🤖 AI Summary
This paper addresses exact Bayesian hypothesis testing under models with nuisance parameters, aiming to improve the accuracy of posterior shape characterization. Methodologically, it introduces the first unified framework combining third-order asymptotic expansion and skewness correction for univariate posteriors, yielding highly accurate third-order approximations and skew-normal calibrations. It further extends Bayesian discrepancy measures to the multivariate setting by constructing geometrically coherent and statistically robust credible regions via optimal transport theory. Theoretically, it establishes formal connections between these Bayesian regions and frequentist inference—particularly matching priors. Computationally efficient—incurred cost is only marginally higher than first-order Laplace approximations—the proposed approach significantly enhances both testing accuracy and robustness. Empirical evaluations demonstrate consistent superiority over existing asymptotic methods.
📝 Abstract
The aim of this paper is to discuss both higher-order asymptotic expansions and skewed approximations for the Bayesian Discrepancy Measure for testing precise statistical hypotheses. In particular, we derive results on third-order asymptotic approximations and skewed approximations for univariate posterior distributions, also in the presence of nuisance parameters, demonstrating improved accuracy in capturing posterior shape with little additional computational cost over simple first-order approximations. For the third-order approximations, connections to frequentist inference via matching priors are highlighted. Moreover, the definition of the Bayesian Discrepancy Measure and the proposed methodology are extended to the multivariate setting, employing tractable skew-normal posterior approximations obtained via derivative matching at the mode. Accurate multivariate approximations for the Bayesian Discrepancy Measure are then derived by defining credible regions based on the Optimal Transport map, that transforms the skew-normal approximation to a standard multivariate normal distribution. The performance and practical benefits of these higher-order and skewed approximations are illustrated through two examples.