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.
翻译:我们从巴伊西亚高斯混合模型的能量后部分布中研究取样问题,这是古典后部的可靠版本。 这种能量后部已知是非log-concave和多式的,导致某些标准的MCMC算法的指数混合时间。 我们引入并研究用于取样的反射大都会-Hasting随机行走(RMRRW)算法。 对于对称两个成分的高斯混合混合物, 我们证明它的混合时间与美元=1.5}(d+\Vert\theta ⁇ 0}\Vert ⁇ 2}4.5}美元有关,只要样本大小为美元(d +\Vert \theta ⁇ 0}\Vert}\Vert ⁇ 2)的顺序。 值得注意的是,这一结果不需要两种方法分离的条件。 在证明这一约束的道路上,我们确定一些可能的独立利益的新结果,允许将Poincar\ {e} 不平等合并为有条件和边缘密度。