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

arXiv: https://arxiv.org/abs/1912.05153 · HTML: https://mlsanity.com/paper/1912.05153/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Dec 2019.

## Built on by (most cited first, 5 of 5)

- [On Sampling with Approximate Transport Maps](https://mlsanity.com/paper/2302.04763/) (2023)
- [Sampling Approximately Low-Rank Ising Models: MCMC meets Variational Methods](https://mlsanity.com/paper/2202.08907/) (2022)
- [Fast Conditional Mixing of MCMC Algorithms for Non-log-concave Distributions](https://mlsanity.com/paper/2306.10506/) (2023)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [Fast Mixing of Data Augmentation Algorithms: Bayesian Probit, Logit, and Lasso Regression](https://mlsanity.com/paper/2412.07999/) (2024)

## Abstract (arXiv)

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 d1.5(d + Vert θ0 Vert2)4.5 as long as the sample size n is of the order d (d + Vert θ0 Vert2). 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.