Analyzing the Relationship Between Prior Sensitivity and Posterior Stability in Bayesian Model Evaluation

Posted: Jul 15, 2023

• Jayden Adams
• Jeremy Cox
• Jessica Perry

Abstract

Bayesian statistics has revolutionized statistical inference by providing a coherent framework for incorporating prior knowledge and quantifying uncertainty. The fundamental Bayesian paradigm combines prior distributions with observed data through Bayes' theorem to yield posterior distributions, which form the basis for inference and decision-making. However, a persistent challenge in Bayesian methodology concerns the specification of prior distributions and their impact on posterior inferences. While the influence of prior choices on posterior results has been extensively studied through sensitivity analysis, the relationship between prior sensitivity and posterior stability remains poorly understood. This paper makes several original contributions to the Bayesian literature. First, we introduce a novel theoretical framework that formally characterizes the relationship between prior sensitivity and posterior stability. Second, we develop the Sensitivity-Stability Trade-off Index (SSTI), a quantitative measure that captures the dynamic interplay between these two fundamental aspects of Bayesian inference. Third, we identify and mathematically characterize three distinct regimes in the sensitivity-stability relationship: compensatory, antagonistic, and synergistic. Fourth, we provide practical diagnostic tools and guidelines for model builders to navigate these regimes effectively.