2015/08/31 by Yves F. Atchadé, Atchadé, Yves F.
Engineering · Mathematics · #FOS: Mathematics #Sparse and Compressive Sensing Techniques #Statistical Methods and Inference #Statistics Theory (math.ST) #Tensor decomposition and applications #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1508.07929
38 pages. Minor modifications from previous version
openalex publication_date 2015/08/31 · arxiv created 2016/11/19 · arxiv updated 2016/11/22 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
We study the contraction properties of a quasi-posterior distribution \checkΠn,d obtained by combining a quasi-likelihood function and a sparsity inducing prior distribution on \rsetd, 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 \checkΠn,d puts increasingly high probability on sparse subsets of \rsetd, and contracts towards 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(√(s_⋆log(d)/n)), where s_⋆ is the number of non-zero 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 \citemeinshausen06 contracts in the Frobenius norm at the rate O(√((p+S)log(p)/n)), where p is the number of nodes, and S the number of edges of the true graph.