On Consistency of Nonparametric Normal Mixtures for Bayesian Density Estimation
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贝叶斯密度估计中非参数正态混合的一致性
- 研究问题
- 基于 Dirichlet 过程及更一般非参数先验的正态混合密度估计，在什么条件下能够保证贝叶斯程序的强一致性？
- 方法与思路
- 作者结合 Kullback–Leibler 支撑条件、基测度均值有限性及标准差先验的指数衰减条件，建立混合模型一致性结果。
- 材料陈述的贡献
- 摘要称，结果在较一般条件下改进既有 Dirichlet 过程混合的一致性理论，并将相同充分条件扩展到更一般的先验。
- 阅读关注点
- 应核对强一致性采用的距离、各项先验条件的精确定义、标准差先验指数衰减的位置，以及可推广先验的具体类别。
- 阅读边界
- 所据 OpenAlex 索引摘要未注明稿件版本；摘要未明确一致性的距离及指数衰减的具体范围，也未提供收敛速度或计算表现。
原文摘要
The past decade has seen a remarkable development in the area of Bayesian nonparametric inference from both theoretical and applied perspectives. As for the latter, the celebrated Dirichlet process has been successfully exploited within Bayesian mixture models, leading to many interesting applications. As for the former, some new discrete nonparametric priors have been recently proposed in the literature that have natural use as alternatives to the Dirichlet process in a Bayesian hierarchical model for density estimation. When using such models for concrete applications, an investigation of their statistical properties is mandatory. Of these properties, a prominent role is to be assigned to consistency. Indeed, strong consistency of Bayesian nonparametric procedures for density estimation has been the focus of a considerable amount of research; in particular, much attention has been devoted to the normal mixture of Dirichlet process. In this article we improve on previous contributions by establishing strong consistency of the mixture of Dirichlet process under fairly general conditions. Besides the usual Kullback–Leibler support condition, consistency is achieved by finiteness of the mean of the base measure of the Dirichlet process and an exponential decay of the prior on the standard deviation. We show that the same conditions are also sufficient for mixtures based on priors more general than the Dirichlet process. This leads to the easy establishment of consistency for many recently proposed mixture models.