Posterior Simulation with Priors Specified on Functionals

Abstract

Many Bayesian analyses use Markov chain Monte Carlo (MCMC) techniques. MCMC techniques work fastest (per iteration) when the prior distribution of the parameters is chosen conveniently, such as a conjugate prior. However, this is sometimes at odds with the prior desired by the investigator. We describe two motivating examples where nonconjugate priors are preferred. One is a Dirichlet process where it is difficult to implement alternative, nonconjugate priors. We develop a method that allows computation to be done with a convenient prior but adjusts the equilibrium distribution of the Markov chain to be the posterior distribution from the desired prior. In addition to allowing more freedom in choosing prior distributions, the method enables the investigator to perform quick sensitivity analyses, even in nonparametric settings.