Editing Bayesian statistics (section)

== Bayes's theorem ==
{{Main|Bayes's theorem}}
Bayes's theorem is used in Bayesian methods to update probabilities, which are degrees of belief, after obtaining new data. Given two events <math>A</math> and <math>B</math>, the conditional probability of <math>A</math> given that <math>B</math> is true is expressed as follows:<ref name="grinsteadsnell2006">{{cite book |last1=Grinstead |first1=Charles M. |last2=Snell |first2=J. Laurie |title=Introduction to probability |date=2006 |publisher=American Mathematical Society |location=Providence, RI |isbn=978-0-8218-9414-9 |edition=2nd}}</ref>

<math display="block">P(A \mid B) = \frac{P(B \mid A)P(A)}{P(B)}</math>

where <math>P(B) \neq 0</math>. Although Bayes's theorem is a fundamental result of [[probability theory]], it has a specific interpretation in Bayesian statistics. In the above equation, <math>A</math> usually represents a [[proposition]] (such as the statement that a coin lands on heads fifty percent of the time) and <math>B</math> represents the evidence, or new data that is to be taken into account (such as the result of a series of coin flips). <math>P(A)</math> is the [[prior probability]] of <math>A</math> which expresses one's beliefs about <math>A</math> before evidence is taken into account. The prior probability may also quantify prior knowledge or information about <math>A</math>. <math>P(B \mid A)</math> is the [[likelihood function]], which can be interpreted as the probability of the evidence <math>B</math> given that <math>A</math> is true. The likelihood quantifies the extent to which the evidence <math>B</math> supports the proposition <math>A</math>. <math>P(A \mid B)</math> is the [[posterior probability]], the probability of the proposition <math>A</math> after taking the evidence <math>B</math> into account. Essentially, Bayes's theorem updates one's prior beliefs <math>P(A)</math> after considering the new evidence <math>B</math>.<ref name="bda" />

The probability of the evidence <math>P(B)</math> can be calculated using the [[law of total probability]]. If <math>\{A_1, A_2, \dots, A_n\}</math> is a [[Partition of a set|partition]] of the [[sample space]], which is the set of all [[Outcome (probability)|outcomes]] of an experiment, then,<ref name="bda" /><ref name="grinsteadsnell2006" />

<math display="block">P(B) = P(B \mid A_1)P(A_1) + P(B \mid A_2)P(A_2) + \dots + P(B \mid A_n)P(A_n) = \sum_i P(B \mid A_i)P(A_i)</math>

When there are an infinite number of outcomes, it is necessary to [[Integral|integrate]] over all outcomes to calculate <math>P(B)</math> using the law of total probability. Often, <math>P(B)</math> is difficult to calculate as the calculation would involve sums or integrals that would be time-consuming to evaluate, so often only the product of the prior and likelihood is considered, since the evidence does not change in the same analysis. The posterior is proportional to this product:<ref name="bda" />

<math display="block">P(A \mid B) \propto P(B \mid A)P(A)</math>

The [[maximum a posteriori]], which is the [[Mode (statistics)|mode]] of the posterior and is often computed in Bayesian statistics using [[mathematical optimization]] methods, remains the same. The posterior can be approximated even without computing the exact value of <math>P(B)</math> with methods such as [[Markov chain Monte Carlo]] or [[variational Bayesian methods]].<ref name="bda" />