Adaptive posterior convergence in sparse high dimensional clipped generalized linear models

Abstract

We develop a framework to study posterior contraction rates in sparse high dimensional generalized linear models (GLM). We introduce a new family of GLMs, denoted by clipped GLM, which subsumes many standard GLMs and makes minor modification of the rest. With a sparsity inducing prior on the regression coefficients, we delineate sufficient conditions on true data generating density that leads to minimax optimal rates of posterior contraction of the coefficients in $\ell_1$ norm. Our key contribution is to develop sufficient conditions commensurate with the geometry of the clipped GLM family, propose prior distributions which do not require any knowledge of the true parameters and avoid any assumption on the growth rate of the true coefficient vector.