Independence Concepts for Convex Sets of Probabilities
This work addresses foundational issues in imprecise probability theory, which is incremental but important for modeling uncertainty in domains like AI and decision-making.
The paper tackles the problem of defining independence for convex sets of probabilities, proposing two concepts based on irrelevance and factorization, and finds they are not equivalent unlike in standard probability theory.
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.