High-dimensional Bernstein–von Mises theorem for the Diaconis–Ylvisaker prior

IF 1.4 3区 数学 Q2 STATISTICS & PROBABILITY
Xin Jin , Anirban Bhattacharya , Riddhi Pratim Ghosh
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引用次数: 0

Abstract

We study the asymptotic normality of the posterior distribution of canonical parameter in the exponential family under the Diaconis–Ylvisaker prior which is a conjugate prior when the dimension of parameter space increases with the sample size. We prove under mild conditions on the true parameter value θ0 and hyperparameters of priors, the difference between the posterior distribution and a normal distribution centered at the maximum likelihood estimator, and variance equal to the inverse of the Fisher information matrix goes to 0 in the expected total variation distance. The proof assumes dimension of parameter space d grows linearly with sample size n only requiring d=o(n). En route, we derive a concentration inequality of the quadratic form of the maximum likelihood estimator without any specific assumption such as sub-Gaussianity. A specific illustration is provided for the Multinomial-Dirichlet model with an extension to the density estimation and Normal mean estimation problems.

Diaconis-Ylvisaker先验的高维Bernstein-von Mises定理
当参数空间的维数随样本量的增加而增加时,我们研究了指数族中典型参数后验分布在Diaconis-Ylvisaker先验下的渐近正态性。我们证明了在先验真参数值θ0和超参数的温和条件下,后验分布与以极大似然估计量为中心的正态分布之间的差和等于Fisher信息矩阵逆的方差在期望总变异距离中趋于0。证明假设参数空间d的维数随样本量n线性增长,只要求d=o(n)。在此过程中,我们导出了最大似然估计量的二次形式的浓度不等式,而没有任何特定的假设,如次高斯性。给出了多项式-狄利克雷模型的具体实例,并将其推广到密度估计和正态均值估计问题。
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来源期刊
Journal of Multivariate Analysis
Journal of Multivariate Analysis 数学-统计学与概率论
CiteScore
2.40
自引率
25.00%
发文量
108
审稿时长
74 days
期刊介绍: Founded in 1971, the Journal of Multivariate Analysis (JMVA) is the central venue for the publication of new, relevant methodology and particularly innovative applications pertaining to the analysis and interpretation of multidimensional data. The journal welcomes contributions to all aspects of multivariate data analysis and modeling, including cluster analysis, discriminant analysis, factor analysis, and multidimensional continuous or discrete distribution theory. Topics of current interest include, but are not limited to, inferential aspects of Copula modeling Functional data analysis Graphical modeling High-dimensional data analysis Image analysis Multivariate extreme-value theory Sparse modeling Spatial statistics.
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