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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:
gives the posterior probability density function for a random variable X given the data Y = y, where
is the prior density of X,
is the likelihood function as a function of x,
is the normalizing constant, and
is the posterior density of X given the data Y = y.
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