Normalized random measures with independent increments represent a large class of Bayesian nonaprametric priors and are widely used in the Bayesian nonparametric framework. In this paper, we provide the posterior consistency analysis for normalized random measures with independent increments (NRMIs) through the corresponding Levy intensities used to characterize the completely random measures in the construction of NRMIs. Assumptions are introduced on the Levy intensities to analyze the posterior consistency of NRMIs and are verified with multiple interesting examples. A focus of the paper is the Bernstein-von Mises theorem for the normalized generalized gamma process (NGGP) when the true distribution of the sample is discrete or continuous. When the Bernstein-von Mises theorem is applied to construct credible sets, in addition to the usual form there will be an additional bias term on the left endpoint closely related to the number of atoms of the true distribution when it is discrete. We also discuss the affect of the estimators for the model parameters of the NGGP under the Bernstein-von Mises convergences. Finally, to further explain the necessity of adding the bias correction in constructing credible sets, we illustrate numerically how the bias correction affects the coverage of the true value by the credible sets when the true distribution is discrete.
Tree-Guided MCMC Inference for Normalized Random Measure Mixture Models
Pohang University of Science and Technology（浦项科技大学）
标准化随机测度（Normalized Random Measures，NRM）是一类离散随机测度，通常用作贝叶斯非参数模型的先验。Dirichlet过程是NRM的一个众所周知的例子。大多数NRM混合模型的后验推理方法依赖于MCMC方法，因为它们易于实现并且其收敛性已经得到了很好的研究。然而，MCMC经常遭受缓慢的收敛时，接受率低。基于树的推理是另一种确定性后验推理方法，其中贝叶斯层次聚类（BHC）或增量贝叶斯层次聚类（IBHC）已分别为DP或NRM混合（NRMM）模型开发。虽然IBHC是NRMM模型后验推理的一种很有前途的方法，因为它的效率和在线推理的适用性，但它的收敛性不能保证，因为它使用了在多次试验后简单地选择最佳解决方案的算法。本文结合MCMC和IBHC的优点，提出了一种NRMM模型的混合推理算法。IBHC构建的树概述了数据的分区，这指导了Metropolis-Hastings程序采用适当的建议。继承MCMC的性质，我们的树引导MCMC（tgMCMC）是保证收敛，并享受快速收敛由于有效的建议指导树。在合成数据集和真实数据集上的实验证明了我们方法的优点。
Non-asymptotic two-sample kernel testing with the spectrally truncated normalized MMD
Nantes Université（南特大学）；Laboratoire de Mathématiques Jean Leray, LMJL（让·勒雷数学实验室）；Ecole Normale Supérieure de Lyon（里昂高等师范学院）；Université Claude Bernard Lyon 1（里昂第一大学）；Laboratoire de Biologie et Modélisation de la Cellule（细胞生物学与建模实验室）；CEREMADE, CNRS, Université Paris-Dauphine（巴黎道芬大学数学与计算机科学研究与应用中心）；Université PSL（巴黎文理研究大学）；Université Paris-Saclay（巴黎萨克雷大学）；Inria, LMO（法国国家信息与自动化研究所，数学与优化实验室）
