Lovric (2025) The Bayes Factor Reversal Paradox

Summary

Lovric identifies a new paradox — a conflict within Bayesian inference, unlike Lindley's (1957) frequentist-versus-Bayesian paradox. In the normal model with known variance, he proves that for any two-sided result significant at the 0.05 level there exist prior variances such that the Bayes factor indicates evidence for the alternative under one choice and evidence for the null under another. Thus the same data, testing the same hypothesis, can yield opposite conclusions depending solely on the prior — and, unlike the Jeffreys–Lindley paradox, this occurs at realistic sample sizes. The result formalizes Robert's (2016) concern about the arbitrariness of the prior scale in default Bayes-factor testing.

Key Claims

• A within-Bayesian paradox. Whereas the Jeffreys–Lindley paradox pits frequentist against Bayesian conclusions as $n\to\infty$, the Bayes factor reversal (BFR) paradox is a conflict inside Bayesian inference driven purely by the prior scale.
• Closed-form reversal. Reparametrizing by $k=n\tau^2$ (effective prior precision relative to the data), $\mathrm{BF}_{01}(z;k)=\sqrt{1+k}\,\exp\!\big(-\tfrac{z^2 k}{2(1+k)}\big)$. At $k=0$, $\mathrm{BF}_{01}=1$ (neutral); as $k\to\infty$, $\mathrm{BF}_{01}\to\infty$ (favours the null — the Jeffreys–Lindley effect).
• Flip point. For any $z$ significant at the two-sided 0.05 level there is a flip point in $k$ where $\mathrm{BF}_{01}$ crosses $1$, so the direction of the evidence reverses as the prior variance is varied — not just its magnitude.
• Realistic sample sizes. Unlike Jeffreys–Lindley, the reversal does not require $n\to\infty$; it arises at ordinary sample sizes.
• Answers Robert (2016). It substantiates "The expected demise of the Bayes factor" critique that the prior-scale choice introduces a high degree of arbitrariness into default Bayes-factor hypothesis testing.

Concepts Introduced or Extended

• Bayes Factor

Entities Mentioned

• Miodrag M. Lovric

Quotes

"For any two-sided statistically significant result at the 0.05 level there exist prior variances such that the Bayes factor indicates evidence for the alternative with one choice while indicating evidence for the null with another. Thus, the same data, testing the same hypothesis, can yield opposite conclusions depending solely on prior choice."

My Take

A pointed sharpening of the prior-sensitivity caveat every Bayes-factor user is told to worry about: Lovric shows the sensitivity is bad enough to flip the sign of the conclusion at realistic $n$, not merely inflate or deflate the strength of evidence. The $k=n\tau^2$ reparametrization is the clarifying move — it collapses sample size and prior variance into one knob and makes the flip point explicit. It is the natural adversarial footnote to Kass-Raftery (1995) and the BIC-based advocacy of Raftery (1995): default/objective Bayes-factor testing cannot escape a prior-scale choice, and that choice can decide the answer. The honest scope limit is that the proof is for the normal-known-variance point-null case; how far the reversal generalizes (and whether principled scale choices avoid it) is the open question.