Order-distance and other metric-like functions on jointly distributed random variables

We construct a class of real-valued nonnegative binary functions on a set of jointly distributed random variables, which satisfy the triangle inequality and vanish at identical arguments (pseudo-quasi-metrics). We apply these functions to the problem of selective probabilistic causality encountered in behavioral sciences and in quantum physics. The problem reduces to that of ascertaining the existence of a joint distribution for a set of variables with known distributions of certain subsets of this set. Any violation of the triangle inequality by one of our functions when applied to such a set rules out the existence of the joint distribution. We focus on an especially versatile and widely applicable class of pseudo-quasi-metrics called order-distances. We show, in particular, that the Bell-CHSH-Fine inequalties of quantum physics follow from the triangle inequalities for appropriately defined order-distances. We show how certain metric-like functions on jointly distributed random variables (pseudo-quasimetrics introduced in Section 1) can be used in dealing with the problem of selective probabilistic causality (introduced in Section 2), illustrating this on examples taken from behavioral sciences and quantum physics (Section 3). Although most of Section 2 applies to arbitrary pseudo-quasimetrics on jointly distributed random variables, we single out one, termed order-distance, which is especially useful due to its versatility. We discuss examples of other pseudo-quasi-metrics and rules for their construction in Section 4. 1. Order p.q.-metrics Random variables in this paper are understood in the broadest sense, as measurable functions X : Vs → V , no restrictions being imposed on the sample spaces (Vs,Σs, μs) and the induced probability spaces, (V,Σ, μ), with the usual meaning of the terms (sets of values Vs, V , sigmaalgebras Σs,Σ, and probability measures μs, μ). In particular, any set X of jointly distributed random variables (functions on the same sample space) is a random variable, and its induced probability space (or, simply, distribution) X = (V,Σ, μ) is referred to as the joint distribution of its elements. Given a class of random variables X , not necessarily jointly distributed, let X ∗ be the class of distributions X for all X ∈ X . For any class function f∗ : X ∗ → R (reals), the function f : X → R defined by f (X) = f∗ ( X ) is called observable (as it does not depend on sample spaces, typically unobservable). We will conveniently confuse f and f∗ for observable functions, so that if 2000 Mathematics Subject Classification. Primary 60B99, Secondary 81Q99, 91E45.