ON THE CONTRACTION PROPERTIES OF SOME HIGH-DIMENSIONAL QUASI-POSTERIOR DISTRIBUTIONS

• 成果类型: Article
• 署名作者: Atchade, Yves A.
• 署名单位: University of Michigan System; University of Michigan
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/16-AOS1526
• 发表日期: 2017
• 页码: 2248-2273
• 关键词: ising-model selection; variable selection; regression

摘要

We study the contraction properties of a quasi-posterior distribution (sic)(n, d) obtained by combining a quasi-likelihood function and a sparsity inducing prior distribution on R-d, as both n (the sample size), and d (the dimension of the parameter) increase. We derive some general results that highlight a set of sufficient conditions under which (sic)(n, d) puts increasingly high probability on sparse subsets of R-d, and contracts toward the true value of the parameter. We apply these results to the analysis of logistic regression models, and binary graphical models, in high-dimensional settings. For the logistic regression model, we shows that for well-behaved design matrices, the posterior distribution contracts at the rate O(root s(star)log(d)/n), where s(star) is the number of nonzero components of the parameter. For the binary graphical model, under some regularity conditions, we show that a quasi-posterior analog of the neighborhood selection of [Ann. Statist. 34 (2006) 1436-1462] contracts in the Frobenius norm at the rate O(root(p + S) log(p)/n), where p is the number of nodes, and S the number of edges of the true graph.