Empirical Bayes inference in sparse high-dimensional generalized linear models
arxiv(2023)
摘要
High-dimensional linear models have been widely studied, but the developments
in high-dimensional generalized linear models, or GLMs, have been slower. In
this paper, we propose an empirical or data-driven prior leading to an
empirical Bayes posterior distribution which can be used for estimation of and
inference on the coefficient vector in a high-dimensional GLM, as well as for
variable selection. We prove that our proposed posterior concentrates around
the true/sparse coefficient vector at the optimal rate, provide conditions
under which the posterior can achieve variable selection consistency, and prove
a Bernstein–von Mises theorem that implies asymptotically valid uncertainty
quantification. Computation of the proposed empirical Bayes posterior is simple
and efficient, and is shown to perform well in simulations compared to existing
Bayesian and non-Bayesian methods in terms of estimation and variable
selection.
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