Bayesian Discrepancy Measure: Higher-Order and Skewed Approximations

Area of Science:

• Statistics
• Bayesian Inference
• Hypothesis Testing

Background:

• Accurate approximation of posterior distributions is crucial for Bayesian hypothesis testing.
• Existing first-order approximations may lack sufficient precision for complex statistical models.
• The Bayesian discrepancy measure is a key tool for precise hypothesis testing.

Purpose of the Study:

• To develop and evaluate higher-order asymptotic and skewed approximations for the Bayesian discrepancy measure.
• To extend these approximations to both univariate and multivariate settings, including nuisance parameters.
• To demonstrate improved accuracy and practical benefits over existing methods.

Main Methods:

• Derivation of third-order asymptotic approximations for univariate posterior distributions.
• Development of skewed approximations using skew-normal distributions via derivative matching.
• Extension to multivariate settings using optimal transport maps for accurate credible regions.

Main Results:

• Third-order and skewed approximations show improved accuracy in capturing posterior shape.
• These advanced approximations offer greater precision with little additional computational cost.
• Connections between Bayesian higher-order approximations and frequentist inference are established.

Conclusions:

• Higher-order asymptotic and skewed approximations provide a significant improvement for Bayesian hypothesis testing.
• The proposed methods are computationally efficient and practically beneficial.
• The study successfully extends accurate approximation techniques to complex multivariate scenarios.