Efficient Bernoulli factory Markov chain Monte Carlo for intractable posteriors

Abstract

Summary Accept-reject-based Markov chain Monte Carlo algorithms have traditionally utilized acceptance probabilities that can be explicitly written as a function of the ratio of the target density at the two contested points. This feature is rendered almost useless in Bayesian posteriors with unknown functional forms. We introduce a new family of Markov chain Monte Carlo acceptance probabilities that has the distinguishing feature of not being a function of the ratio of the target density at the two points. We present two stable Bernoulli factories that generate events within this class of acceptance probabilities. The efficiency of our methods relies on obtaining reasonable local upper or lower bounds on the target density, and we present two classes of problems where such bounds are viable: Bayesian inference for diffusions, and Markov chain Monte Carlo on constrained spaces. The resulting portkey Barker’s algorithms are exact and computationally more efficient that the current state of the art.