Empirical priors inference in sparse high-dimensional generalized linear models

High-dimensional linear models have been extensively studied in the recent literature, but the developments in high-dimensional generalized linear models, or GLMs, have been much slower. In this talk, I will discuss our work using a novel empirical or data-driven priors framework for inference on the coefficient vector and for variable selection in high-dimensional GLM. In this framework, we use the data to appropriately center the prior distribution, leading to an empirical Bayes posterior distribution. For our proposed method, we prove that the posterior distribution concentrates around the true/sparse coefficient vector at the optimal rate and provide conditions under which the posterior can achieve variable selection consistency. Computation of the proposed empirical Bayes posterior is simple and efficient, and is shown to perform well in simulations compared to existing methods in terms of variable selection in logistic and Poisson regression. I will also briefly discuss our more recent work considering a computational speed-up using a suitable variational approximation.