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

TL;DR

The paper identifies the half-order exponent $t=\tfrac{1}{2}$ as the unique, minimax-stable choice for validating Bayes-factor computations via the Turing--Good identities, by equalizing and bounding the second moments of the two diagnostic sides under mutual absolute continuity. It derives exact variance expressions in terms of the overlap $\rho=I(\tfrac{1}{2})$ and proves that the two-sided, split-sample half-order diagnostic has uniformly finite variance and, in the small-overlap regime, outperforms the standard one-sided Turing check. The work extends the half-order idea to a geometric bridge for normalizing-constant ratios and to importance-sampling diagnostics, where $\rho$ provides a distribution-free, interpretable measure of weight concentration. Simulations on binomial Bayes-factor workflows illustrate stable finite-sample behavior and sensitivity to simulator--evaluator mismatches, and the results motivate a practical default design: a symmetric two-sided half-order check with overlap estimation for robust, cost-aware diagnostics in Bayesian computation. Overall, the half-order framework offers a principled, stable basis for validating and designing ratio-estimation procedures under tail-robustness constraints.

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.

Figures (4)

• Figure 1: Simulation study 1A (binomial example, correctly specified; $n=50$): fan charts of running Monte Carlo discrepancies as a function of $m$ (number of simulated datasets per hypothesis) across $R=10{,}000$ independent runs. Panel (A) shows $\widehat{\Delta}_{1/2}(m)=\widehat{\mathbb{E}}_{\mathcal{H}_2,\le m}[B^{1/2}]-\widehat{\mathbb{E}}_{\mathcal{H}_1,\le m}[B^{-1/2}]$. Panel (B) shows the reverse Turing discrepancy $\widehat{\mathbb{E}}_{\mathcal{H}_1,\le m}[B^{-1}]-1$. Panel (C) shows the reverse Good discrepancy $\widehat{\mathbb{E}}_{\mathcal{H}_1,\le m}[B^{-2}]-\widehat{\mathbb{E}}_{\mathcal{H}_2,\le m}[B^{-1}]$. Solid curves are means; shaded bands are central 50% (dark) and 90% (light) intervals; the horizontal dotted line indicates $0$ (the exact identity).
• Figure 2: Simulation study 1B (binomial example; simulator--evaluator mismatch; $n=50$): fan charts of running Monte Carlo discrepancies across $R=10{,}000$ runs. Panels (A)--(C) match Figure \ref{['fig:sim1_outliers_sqrt']}. The vertical blue line indicates the first $m$ at which the central 90% band excludes $0$ (on the plotted $m$ grid).
• Figure 3: Toy normalizing-constant ratio estimation under tail mismatch in the $\mathrm{Beta}(a,1)$ vs. $\mathrm{Unif}(0,1)$ family (Simulation study 3). We report $\log_{10}(\widehat{r}/r)$ over $R=2000$ independent repetitions at matched computational cost $N_{\text{total}}=2000$. Top row ($a=\tfrac{1}{2}$): forward breakdown. Panels (A)--(B) compare the forward one-sided estimator $\widehat{r}_{\mathrm{F}}=\frac{1}{N}\sum_{i=1}^N w(X_{2i})$ ($X_{2i}\sim p_2$; $N=2000$) to the half-order bridge $\widehat{r}_{1/2}$ ($n_1=n_2=1000$). Because $\mathbb{E}_{p_2}[w^2]=\infty$ at $a=\tfrac{1}{2}$, the forward estimator exhibits a pronounced right tail and many outliers. Bottom row ($a=3$): reverse breakdown. Panels (C)--(D) compare the reverse one-sided estimator $\widehat{r}_{\mathrm{R}}=\{\frac{1}{N}\sum_{i=1}^N w(X_{1i})^{-1}\}^{-1}$ ($X_{1i}\sim p_1$; $N=2000$) to the same half-order bridge estimator ($n_1=n_2=1000$). Because $\mathbb{E}_{p_1}[w^{-2}]=\infty$ for $a\ge 2$, the reverse estimator develops a heavy left tail (extreme underestimation on the log scale). In all panels, the half-order bridge remains tightly concentrated around $0$. Panel headers report the closed-form overlap $\rho^2=4a/(a+1)^2$ and the corresponding asymptotic RSD prediction from Proposition \ref{['prop:halforder-bridge-stability']}.
• Figure 4: Importance-sampling weight concentration away from $t=\tfrac{1}{2}$ and stability anchored by the half-order overlap in the $\mathrm{Beta}(a,1)$ vs. $\mathrm{Unif}(0,1)$ family ($R=500$ replicates, $N=2000$). Top row (proposal-side): $X_i\sim p_2$ with $a=1/2$, comparing weights $w=W^t$ for $t\in\{1,\tfrac{1}{2}\}$. Bottom row (target-side): $X_i\sim p_1$ with $a=3$, comparing reverse weights $w=W^{t-1}$ for $t\in\{\tfrac{1}{4},\tfrac{1}{2}\}$. Left panels show Lorenz-type concentration curves $C(p)=\sum_{i=1}^{\lceil pN\rceil}\tilde{w}_{(i)}$ (median and 10/90% envelopes across replicates) on a log-$p$ scale; the diagonal $C(p)=p$ corresponds to uniform weights. Middle panels show boxplots of the normalized reciprocal squared-weight concentration $\kappa_N := 1/(N\sum_{i=1}^N \tilde{w}_i^2)$ (commonly reported as $\mathrm{ESS}/N$), with dotted reference lines at the half-order overlap benchmark $\rho^2=4a/(a+1)^2$. Right panels show the cumulative weight share carried by the largest 1% of weights ($S_{0.01}$); the dotted line is the uniform baseline $0.01$. In both stress tests, the half-order weights exhibit mild concentration and $\kappa_N$ close to $\rho^2$, whereas the tail-sensitive exponents yield substantial concentration and degraded squared-weight summaries.

Theorems & Definitions (26)

• Theorem 2.1: Turing--Good moment identity; Good1985
• proof
• Corollary 2.2: Turing's theorem
• Lemma 2.3: Basic bounds and equality characterization
• proof
• Lemma 2.4: Variance identities
• proof
• Corollary 2.5: Half-order equalization
• proof
• Lemma 2.6: CGF of $\log B$ and log-convexity of $I$