Approximate Sampling using an Accelerated Metropolis-Hastings based on Bayesian Optimization and Gaussian Processes

TL;DR

This paper introduces an accelerated Metropolis-Hastings algorithm that leverages Bayesian optimization and Gaussian processes to reduce computational costs and improve sampling efficiency for unimodal distributions.

Contribution

It proposes a novel MH algorithm that uses Bayesian optimization and Gaussian process-based Laplace approximation to enhance proposal distribution quality and reduce expensive evaluations.

Findings

Significant reduction in function evaluations compared to standard MH

Shorter burn-in periods observed in experiments

Improved sampling accuracy demonstrated through statistical tests

Abstract

Markov Chain Monte Carlo (MCMC) methods have a drawback when working with a target distribution or likelihood function that is computationally expensive to evaluate, specially when working with big data. This paper focuses on Metropolis-Hastings (MH) algorithm for unimodal distributions. Here, an enhanced MH algorithm is proposed that requires less number of expensive function evaluations, has shorter burn-in period, and uses a better proposal distribution. The main innovations include the use of Bayesian optimization to reach the high probability region quickly, emulating the target distribution using Gaussian processes (GP), and using Laplace approximation of the GP to build a proposal distribution that captures the underlying correlation better. The experiments show significant improvement over the regular MH. Statistical comparison between the results from two algorithms is presented.