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October 2017 On the contraction properties of some high-dimensional quasi-posterior distributions
Yves A. Atchadé
Ann. Statist. 45(5): 2248-2273 (October 2017). DOI: 10.1214/16-AOS1526

Abstract

We study the contraction properties of a quasi-posterior distribution $\check{\Pi}_{n,d}$ obtained by combining a quasi-likelihood function and a sparsity inducing prior distribution on $\mathbb{R}^{d}$, 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 $\check{\Pi}_{n,d}$ puts increasingly high probability on sparse subsets of $\mathbb{R}^{d}$, 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 $O(\sqrt{s_{\star}\log(d)/n})$, 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 $O(\sqrt{(p+S)\log(p)/n})$, where $p$ is the number of nodes, and $S$ the number of edges of the true graph.

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Yves A. Atchadé. "On the contraction properties of some high-dimensional quasi-posterior distributions." Ann. Statist. 45 (5) 2248 - 2273, October 2017. https://doi.org/10.1214/16-AOS1526

Information

Received: 1 September 2015; Revised: 1 September 2016; Published: October 2017
First available in Project Euclid: 31 October 2017

zbMATH: 1383.62058
MathSciNet: MR3718168
Digital Object Identifier: 10.1214/16-AOS1526

Subjects:
Primary: 62F15, 62Jxx

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 5 • October 2017
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