Kass-Raftery (1995) Bayes Factors

Summary

Kass and Raftery's review is the standard reference on Bayes factors — the ratio of marginal likelihoods that quantifies the evidence one model or hypothesis provides over another, introduced by Jeffreys. It covers the definition and interpretation of Bayes factors, an interpretive scale for strength of evidence, the full menu of computation methods (from exact integration to the Schwarz/BIC approximation), the crucial issue of prior sensitivity, and five scientific applications (genetics, sports, ecology, sociology, psychology).

Key Claims

• Definition. The Bayes factor $B_{10}=p(D\mid M_1)/p(D\mid M_0)$ is the ratio of the models' marginal (integrated) likelihoods; it equals the posterior odds divided by the prior odds, so it summarizes the data's contribution to the comparison. General — the models need not be nested.
• Evidence, including for the null. Unlike classical significance tests, Bayes factors can provide evidence in favor of a null hypothesis and incorporate external information through the priors.
• Interpretive scale (after Jeffreys): reading $2\ln B_{10}$, roughly 0–2 is "not worth more than a bare mention", 2–6 "positive", 6–10 "strong", and $>10$ "very strong/decisive".
• Computation. Several techniques: exact/analytic evaluation, the Laplace approximation (asymptotic, easy from likelihood-maximizing output), importance sampling and Monte Carlo integration/quadrature, and MCMC; the Schwarz criterion (BIC) gives a rough, easy approximation to the log Bayes factor.
• Nonstandard models. Where regularity conditions fail, computing a Bayes factor can be technically simpler than constructing a non-Bayesian significance test.
• Prior sensitivity is central. Bayes factors depend on the priors (diffuse priors introduce arbitrary constants — Lindley/Bartlett paradox); it is important and feasible to assess the sensitivity of conclusions to the priors used.

Concepts Introduced or Extended

• Bayes Factor
• Model Comparison

Entities Mentioned

• Robert E. Kass
• Adrian E. Raftery

Quotes

"Several techniques are available for computing Bayes factors, including asymptotic approximations that are easy to compute using the output from standard packages that maximize likelihoods … The Schwarz criterion (or BIC) gives a rough approximation to the logarithm of the Bayes factor."

My Take

The paper that made Bayes factors usable: it collected the scattered computational tools into one practical menu and, just as importantly, told practitioners how to read the number (the Jeffreys scale) and how to worry about it (prior sensitivity). Its most consequential single takeaway is the BIC-as-approximate-Bayes-factor shortcut, which let people compute an evidence measure from any likelihood-maximizing package — the bridge to Raftery's and Raftery's (1995) BIC-for-model-selection program, and the reason BIC is everywhere. The honest caveat it foregrounds — Bayes factors' dependence on the prior, unlike posterior estimates — is the recurring gotcha the wiki's bayes-factor and marginal-likelihood pages keep returning to.