On the Ubiquity of Information Inconsistency for Conjugate Priors

TL;DR

This paper demonstrates that conjugate priors often lead to non-definitive Bayesian conclusions in hypothesis testing, highlighting the need for alternative priors that ensure information consistency.

Contribution

It reveals the widespread occurrence of information inconsistency with conjugate priors and advocates for the use of theoretically recommended, information consistent priors in Bayesian testing.

Findings

Conjugate priors often cause bounded Bayes factors despite strong evidence.

Scale mixtures and adaptive priors are shown to be information consistent.

Using conjugate priors in hypothesis testing can lead to misleading conclusions.

Abstract

Informally, "Information Inconsistency" is the property that has been observed in many Bayesian hypothesis testing and model selection procedures whereby the Bayesian conclusion does not become definitive when the data seems to become definitive. An example is that, when performing a t-test using standard conjugate priors, the Bayes factor of the alternative hypothesis to the null hypothesis remains bounded as the t statistic grows to infinity. This paper shows that information inconsistency is ubiquitous in Bayesian hypothesis testing under conjugate priors. Yet the title does not fully describe the paper, since we also show that theoretically recommended priors, including scale mixtures of conjugate priors and adaptive priors, are information consistent. Hence the paper is simply a forceful warning that use of conjugate priors in testing and model selection is highly problematical, and should be replaced by the information consistent alternatives.