Dilation Properties of Coherent Nearly-Linear Models

TL;DR

This paper investigates the dilation phenomenon in Imprecise Probability theory within coherent Nearly-Linear models, characterizing when dilation occurs and analyzing properties of a key subfamily called Vertical Barrier Models.

Contribution

It provides a detailed characterization of dilation in coherent Nearly-Linear models, including the coarsening property, extent, and constriction, extending previous results for special cases.

Findings

Dilation occurs under specific conditions in NL models.

Vertical Barrier Models exhibit particular dilation properties.

Logical independence and extreme evaluations influence dilation behavior.

Abstract

Dilation is a puzzling phenomenon within Imprecise Probability theory: when it obtains, our uncertainty evaluation on event $A$ is vaguer after conditioning $A$ on $B$, whatever is event $B$ in a given partition $\mathcal{B}$. In this paper we investigate dilation with coherent Nearly-Linear (NL) models. These are a family of neighbourhood models, obtaining lower/upper probabilities by linear affine transformations (with barriers) of a given probability, and encompass several well-known models, such as the Pari-Mutuel Model, the $\varepsilon$-contamination model, the Total Variation Model, and others. We first recall results we recently obtained for conditioning NL model with the standard procedure of natural extension and separately discuss the role of the alternative regular extension. Then, we characterise dilation for coherent NL models. For their most relevant subfamily, Vertical Barrier Models (VBM), we study the coarsening property of dilation, the extent of dilation, and constriction. The results generalise existing ones established for special VBMs. As an interesting aside, we discuss in a general framework how logical (in)dependence of $A$ from $\mathcal{B}$ or extreme evaluations for $A$ influence dilation.