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

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.