Bayesian Repulsive Gaussian Mixture Model

We develop a general class of Bayesian repulsive Gaussian mixture models that encourage well-separated clusters, aiming at reducing potentially redundant components produced by independent priors for locations (such as the Dirichlet process). The asymptotic results for the posterior distribution of the proposed models are derived, including posterior consistency and posterior contraction rate in the context of nonparametric density estimation. More importantly, we show that compared to the independent prior on the component centers, the repulsive prior introduces additional shrinkage effect on the tail probability of the posterior number of components, which serves as a measurement of the model complexity. In addition, a generalized urn model that allows a random number of components and correlated component centers is developed based on the exchangeable partition distribution, which gives rise to the corresponding blocked-collapsed Gibbs sampler for posterior inference. We evaluate the performance and demonstrate the advantages of the proposed methodology through extensive simulation studies and real data analysis.

本文提出一类通用的贝叶斯斥力高斯混合模型（Bayesian repulsive Gaussian mixture models），该模型旨在促进簇间良好分离，以减少由位置参数独立先验（independent priors for locations，如狄利克雷过程（Dirichlet Process））所生成的潜在冗余分量。我们推导了所提模型后验分布（posterior distribution）的渐近性质，包括非参数密度估计（nonparametric density estimation）场景下的后验一致性（posterior consistency）与后验收缩率（posterior contraction rate）。更重要的是，相较于分量中心（component centers）的独立先验，斥力先验（repulsive prior）会对作为模型复杂度（model complexity）衡量指标的后验分量数（posterior number of components）的尾部概率（tail probability）引入额外的收缩效应（shrinkage effect）。此外，基于可交换划分分布（exchangeable partition distribution），我们构建了一种支持随机分量数（random number of components）与相关分量中心（correlated component centers）的广义瓮模型（generalized urn model），并据此导出了用于后验推断（posterior inference）的分块折叠吉布斯采样器（blocked-collapsed Gibbs sampler）。我们通过大规模仿真实验（extensive simulation studies）与真实数据分析（real data analysis），评估了所提方法的性能并验证了其优势。