Optimality of the Half-Order Exponent in the Turing-Good Identities for Bayes Factors

TL;DR

This paper demonstrates that using a half-order (square-root) power in the Turing-Good identities for Bayes factors provides a uniquely stable and balanced diagnostic, improving Monte Carlo evaluation reliability across models.

Contribution

It introduces a nonasymptotic variance theory showing the half-order power as minimax-stable, ensuring balanced variability and finite moments in Bayes factor diagnostics.

Findings

Half-order power minimizes variance imbalance.

Balanced two-sample diagnostic with uniform variance bounds.

Stable finite-sample behavior demonstrated in simulations.

Abstract

Bayes factors are widely computed by Monte Carlo, yet heavy-tailed sampling distributions can make numerical validation unreliable. The Turing--Good identities provide exact moment equalities for powers of a Bayes factor (a density ratio). When these identities are used as Good-check diagnostics, the power choice becomes a statistical design parameter. We develop a nonasymptotic variance theory for Monte Carlo evaluation of the identities and show that the half-order (square-root) power is uniquely minimax-stable: it equalizes variability across the two model orientations and is the only choice that guarantees finite second moments in a distribution-free worst-case sense over all mutually absolutely continuous model pairs. This yields a balanced two-sample half-order diagnostic that is symmetric in model labeling and has a uniform variance bound at fixed computational budget; in small-overlap regimes it is guaranteed to be no less efficient than the standard one-sided Turing check. Simulations for binomial Bayes factor workflows illustrate stable finite-sample behavior and sensitivity to simulator--evaluator mismatches. We further connect the half-order overlap viewpoint to stable primitives for normalizing-constant ratios and importance-sampling degeneracy summaries.