Definition of Term: Conjugate prior

In the Bayesian approach to statistics, a prior distribution expresses prior knowledge about parameters in a statistical analysis. The prior distribution is then combined with the information in the data using Bayes’ theorem, and the resulting posterior distribution is used for making inference statements (such as estimates or credible intervals) about the parameters. Specification of the prior distribution is therefore an important task in Bayesian statistics. Two simplifying techniques that are often used are to employ weak prior distributions (that represent prior information that is supposed to be weak relative to the information in the data) and conjugate prior distributions.

A conjugate distribution is of a mathematical form that combines conveniently with the information in the data, so that the posterior distribution is easy to work with. The specification of prior information is generally an imprecise process, and the particular choice of distributional form is to some extent arbitrary. (It is this arbitrariness that is objected to by those who advocate the Bayes linear approach rather than the fully Bayesian approach; both may be found in the toolkit.) So as long as the conjugate form would not obviously be an inappropriate representation of prior information it is sensible to use a conjugate prior distribution.