Exchangeability and Conditionally Identical Common Cause Systems

Abstract

A pair (A, B) of events in a classical probability measure space (Ω, p) is called exchangeable iff p(A\(\bar B \)) = p(\( \bar A \)B). Conditionally identical common cause system of size n for the correlation is an n-partition of Ω such that (i) any member of the partition screens the correlation off and (ii) for any member {Ci}i ∊ I of the partition p(A|Ci) = p(B|Ci). The common cause system is called proper if p(A|Ci)≠(A|Cj) for some i ≠ j. In the paper it is shown that exchangeable correlations be explained by proper conditionally identical common cause systems in the following sense. (i) Given a proper conditionally identical common cause system of size n for the two events A and B in Ω, then the pair (A, B) will be an exchangeable (positively) correlating pair. (ii) Given any exchangeable (positively) correlating pair of events in Ω and given any finite number n > 2, then the probability space can be embedded into a larger probability space in such a way that the larger space contains a proper conditionally identical common cause system of size n for the correlation.