Dirichlet process mixtures of block $g$ priors for model selection and prediction in linear models

TL;DR

This paper proposes Dirichlet process mixtures of block g priors for improved model selection and prediction in linear models, effectively handling predictor correlation and avoiding paradoxes.

Contribution

It introduces a novel extension of g priors that allows differential shrinkage and accounts for predictor correlation, with a scalable MCMC inference method.

Findings

Consistent in various senses and avoids the Lindley paradox.

Enhances power to detect small effects with minimal false discoveries.

Performs well on real and simulated datasets.

Abstract

This paper introduces Dirichlet process mixtures of block $g$ priors for model selection and prediction in linear models. These priors are extensions of traditional mixtures of $g$ priors that allow for differential shrinkage for various (data-selected) blocks of parameters while fully accounting for the predictors' correlation structure, providing a bridge between the literatures on model selection and continuous shrinkage priors. We show that Dirichlet process mixtures of block $g$ priors are consistent in various senses and, in particular, that they avoid the conditional Lindley ``paradox'' highlighted by Som et al. (2016). Further, we develop a Markov chain Monte Carlo algorithm for posterior inference that requires only minimal ad-hoc tuning. Finally, we investigate the empirical performance of the prior in various real and simulated datasets. In the presence of a small number of very large effects, Dirichlet process mixtures of block $g$ priors lead to higher power for detecting smaller but significant effects without only a minimal increase in the number of false discoveries.