Editing Posterior probability (section)

==Definition in the distributional case==

In Bayesian statistics, the posterior probability is the probability of the parameters <math>\theta</math> given the evidence <math>X</math>, and is denoted <math>p(\theta |X)</math>.

It contrasts with the [[likelihood function]], which is the probability of the evidence given the parameters: <math>p(X|\theta)</math>.

The two are related as follows:

Given a [[prior probability|prior]] belief that a [[probability density function|probability distribution function]] is <math>p(\theta)</math> and that the observations <math>x</math> have a likelihood <math>p(x|\theta)</math>, then the posterior probability is defined as 
:<math>p(\theta|x) = \frac{p(x|\theta)}{p(x)}p(\theta)</math>,<ref>{{cite book| title=Pattern Recognition and Machine Learning| author=Christopher M. Bishop| publisher=Springer| year=2006| isbn=978-0-387-31073-2| pages=21–24}}</ref>
where <math>p(x)</math> is the normalizing constant and is calculated as
:<math> p(x) = \int p(x|\theta)p(\theta)d\theta</math>
for continuous <math>\theta</math>, 
or by summing <math>p(x|\theta)p(\theta)</math>
over all possible values of <math>\theta</math> for discrete <math>\theta</math>.<ref name="BDA">{{cite book | title=Bayesian Data Analysis| author=Andrew Gelman, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari and Donald B. Rubin | publisher=CRC Press | year=2014 | isbn=978-1-4398-4095-5 | pages=7}}</ref>

The posterior probability is therefore [[direct proportionality|proportional to]] the product ''Likelihood · Prior probability''.<ref>{{Cite book |last=Ross |first=Kevin |url=https://bookdown.org/kevin_davisross/bayesian-reasoning-and-methods/continuous.html |title=Chapter 8 Introduction to Continuous Prior and Posterior Distributions {{!}} An Introduction to Bayesian Reasoning and Methods}}</ref>