Marczyk, Michal and Wronski, Leszek
(2009)
Exhaustive Classication of Finite Classical Probability Spaces with Regard to the Notion of Causal Uptonclosedness.
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Abstract
Extending the ideas from (HoferSzabó and Rédei [2006]), we introduce the notion of causal uptonclosedness of probability spaces. A probability space is said to be causally uptonclosed 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 upto3closed w.r.t. the relation of logical independence iff its probability measure is constant on the set of atoms of non0 probability. (The latter condition is a weakening of the notion of measure uniformity.) Other independence relations are also considered.
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