Tractable Bayesian variable selection: beyond normality

Bayesian variable selection for continuous outcomes often assumes normality, and so do its theoretical studies. There are sound reasons behind this assumption, particularly for large $p$: ease of interpretation, analytical and computational convenience. More flexible frameworks exist, including semi- or non-parametric models, often at the cost of losing some computational or theoretical tractability. We propose a simple extension of the Normal model that allows for skewness and thicker-than-normal tails but preserves its tractability. We show that a classical strategy to induce asymmetric Normal and Laplace errors via two-piece distributions leads to easy interpretation and a log-concave likelihood that greatly facilitates optimization and integration. We also characterize asymptotically its maximum likelihood estimator and Bayes factor rates under model misspecification. Our work focuses on the likelihood and can thus be combined with any likelihood penalty or prior, but here we adopt non-local priors, a family that induces extra sparsity and which we characterize under misspecification for the first time. Under suitable conditions Bayes factor rates are of the same order as those that would be obtained under the correct model, but we point out a potential loss of sensitivity to detect truly active covariates. Our examples show how a novel approach to infer the error distribution leads to substantial gains in sensitivity, thus warranting the effort to go beyond normality, whereas for near-normal data one can get substantial speedups relative to assuming unnecessarily flexible models. The methodology is available as part of R package mombf.