On Exact Bayesian Credible Sets for Classification and Pattern Recognition

TL;DR

This paper introduces a method to construct exact Bayesian credible sets for classification problems, overcoming the limitations of traditional credible sets that cannot achieve arbitrary levels, by connecting Bayesian and Neyman--Pearson approaches.

Contribution

It proposes a generalized credible set that attains any preassigned level and introduces a randomized decision rule to address discrete credible levels, enhancing Bayesian inference for classification.

Findings

Developed a randomized decision rule for exact credible levels.

Introduced the Steering Wheel Plot for visualizing uncertainty.

Connected Bayesian credible sets with Neyman--Pearson lemma.

Abstract

The current definition of a Bayesian credible set cannot, in general, achieve an arbitrarily preassigned credible level. This drawback is particularly acute for classification problems, where there are only a finite number of achievable credible levels. As a result, there is as of today no general way to construct an exact credible set for classification. In this paper, we introduce a generalized credible set that can achieve any preassigned credible level. The key insight is a simple connection between the Bayesian highest posterior density credible set and the Neyman--Pearson lemma, which, as far as we know, hasn't been noticed before. Using this connection, we introduce a randomized decision rule to fill the gaps among the discrete credible levels. Accompanying this methodology, we also develop the Steering Wheel Plot to represent the credible set, which is useful in visualizing the uncertainty in classification. By developing the exact credible set for discrete parameters, we make the theory of Bayesian inference more complete.