Uncertainty, Imprecise Probabilities and Interval Capacity Measures on a Product Space

TL;DR

This paper extends the framework of interval probability measures and imprecise probabilities on a product space, introducing new concepts like weak complementation and an updating rule that accounts for independence and dependence.

Contribution

It provides a comprehensive analysis of interval probability measures and introduces an updating rule within a product space framework, advancing the modeling of uncertainty.

Findings

New concepts of weak complementation and interval probability measures introduced.

Analysis of these concepts within a specific probability space.

An updating rule for events incorporating independence and dependence.

Abstract

In Basili and Pratelli (2024), a novel and coherent concept of interval probability measures has been introduced, providing a method for representing imprecise probabilities and uncertainty. Within the framework of set algebra, we introduced the concepts of weak complementation and interval probability measures associated with a family of random variables, which effectively capture the inherent uncertainty in any event. This paper conducts a comprehensive analysis of these concepts within a specific probability space. Additionally, we elaborate on an updating rule for events, integrating essential concepts of statistical independence, dependence, and stochastic dominance.