Posterior Distribution

The term posterior distribution refers to a probability density function for an unobserved or latent variable, θ, based on the observed data, x, used in Bayesian statistical analysis. In Bayesian data analysis, inferences are made on the basis of the posterior distributions of parameters. Bayes' theorem is generally phrased in terms of distributions of observed and unobserved variables:

By using this formulation of Bayes' theorem, statisticians are able to make inferences about some parameter of interest, θ, given the observed data, x. Therefore, under the framework of a Bayesian data analysis, statistical inferences are based on a quantity that is of direct interest to the analyst (i.e., θ), not some proxy for that quantity of interest (i.e., the data, x).

In order to estimate the posterior distribution and make inferences about it, three pieces of information are required: (a) estimation of the function, f(x|∘), often termed the “likelihood function,” which represents a statistical model that has been fit to the distribution of observed data, x, given the underlying parameter, θ; (b) estimation of f(θ), referred to as the “prior distribution,” which represents either the empirical or expected distribution of the parameter, θ, in the population, and (c) estimation of f(x), which represents the empirical distribution of the observed data, x.

Inferences from the posterior distribution are typically made by determining point estimates for θ, either by finding the mean of the posterior distribution (referred to as an “expected a posteriori” estimate) or by determining the mode of the posterior distribution (referred to as a “modal a posteriori” estimate). The standard error of θ is determined by estimating the standard deviation of the posterior distribution.