Adaptive posterior convergence in sparse high dimensional clipped generalized linear models

TL;DR

This paper introduces a new framework for analyzing posterior contraction in sparse high-dimensional clipped GLMs, achieving minimax optimal rates without prior knowledge of true parameters.

Contribution

It develops a novel family of clipped GLMs and establishes conditions for optimal posterior contraction rates with priors that do not depend on true parameters.

Findings

Achieves minimax optimal posterior contraction rates in $\,\ell_1$ norm.

Proposes prior distributions independent of true parameters.

Provides sufficient conditions aligned with clipped GLM geometry.

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.