文章基本信息

标题：Optimal Bayesian Minimax Rates for Unconstrained Large Covariance Matrices
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作者：Kyoungjae Lee ; Jaeyong Lee
期刊名称：Bayesian Analysis
印刷版ISSN：1931-6690
电子版ISSN：1936-0975
出版年度：2018
卷号：13
期号：4
页码：1215-1233
DOI：10.1214/18-BA1094
语种：English
出版社：International Society for Bayesian Analysis
摘要：We obtain the optimal Bayesian minimax rate for the unconstrained large covariance matrix of multivariate normal sample with mean zero, when both the sample size, n, and the dimension, p, of the covariance matrix tend to infinity. Traditionally the posterior convergence rate is used to compare the frequentist asymptotic performance of priors, but defining the optimality with it is elusive. We propose a new decision theoretic framework for prior selection and define Bayesian minimax rate. Under the proposed framework, we obtain the optimal Bayesian minimax rate for the spectral norm for all rates of p. We also considered Frobenius norm, Bregman divergence and squared log-determinant loss and obtain the optimal Bayesian minimax rate under certain rate conditions on p. A simulation study is conducted to support the theoretical results.
关键词：Bayesian minimax rate; convergence rate; decision theoretic prior selection; unconstrained covariance.