Kuznetsov independence for interval-valued expectations and sets of probability distributions: Properties and algorithms

Abstract

Kuznetsov independence of variables X and Y means that, for any pair of bounded functions f(X)f(X) and g(Y)g(Y), E[f(X)g(Y)]=E[f(X)]⊠E[g(Y)]E[f(X)g(Y)]=E[f(X)]⊠E[g(Y)], where E[⋅]E[⋅] denotes interval-valued expectation and ⊠ denotes interval multiplication. We present properties of Kuznetsov independence for several variables, and connect it with other concepts of independence in the literature; in particular we show that strong extensions are always included in sets of probability distributions whose lower and upper expectations satisfy Kuznetsov independence. We introduce an algorithm that computes lower expectations subject to judgments of Kuznetsov independence by mixing column generation techniques with nonlinear programming. Finally, we define a concept of conditional Kuznetsov independence, and study its graphoid properties.