Bayes Risk for Goodness of Fit Tests

TL;DR

This paper introduces a Bayesian risk-based framework for goodness-of-fit tests, revealing that optimal calibration occurs on the moderate-deviation scale, unifying classical and information-theoretic approaches.

Contribution

It formalizes the risk calibration for GOF tests, connects Bayesian risk with Sanov asymptotics, and applies the framework to various statistical testing scenarios.

Findings

Bayes-risk optimal thresholds operate on the moderate-deviation scale.

Explicit connections between Bayesian risk and Sanov information asymptotics are established.

Applications include location testing, shape testing, and links to Fisher information geometry.

Abstract

We develop a unified framework for goodness-of-fit (GOF) testing through the lens of Bayes risk. Classical GOF procedures are commonly calibrated either at fixed significance level (CLT scale) or through exponential error exponents (LDP scale). We establish that Bayes-risk optimal calibration operates on the moderate-deviation (MDP) scale, producing canonical $\sqrt{\log n}$ inflation of rejection thresholds and polynomially decaying Type I error. Our main contributions are: (i) we formalise the Rubin--Sethuraman program for KS-type statistics as a risk-calibration theorem with explicit regularity conditions on priors and empirical-process functionals; (ii) we develop the precise connection between Bayes-risk expansions and Sanov information asymptotics, showing how $\log n$-order truncations arise naturally when risk, rather than pure exponents, is the evaluation criterion; (iii) we provide detailed applications to location testing under Laplace families, shape testing via Bayes factors, and connections to Fisher information geometry. The organizing principle throughout is that sample size enters Bayes-optimal GOF cutoffs through the MDP scale, unifying KS-based and Sanov-based perspectives under a single risk criterion.