  A desired closure property in Bayesian probability is that an updated
posterior distribution be in the same class of distributions --- say Gaussians
--- as the prior distribution. When the updating takes place via a statistical
model, one calls the class of prior distributions the `conjugate priors' of the
model. This paper gives (1) an abstract formulation of this notion of conjugate
prior, using channels, in a graphical language, (2) a simple abstract proof
that such conjugate priors yield Bayesian inversions, and (3) a logical
description of conjugate priors that highlights the required closure of the
priors under updating. The theory is illustrated with several standard
examples, also covering multiple updating.
