Covariate-dependent Nonparametric $g$-modeling for regression via infinite Mixture-of-Expertizing class

Abstract

Empirical Bayes $g$-modeling captures unit-level heterogeneity by estimating a latent prior distribution from observed data. In the existing formulations, however, the prior is shared by all units. In this paper, we develop a covariate-dependent g-modeling framework for regression in which the entire prior distribution of the regression coefficients is allowed to depend on covariates. We formulate the estimation of the prior as nonparametric maximum likelihood estimation (NPMLE) of the covariate-dependent prior, and show that the unrestricted problem is ill-posed. To resolve this, we introduce the infinite Mixture-of-Expertizing class of conditional priors, under which the NPMLE is precisely a softmax-gated Mixture of Experts (MoE) whose number of experts is not fixed in advance but is determined by the data. Building on a first-order optimality condition, we propose two exemplar-based estimation algorithms that select experts automatically, together with a post-hoc aggregation of experts for interpretation. On the theoretical side, we show that every conditional prior in the class is Lipschitz continuous in the covariates, and that aggregated softmax gates can approximate any continuous gate function. The effectiveness of the proposed NPMLE is shown through application to synthetic datasets and real datasets.