Probabilistic Dependence Between Events
Ruma Falk, Maya Bar‐Hillel · 1983
Two events, A and B, in a probability space (S, P) are customarily classified as being either independent [i.e., P(A n B) = P(A)P(B), or P(B IA) = P(B)] or dependent [i.e., P(A n B) #/ P(A)P(B), or P(B I A) #/ P(B)]. The relationship of dependence, however, lends itself very naturally to a further classification, namely P(B I A) > P(B) versus P(B I A) P(A)P(B) versus P(A n B) < P(A)P(B)]. The former might be called and the latter relevance; in this light, independence might also be called irrelevance. If the distinction between dependence and independence is deemed important, certainly the distinction between positive and negative relevance is even more so. Evidence tells us little if we know that it is relevant to some hypothesis, but do not know the direction in which it affects the probability. Nonetheless, the distinction between positive and negative relevance is so rarely encountered, that it even lacks an accepted label or term. In the present paper we first suggest a threefold classification of dependence relationships between pairs of events, then point out some misconceptions concerning these relationships, and, lastly, speculate as to the reasons that it is not customarily employed.