# Bayesian Discrepancy Measure: Higher-Order and Skewed Approximations

## TL;DR

This paper introduces improved Bayesian methods for hypothesis testing using higher-order and skewed approximations, offering better accuracy with minimal computational cost.

## Contribution

The paper presents novel third-order and skewed approximations for the Bayesian discrepancy measure, extending to multivariate settings with optimal transport maps.

## Key findings

- Third-order approximations improve posterior shape accuracy with low computational cost.
- Skewed approximations enhance hypothesis testing in univariate and multivariate settings.
- Optimal transport maps enable accurate multivariate credible regions from skew-normal approximations.

## Abstract

The aim of this paper is to discuss both higher-order asymptotic expansions and skewed approximations for the Bayesian discrepancy measure used in testing precise statistical hypotheses. In particular, we derive results on third-order asymptotic approximations and skewed approximations for univariate posterior distributions, including cases with nuisance parameters, demonstrating improved accuracy in capturing posterior shape with little additional computational cost over simple first-order approximations. For 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 an 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.