Closed-form approximations of the two-sample Pearson Bayes factor

TL;DR

The paper tackles computing Bayes factors for two-sample mean differences from minimal summary statistics by approximating the gamma-quotient $C_{\nu}=\frac{\Gamma(\nu/2)}{\Gamma(\nu/2+1/2)}$ in the Pearson Bayes factor $\mathrm{PBF}_{10}=C_{\nu}\sqrt{\frac{1}{\pi}\left(1+\frac{t^2}{\nu}\right)^{\nu-1}}$. It presents three closed-form approximations—Wendel's asymptotic, Stirling's formula, and Frame's quotient formula—to replace the problematic gamma ratio with elementary expressions, enabling calculation from $t$ and $\nu$ alone. Empirical examples and simulations show these approximations substantially improve over the classic BIC-based method, with Frame's approach achieving the highest accuracy (often below 0.01% error for moderate sample sizes) and all three methods outperforming BIC across a range of $N$. The work offers practical, calculator-friendly tools for Bayesian evidential assessment in two-sample designs, while noting the dependence on the Pearson Type VI prior and aligning this with the information-prior spirit of BIC-based approaches.

Abstract

In this paper, I present three closed-form approximations of the two-sample Pearson Bayes factor, a recently developed index of evidential value for data in two-group designs. The techniques rely on some classical asymptotic results about Gamma functions. These approximations permit simple closed-form calculation of the Pearson Bayes factor in cases where only minimal summary statistics are available (i.e., the t-score and degrees of freedom). Moreover, these approximations vastly outperform the classic BIC method for approximating Bayes factors from experimental designs.

Figures (3)

• Figure 1: A Pearson Type VI prior for $\tau$, plotted as a function of shape parameter $\alpha$.
• Figure 2: Average percent error of the Wendel, Stirling, and Frame methods (compared to analytic Bayes factor) for values of total sample size $N$ ranging from 4 to 100.
• Figure 3: Average percent error of the BIC method and the Wendel method (each compared to the analytic Pearson Bayes factor) for values of total sample size $N$ ranging from 4 to 100.