ABSTRACT

Bhattacharya [Gibbs sampling based Bayesian analysis of mixtures with unknown number of components. Sankhya B. 2008;70:133–155] introduced a mixture model based on the Dirichlet process, where an upper bound on the unknown number of components is to be specified. Defining a Bayesian analogue of the mean integrated squared error (Bayesian MISE), here we consider a Bayesian asymptotic density estimation framework for objectively specifying the upper bound, as well as the precision parameter of the Dirichlet process, such that the Bayesian MISE converges at a desired rate. As a byproduct of our approach, we also investigate Bayesian MISE convergence rate of the traditional Dirichlet process mixture model, which leads to asymptotic specification of the precision parameter. Various asymptotic issues related to the two aforementioned mixtures, including comparative performances, are also investigated. The theoretical studies, supplemented with simulation experiments, bring out the superiority of the approach of Bhattacharya (2008).

摘要

Bhattacharya [基于Gibbs抽样的未知成分数量的混合物的贝叶斯分析。Sankhya B. 2008; 70：133–155]引入了一种基于Dirichlet过程的混合模型，其中将指定未知数量的组分的上限。定义平均积分平方误差的贝叶斯类似物（贝叶斯MISE），在此我们考虑一个贝叶斯渐近密度估计框架，用于客观地指定上限和Dirichlet过程的精度参数，以使贝叶斯MISE收敛于期望的速率。作为我们方法的副产品，我们还研究了传统Dirichlet过程混合模型的贝叶斯MISE收敛速度，这导致了精度参数的渐近规范。与上述两种混合相关的各种渐近问题，包括比较性能，也进行了调查。理论研究和仿真实验相辅相成，证明了Bhattacharya（2008）方法的优越性。