Local and global robustness in conjugate Bayesian analysis

TL;DR

This paper explores how small and large changes in conjugate priors affect Bayesian inference, using a geometric approach to analyze sensitivity and robustness in posterior results.

Contribution

It introduces a flexible perturbation framework within local mixture models and employs geometric methods for sensitivity analysis in Bayesian conjugate models.

Findings

Perturbations can be systematically analyzed using a geometric approach.

Sensitivity measures are defined on a smooth convex manifold.

The framework generalizes previous linear perturbation methods.

Abstract

This paper studies the influence of perturbations of conjugate priors in Bayesian inference. A perturbed prior is defined inside a larger family, local mixture models, and the effect on posterior inference is studied. The perturbation, in some sense, generalizes the linear perturbation studied in \cite{Gustafson1996}. It is intuitive, naturally normalized and is flexible for statistical applications. Both global and local sensitivity analyses are considered. A geometric approach is employed for optimizing the sensitivity direction function, the difference between posterior means and the divergence function between posterior predictive models. All the sensitivity measure functions are defined on a convex space with non-trivial boundary which is shown to be a smooth manifold.