In Bayesian probability theory, the posterior probability is the conditional probability of some event or proposition, taking empirical data into account. Compare with prior probability, which may be assessed in the absence of empirical data, or which may incorporate preexisting data and information. The posterior probability can be calculated by Bayes' theorem from the prior probability and the likelihood. Similarly a posterior probability distribution is the conditional probability distribution of the uncertain quantity given the data. It can be calculated by multiplying the prior probability distribution by the likelihood function, and then dividing by the normalizing constant. For example 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,
 is the likelihood function as a function of x,
 is the normalizing constant, and
 is the posterior density of X.
