On the Contraction Properties of Some High-dimensional Quasi-posterior Distributions

We study the contraction properties of a class of quasi-posterior distributions Π̌n,d obtained by combining a quasi-likelihood function and a sparsity inducing prior distribution on R, as both n (the sample size), and d (the dimension of the parameter) increase. We derive two general results that highlight a set of sufficient conditions under which Π̌n,d puts increasingly high probability on sparse subsets of R, and contracts towards the true value of the parameter. We apply these results to the analysis of high-dimensional linear and logistic regression models. For the linear regression model, we shows that for well-behaved design matrices, the proposed quasi-posterior distribution contracts at the same rate O( √ s? log(d)/n) as the well-known LASSO procedure, where s? is the number of non-zero components of the parameter. Our results also show that the behavior of the quasi-posterior distribution is robust to the misspecification of the noise distribution, particularly when a large regularization parameter value is used in the prior distribution.