mpirical BayesD, andDBayes FactorD.

• Bayesian 5tatistics summary from 5cholarpedia.

• Weisstein, &ric W., DBayes' TheoremD from (athWorld.

• Bayes' theorem at Planet(ath

• ) tutorial on probability and BayesO theorem devised for %ford 8niversity psychology

students

• ) graphical e%planation of the Bayes theorem.

4etrieved from Dhttp:LLen.ikipedia.orgLLinde%.phpEtitleRBayes

@71StheoremoldidR?003?10


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