Neyman-Pearson lemma for Bayes factors

TL;DR

This paper demonstrates that the Neyman-Pearson lemma can be applied to Bayes factors when considering expected error rates, providing a bridge between Bayesian and frequentist hypothesis testing.

Contribution

It extends the Neyman-Pearson lemma to Bayes factors, linking Bayesian and frequentist frameworks through expected error rates.

Findings

Bayes factors maximize expected power for fixed expected type-1 error.

Expected type-1 error rate for simple null hypotheses aligns with frequentist error.

Connections between Karlin-Rubin theorem and Bayes factors are discussed.

Abstract

We point out that the Neyman-Pearson lemma applies to Bayes factors if we consider expected type-1 and type-2 error rates. That is, the Bayes factor is the test statistic that maximises the expected power for a fixed expected type-1 error rate. For Bayes factors involving a simple null hypothesis, the expected type-1 error rate is just the completely frequentist type-1 error rate. Lastly we remark on connections between the Karlin-Rubin theorem and uniformly most powerful tests, and Bayes factors. This provides frequentist motivations for computing the Bayes factor and could help reconcile Bayesians and frequentists.