Reichenbachian Common Cause Systems

A partition {Ci}i∈I of a Boolean algebra S in a probability measure space (S, p) is called a Reichenbachian common cause system for the correlated pair A, B of events in S if any two elements in the partition behave like a Reichenbachian common cause and its complement, the cardinality of the index set I is called the size of the common cause system. It is shown that given any correlation in (S, p), and given any finite size n > 2, the probability space (S, p) can be embedded into a larger probability space in such a manner that the larger space contains a Reichenbachian common cause system of size n for the correlation. It also is shown that every totally ordered subset in the partially ordered set of all partitions of S contains only one Reichenbachian common cause system. Some open problems concerning Reichenbachian common cause systems are formulated. 1 Reichenbach’s notion of common cause Let (S, p) be a classical probability space with Boolean algebra S of random events and probability measure p on S. If the joint probability p(A∩B) of A and B is greater than the product of the single probabilities, i.e. if p(A ∩B) > p(A)p(B) (1) then the events A and B are said to be (positively) correlated and the quantity Corr(A, B) ≡ p(A ∩B)− p(A)p(B) (2) is called the correlation of A and B. According to Reichenbach [13], Section 19, a probabilistic common cause of a correlation such as (1) is an event C (common cause) that satisfies the four conditions specified in the next definition. Definition 1 C is a Reichenbachian common cause of the correlation (1) if the following (independent) conditions hold: p(A ∩B|C) = p(A|C)p(B|C) (3) p(A ∩B|C⊥) = p(A|C⊥)p(B|C⊥) (4) p(A|C) > p(A|C⊥) (5) p(B|C) > p(B|C⊥) (6) where p(X|Y ) = p(X ∩Y )/p(Y ) denotes the conditional probability of X on condition Y , C⊥ denotes the complement of C and it is assumed that none of the probabilities p(X), (X = A, B, C, C⊥) is equal to zero.