Bernstein-von Mises Theorem for Sparse Generalized Linear Model

Hanqing Li; Xuewen Lu

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arXiv · 2605.27137

Bernstein-von Mises Theorem for Sparse Generalized Linear Model

Hanqing Li·Xuewen Lu

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Abstract

We study spike-and-slab priors for generalized linear models with possible grouped sparsity. The main result is an oracle Bernstein--von Mises theorem for the fractional posterior under supportwise likelihood assumptions. The proof develops sparse local asymptotic normality and Laplace approximation around support-specific pseudo-true centers, and combines them with fixed-prior mass, support penalization, recovery geometry, and beta-min separation to obtain contraction, support recovery, Gaussian mixture approximation, and collapse to the oracle Gaussian law. Model-entry verifications are given for Gaussian regression and for logistic, Poisson, probit, Gamma log-link, and negative-binomial log-link regression under stated sufficient conditions. The ordinary posterior is treated only through restricted Gaussian and canonical-link extensions, with coverage under additional active-dimension and moment conditions.

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Hanqing Li, Xuewen Lu. 2026-05-26. Bernstein-von Mises Theorem for Sparse Generalized Linear Model. https://arxiv.org/abs/2605.27137

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