Adaptive mixture variational inference for spike-and-slab regression

2026-09-29

AI Summary

This study addresses the limitation of mean-field approximations in capturing joint variable uncertainty under correlated predictors by proposing an adaptive mixture variational inference method. Built upon Gaussian regression with spike-and-slab priors, the approach directly minimizes the reverse Kullback–Leibler divergence to jointly optimize component parameters and mixture weights during mixture growth, thereby eliminating the need for additional divergence penalties. Experimental results on simulated data demonstrate that the proposed method significantly reduces coverage probability and coefficient covariance errors, effectively improving posterior approximation accuracy in sparse regression. These findings validate the superiority of the direct joint refinement strategy over conventional approaches.

Abstract

Correlated predictors can support competing sparse explanations with similar predictions, making joint uncertainty about variable inclusion difficult to capture with mean-field approximations. We develop an adaptive fitting procedure for mixtures of product distributions in Gaussian regression with a point-mass spike-and-slab prior. It minimizes reverse Kullback-Leibler divergence directly on inclusion indicators and active coefficients, jointly refining component parameters and weights as the mixture grows. This avoids an additional divergence penalty on unused latent coefficients under independent augmentation. Our analysis relates approximation accuracy to mixture size, support coverage and dependence within supports, and establishes contraction, selection consistency and a Bernstein-von Mises approximation under explicit conditions on the prior, posterior concentration and variational error. On all 250 simulated datasets with exact posterior references, mixtures reduce errors in inclusion probabilities, grouped support probabilities and coefficient covariance relative to multistart mean field. Comparisons at fixed mixture size and common initialization favor direct joint refinement over augmented or restricted refinement in posterior divergence. Complete stagewise fitting can nevertheless be more accurate near collinearity. The results support direct joint refinement for posterior approximation while showing that local gains do not ensure superiority of the full adaptive search.

Problem

• spike-and-slab regression
• variational inference
• correlated predictors
• mean-field approximation
• posterior approximation

Innovation

• variational inference
• spike-and-slab prior
• mixture approximation
• reverse KL divergence
• sparse regression