On Bayesian Oracle Properties

TL;DR

This paper extends the understanding of Bayesian oracle properties to a broad class of quasi-posteriors, including non-regular models, providing theoretical guarantees for model selection and averaging methods.

Contribution

It introduces a unified framework for establishing Bayesian oracle properties in general quasi-posterior settings, including non-regular models with partial identification.

Findings

Oracle properties hold under general quasi-posterior frameworks.

Guidelines for prior penalty and temperature parameter choices in non-regular models.

Bayesian model averaging can outperform model selection in complex models.

Abstract

When model uncertainty is handled by Bayesian model averaging (BMA) or Bayesian model selection (BMS), the posterior distribution possesses a desirable "oracle property" for parametric inference, if for large enough data it is nearly as good as the oracle posterior, obtained by assuming unrealistically that the true model is known and only the true model is used. We study the oracle properties in a very general context of quasi-posterior, which can accommodate non-regular models with cubic root asymptotics and partial identification. Our approach for proving the oracle properties is based on a unified treatment that bounds the posterior probability of model mis-selection. This theoretical framework can be of interest to Bayesian statisticians who would like to theoretically justify their new model selection or model averaging methods in addition to empirical results. Furthermore, for non-regular models, we obtain nontrivial conclusions on the choice of prior penalty on model complexity, the temperature parameter of the quasi-posterior, and the advantage of BMA over BMS.