Abstract

Given a collection of candidate probability models for an observed data y, a fundamental statistical task is to evaluate which models are more likely to have generated y. Tackling this problem within a Bayesian framework requires one to complement the probability model for y (likelihood) with a prior probability model on the parameters describing each of the candidate models, as well as to specify model prior probabilities and possibly a utility function. We shall review some recent strategies for high-dimensional model choice, and then discuss what we denominate the “model separation principle”. This principle states that the models under consideration should be minimally different from each other, else it becomes hard to distinguish them on the basis of the observed y. In the common setting where some of the models are nested this principle is violated, as say Model 1 is a particular case of Model 2 and thus these models are not well separated. We shall review a class of prior distributions called non-local priors (NLPs) as a way to enforce the model separation principle and some of the NLP properties, focusing on parsimony and accelerated convergence rates in high-dimensional inference. We shall illustrate their use in ongoing work related to regression and robust regression.