Bayesian Discrepancy Measure: Higher-order and Skewed approximations

TL;DR

This work advances Bayesian hypothesis testing by refining the Bayesian Discrepancy Measure (BDM) through higher-order tail-area expansions and skewed posterior approximations. It develops scalar and multivariate extensions, including a center-outward optimal transport framework for multivariate hypotheses and skew-normal based approximations to capture posterior asymmetry with minimal extra computation. The methods are demonstrated on exponential and logistic regression examples, showing that third-order and skewed approaches outperform first-order Gaussian approximations and align well with frequentist concepts under matching priors. The results enhance practical Bayesian testing by providing accurate, computationally efficient tools for tail probabilities, credible regions, and multivariate hypothesis assessment, with clear extensions to high-dimensional settings via SN-based transport maps.

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.

Figures (5)

• Figure 1: First panel: Original SN approximation of a bivariate posterior distribution, with the mode in red and skewness direction indicated by the black line. Second panel: Rotated SN distribution aligning the skewness with the first coordinate; red dashed lines show quantiles of the first rotated component. Third panel: Symmetrized distribution after applying a univariate marginal transformation .Fourth panel: Final standardized and centered Normal distribution. Bottom panel: Visualization of the Optimal Transport (OT) map.
• Figure 2: Exact posterior (in green) and approximate posteriors for $n=6,12$ in the Exponential model (panels 1-2). The blue verical line indicates the posterior median. BDM for a series of parameters (panels 3-4).
• Figure 3: Exact posterior (in green) and approximate posteriors for $n=20,40$ in the Exponential model (panels 1-2). The blue vertical line indicates the posterior median. BDM for a series of parameters (panels 3-4).
• Figure 4: Marginal posterior distributions for the regression parameters of the logistic regression example. The marginal medians are indicated in blue, while the parameters under the null hypothesis are indicated in red.
• Figure 5: Joint posterior for $(\beta_1,\beta_2)$ in the logistic regression example with the first order (IOrder) and skew normal (SN) approximations. The point (0,0) is marked with a cross.