Sampling for Bayesian Mixture Models: MCMC with Polynomial-Time Mixing

TL;DR

This paper introduces a new MCMC algorithm called RMRW for sampling from challenging Bayesian Gaussian mixture models, providing polynomial-time mixing guarantees without requiring mean separation.

Contribution

The paper presents the RMRW algorithm and proves its polynomial mixing time for symmetric two-component Gaussian mixtures, independent of mean separation.

Findings

RMRW achieves polynomial mixing time bounds.

No separation conditions needed for the means.

New inequalities for combining Poincaré inequalities.

Abstract

We study the problem of sampling from the power posterior distribution in Bayesian Gaussian mixture models, a robust version of the classical posterior. This power posterior is known to be non-log-concave and multi-modal, which leads to exponential mixing times for some standard MCMC algorithms. We introduce and study the Reflected Metropolis-Hastings Random Walk (RMRW) algorithm for sampling. For symmetric two-component Gaussian mixtures, we prove that its mixing time is bounded as $d^{1.5}(d + \Vert \theta_{0} \Vert^2)^{4.5}$ as long as the sample size $n$ is of the order $d (d + \Vert \theta_{0} \Vert^2)$. Notably, this result requires no conditions on the separation of the two means. En route to proving this bound, we establish some new results of possible independent interest that allow for combining Poincar\'{e} inequalities for conditional and marginal densities.