Bernstein-von Mises Theorem for Sparse Generalized Linear Model

2026-05-26

AI Summary

This work develops a Bayesian asymptotic theory for group-sparse structures in generalized linear models using spike-and-slab priors. By introducing a support-dependent likelihood condition and sparse local asymptotic normality, and leveraging Laplace approximations centered at a pseudo-true parameter, the authors establish an oracle-type Bernstein–von Mises theorem for the fractional posterior. Under conditions on prior concentration, support penalization, recovery geometry, and beta-min separation, they prove that the posterior contracts at the optimal rate, exactly recovers the true support, and converges to the oracle Gaussian distribution. The methodology is validated across multiple regression settings, including Gaussian, logistic, Poisson, probit, gamma, and negative binomial log-link models.

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.

Problem

• sparse generalized linear model
• Bernstein-von Mises theorem
• spike-and-slab prior
• grouped sparsity
• posterior asymptotics

Innovation

• Bernstein–von Mises theorem
• spike-and-slab priors
• sparse generalized linear models
• fractional posterior
• support recovery