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

Various researchers have studied posterior inference of parameters in Bayesian mixture models [24, 42, 23], so that the statistical behavior of such models is relatively well-understood. In contrast, much less is known about the efficiency of different algorithms for sampling from the posterior distributions that arise from Bayesian mixture models. A standard approach for doing so is via some form of Markov Chain Monte Carlo (MCMC). Many different types of MCMC algorithms have been introduced for various types of Bayesian mixture models, including finite Bayesian mixture models [21, 49, 50, 26, 40], Dirichlet process mixture models [37, 41, 25, 28], and hierarchical and nested Dirichlet process models [52, 47]. Despite the plethora of possible MCMC methods, upper bounds on their mixing times are often challenging to establish. We refer the reader to the papers [27, 3, 55, 48, 57] for non-asymptotic upper bounds on mixing times for certain types of Bayesian models, different from those studied in this paper. In recent years, it has been increasingly common in the Bayesian literature to make use of a fractional likelihood--meaning an ordinary likelihood raised to some fractional power. Combining such a fractional likelihood with a prior distribution in the usual way leads to a class of posteriors known as power posterior or fractional posterior distributions. The power posterior distributions have been shown to have attractive properties in terms of robustness to mis-specification in Bayesian mixture models [39], and have been used in various applications 1 arXiv:1912.05153v1