# Posterior probability

(Redirected from Posterior distribution)

The posterior probability of a random event or an uncertain proposition is the conditional probability that is assigned when the relevant evidence is taken into account.

The posterior probability distribution of one random variable given the value of another can be calculated with Bayes' theorem by multiplying the prior probability distribution by the likelihood function, and then dividing by the normalizing constant, as follows:

${\displaystyle f_{X\mid Y=y}(x)={f_{X}(x)L_{X\mid Y=y}(x) \over {\int _{-\infty }^{\infty }f_{X}(x)L_{X\mid Y=y}(x)\,dx}}}$

gives the posterior probability density function for a random variable X given the data Y = y, where

• ${\displaystyle f_{X}(x)}$ is the prior density of X,
• ${\displaystyle L_{X\mid Y=y}(x)=f_{Y\mid X=x}(y)}$ is the likelihood function as a function of x,
• ${\displaystyle \int _{-\infty }^{\infty }f_{X}(x)L_{X\mid Y=y}(x)\,dx}$ is the normalizing constant, and
• ${\displaystyle f_{X\mid Y=y}(x)}$ is the posterior density of X given the data Y = y.