Conjugacy, Self-Consistency and Bayesian Consensus

Abstract

The notion of “conjugacy” has long been a staple of Bayesian inference, though the use of conjugate priors is much less prominent in current forms of Bayesian analysis than it was, say, twenty years ago. There are a number of reasons for the devaluation of conjugacy over the last two decades. The most obvious ones concern those associated with the major advances in Bayesian computation in the decade of the 1990s which have made possible the approximation of posterior distributions and related quantities for a broad array of prior models, rendering the use of priors which facilitate straightforward, closed-form posterior analysis less necessary, if still convenient. With the development and refinement of the Gibbs sampler and similar Markov chain Monte Carlo algorithms, one can now execute a well-approximated Bayesian analysis based on virtually any prior model.