Adaptive Incremental Mixture Markov chain Monte Carlo

TL;DR

AIMM is a novel adaptive MCMC method that locally updates a mixture of Gaussians proposal to efficiently sample from complex, high-dimensional, and multimodal distributions, with proven convergence.

Contribution

It introduces a semiparametric, locally adaptive mixture proposal that dynamically adds components, improving sampling efficiency over traditional global adaptive methods.

Findings

Performs well on high-dimensional targets

Effective with multimodal distributions

Theoretically converges to the true distribution

Abstract

We propose Adaptive Incremental Mixture Markov chain Monte Carlo (AIMM), a novel approach to sample from challenging probability distributions defined on a general state-space. While adaptive MCMC methods usually update a parametric proposal kernel with a global rule, AIMM locally adapts a semiparametric kernel. AIMM is based on an independent Metropolis-Hastings proposal distribution which takes the form of a finite mixture of Gaussian distributions. Central to this approach is the idea that the proposal distribution adapts to the target by locally adding a mixture component when the discrepancy between the proposal mixture and the target is deemed to be too large. As a result, the number of components in the mixture proposal is not fixed in advance. Theoretically, we prove that there exists a process that can be made arbitrarily close to AIMM and that converges to the correct target distribution. We also illustrate that it performs well in practice in a variety of challenging situations, including high-dimensional and multimodal target distributions.