Independence Concepts for Convex Sets of Probabilities

TL;DR

This paper explores different notions of independence in convex sets of probabilities, comparing irrelevance and factorization, and examines how these concepts influence the construction of joint probability sets from marginals.

Contribution

It introduces and compares two concepts of independence for convex probability sets, highlighting their differences and implications for probabilistic modeling.

Findings

Irrelevance and factorization are related but not equivalent in convex probability sets.

The paper clarifies how independence concepts affect the construction of joint probability sets.

It provides a framework for building global convex sets from marginals based on independence notions.

Abstract

In this paper we study different concepts of independence for convex sets of probabilities. There will be two basic ideas for independence. The first is irrelevance. Two variables are independent when a change on the knowledge about one variable does not affect the other. The second one is factorization. Two variables are independent when the joint convex set of probabilities can be decomposed on the product of marginal convex sets. In the case of the Theory of Probability, these two starting points give rise to the same definition. In the case of convex sets of probabilities, the resulting concepts will be strongly related, but they will not be equivalent. As application of the concept of independence, we shall consider the problem of building a global convex set from marginal convex sets of probabilities.