Restricted type II maximum likelihood priors on regression coefficients

TL;DR

This paper investigates restricted type II maximum likelihood priors for regression coefficients in Bayesian hypothesis testing, showing they often align with BIC results and address issues with arbitrary prior scales.

Contribution

It introduces a method for defining priors by estimating their variances with restrictions, ensuring they remain suitably vague for model uncertainty.

Findings

Type II ML priors often match BIC outcomes.

Restrictions on priors improve model uncertainty assessment.

Method addresses issues with large prior scales in testing.

Abstract

In Bayesian hypothesis testing and model selection, prior distributions must be chosen carefully. For example, setting arbitrarily large prior scales for location parameters, which is common practice in estimation problems, can lead to undesirable behavior in testing (Lindley's paradox). We study the properties of some restricted type II maximum likelihood (type II ML) priors on regression coefficients. In type II ML, hyperparameters are "estimated" by maximizing the marginal likelihood of a model. In this article, we define priors by estimating their variances or covariance matrices, adding restrictions which ensure that the resulting priors are at least as vague as conventional proper priors for model uncertainty. We find that these type II ML priors typically yield results that are close to answers obtained with the Bayesian Information Criterion (BIC).