Bortolato, ElenaBertolino, FrancescoMusio, MonicaVentura, Laura2025-09-082025-09-082025Bortolato E, Bertolino F, Musio M, Ventura L. Bayesian discrepancy measure: higher-order and skewed approximations. Entropy. 2025 Jul;27(7):657. DOI: 10.3390/e270706571099-4300http://hdl.handle.net/10230/71143The 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.application/pdfeng© 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).info:eu-repo/semantics/articlehttp://dx.doi.org/10.3390/e27070657Bayesian discrepancy measureCredible regionsEvidenceHigher-order asymptoticsMatching priorsNuisance parameterOptimal transport mapPrecise null hypothesisSkewed approximationsSkew-normal distributionTail area probabilityinfo:eu-repo/semantics/openAccess