The posterior probability of a random event or an uncertain proposition is the conditional probability it 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 by Bayes' theorem by multiplying the prior probability distribution by the likelihood function, and then dividing by the normalizing constant, as follows:

$f\_\{X\backslash mid\; Y=y\}(x)=\{f\_X(x)\; L\_\{X\backslash mid\; Y=y\}(x)\; \backslash over\; \{\backslash int\_\{-\backslash infty\}^\backslash infty\; f\_X(x)\; L\_\{X\backslash mid\; Y=y\}(x)\backslash ,dx\}\}$

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

• $f\_X(x)$ is the prior density of X,

• $L\_\{X\backslash mid\; Y=y\}(x)\; =\; f\_\{Y\backslash mid\; X=x\}(y)$ is the likelihood function as a function of x,

• $\backslash int\_\{-\backslash infty\}^\backslash infty\; f\_X(x)\; L\_\{X\backslash mid\; Y=y\}(x)\backslash ,dx$ is the normalizing constant, and

• $f\_\{X\backslash mid\; Y=y\}(x)$ is the posterior density of X given the data Y = y.

Statistics