Bayesian and robust Bayesian analysis in a general setting

Abstract

In this paper we introduce a broad family of loss functions based on the concept of Bregman divergence. We deal with both Bayesian estimation and prediction problems and show that all Bayes solutions associated with loss functions belonging to the introduced family of losses satisfy the same equation. We further concentrate on the concept of robust Bayesian analysis and provide one equation that explicitly leads to robust Bayes solutions. The results are model-free and include many existing results in Bayesian and robust Bayesian contexts in the literature.

Keywords:

• Bayesian inference
• Bregman divergence
• loss function
• posterior regret Gamma minimax
• robust Bayesian inference