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Causality, Probability, and Time by Samantha Kleinberg

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Appendix B

Proofs

B.1. Probability Raising

Claim. The following conditions for probabilistic causality are equivalent in non-deterministic cases:

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Proof. Assume that 1 > P(C) > 0 (and thus 1 > PC) > 0). By definition:

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Then, if P(EC) > P(E|C), it must be that P(E|C) < P(E) to maintain the equality, and if P(E|C) > P(EC), then by the same reason P(E|C) > P(E). Thus, if (B.2) is satisfied, (B.1) is satisfied. Conversely, if P(E|C) > P(E), then we must have P(E|C) > P(EC). Thus, if (B.1) is satisfied (B.2) is satisfied and finally we conclude ...

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