Exact MCMC for Intractable Proposals

TL;DR

This paper introduces an exact MCMC method using Bernoulli factory algorithms to handle proposals with unknown normalizing constants, improving Bayesian inference accuracy.

Contribution

It adapts Bernoulli factory MCMC algorithms for proposals with intractable normalizing constants, providing a new exact sampling approach.

Findings

Demonstrates the method on three diverse examples.

Shows the approach yields exact samples despite intractable proposals.

Highlights the versatility of Bernoulli factory algorithms in MCMC.

Abstract

Accept-reject based Markov chain Monte Carlo (MCMC) methods are the workhorse algorithm for Bayesian inference. These algorithms, like Metropolis-Hastings, require choosing a proposal distribution which is typically informed by the desired target distribution. Surprisingly, proposal distributions with unknown normalizing constants are not uncommon, even though for such a choice of a proposal, the Metropolis-Hastings acceptance ratio cannot be evaluated exactly. Across the literature, authors resort to approximation methods that yield inexact MCMC or develop specialized algorithms to combat this problem. We show how Bernoulli factory MCMC algorithms, originally proposed for doubly intractable target distributions, can quite naturally be adapted to yield an exact MCMC sampling method. We present three diverse and relevant examples demonstrating the usefulness of the Bernoulli factory approach to this problem.