Bayes factors and posterior estimation: Two sides of the very same coin

TL;DR

This paper clarifies that Bayes factors and posterior estimation are compatible when priors are consistently defined, emphasizing the importance of aligning prior model odds with estimation procedures.

Contribution

It demonstrates that apparent contradictions between Bayes factors and posterior estimates arise from inconsistent prior definitions, advocating for unified prior use.

Findings

Bayes factors and posterior estimates are compatible with consistent priors.

Incompatibilities are due to inconsistent prior model odds.

Recommendations for reporting both Bayes factors and posterior estimates.

Abstract

Recently, several researchers have claimed that conclusions obtained from a Bayes factor (or the posterior odds) may contradict those obtained from Bayesian posterior estimation. In this short paper, we wish to point out that no such "incompatibility" exists if one is willing to consistently define one's priors and posteriors. The key for compatibility is that the (implied) prior model odds used for testing are the same as those used for estimation. Our recommendation is simple: If one reports a Bayes factor comparing two models, then one should also report posterior estimates which appropriately acknowledge the uncertainty with regards to which of the two models is correct.