Nearly-Linear uncertainty measures

TL;DR

This paper introduces Nearly-Linear (NL) uncertainty measures, a flexible family of imprecise probability models that generalize existing models while maintaining simplicity, and explores their properties, consistency, and applications.

Contribution

It defines and analyzes the properties of NL models, showing their ability to represent diverse belief structures and their relation to other uncertainty models.

Findings

NL measures form three major subfamilies based on properties.

NL models can represent conflicting and irrational belief attitudes.

Comparison with neo-additive capacities and probability intervals highlights differences.

Abstract

Several easy to understand and computationally tractable imprecise probability models, like the Pari-Mutuel model, are derived from a given probability measure P_0. In this paper we investigate a family of such models, called Nearly-Linear (NL). They generalise a number of well-known models, while preserving a simple mathematical structure. In fact, they are linear affine transformations of P_0 as long as the transformation returns a value in [0,1] . We study the properties of NL measures that are (at least) capacities, and show that they can be partitioned into three major subfamilies. We investigate their consistency, which ranges from 2-coherence, the minimal condition satisfied by all, to coherence, and the kind of beliefs they can represent. There is a variety of different situations that NL models can incorporate, from generalisations of the Pari-Mutuel model, the {\epsilon}-contamination model and other models to conflicting attitudes of an agent towards low/high P_0-probability events (both prudential and imprudent at the same time), or to symmetry judgments. The consistency properties vary with the beliefs represented, but not strictly: some conflicting and partly irrational moods may be compatible with coherence. In a final part, we compare NL models with their closest, but only partly overlapping, models, neo-additive capacities and probability intervals.