Bayes Rule

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 $p(\theta )$ and that the observations $x$ have a likelihood $p(x|\theta )$, then the posterior probability is defined as

$p(\theta |x)=\frac{p(x|\theta )}{p(x)}p(\theta )$

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

$p(x)=\int p(x|\theta )p(\theta )d\theta$ for continuous $\theta$ , or by summing $p(x|\theta )p(\theta )$ over all possible values of $\theta$ for discrete $\theta$.

Related

• Prior Probability