# Posterior Probability

Posterior probability is a type of conditional probability in Bayesian statistics.

A posterior probability distribution is a conditional probability distribution obtained by applying the distributional form of Bayes’ Theorem.

Given a prior belief that a probability distribution function is {\displaystyle p(\theta )}p(\theta ) and that the observations {\displaystyle x}x have a likelihood {\displaystyle p(x|\theta )}p(x|\theta ), then the posterior probability is defined as

$p(θ∣x)=p(x)p(x∣θ) p(θ)$

where $p(x)$ is the normalizing constant and is calculated as

$p(x)=∫p(x∣θ)p(θ)dθ$ for continuous $θ$ , or by summing $p(x∣θ)p(θ)$ over all possible values of $θ$ for discrete $θ$.