Properization: Constructing Proper Scoring Rules via Bayes Acts

TL;DR

This paper introduces a method called properization that transforms any scoring rule into a proper one using Bayes acts, enhancing predictive evaluation tools in statistics.

Contribution

It formalizes the properization process, providing conditions for its application, and demonstrates how to create and reinterpret proper scoring rules using Bayes acts.

Findings

Properization can convert any scoring rule into a proper one.

The paper provides sufficient conditions for the existence of Bayes acts.

New proper scoring rules are constructed and existing ones are reinterpreted.

Abstract

Scoring rules serve to quantify predictive performance. A scoring rule is proper if truth telling is an optimal strategy in expectation. Subject to customary regularity conditions, every scoring rule can be made proper, by applying a special case of the Bayes act construction studied by Gr\"unwald and Dawid (2004) and Dawid (2007), to which we refer as properization. We discuss examples from the recent literature and apply the construction to create new types, and reinterpret existing forms, of proper scoring rules and consistent scoring functions. In an abstract setting, we formulate sufficient conditions under which Bayes acts exist and scoring rules can be made proper.