Exhaustive Classication of Finite Classical Probability Spaces with Regard to the Notion of Causal Up-to-n-closedness

Marczyk, Michal and Wronski, Leszek (2009) Exhaustive Classication of Finite Classical Probability Spaces with Regard to the Notion of Causal Up-to-n-closedness.

Abstract

Extending the ideas from (Hofer-Szabó and Rédei [2006]), we introduce the notion of causal up-to-n-closedness of probability spaces. A probability space is said to be causally up-to-n-closed with respect to a relation of independence R_ind iff for any pair of correlated events belonging to R_ind the space provides a common cause or a common cause system of size at most n. We prove that a finite classical probability space is causally up-to-3-closed w.r.t. the relation of logical independence iff its probability measure is constant on the set of atoms of non-0 probability. (The latter condition is a weakening of the notion of measure uniformity.) Other independence relations are also considered.