A mathematically precise definition of conjugate priors is given using channels.

A desired closure property in Bayesian probability is that an updated posterior distribution is in the same class of distributions — say Gaussians — as the prior distribution. When the updating takes place via a likelihood, one then calls the class of prior distributions the `conjugate prior’ of this likelihood. This paper gives (1) an abstract formulation of this notion of conjugate prior, using channels, in a graphical language, and (2) a simple abstract proof that such conjugate priors yield Bayesian inversions. The theory is illustrated with several standard examples.