# Conditional dependence

A Bayesian network enunciating conditional dependence

In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs.[1][2] For example, if A and B are two events that individually affect the happening of a third event C, and do not directly affect each other, then initially (when the event C has not occurred)

$P(A\mid B) = P(A) \text{ or } P(B\mid A) = P(B)$[3][4] (A and B are independent)

Eventually the event C occurs, and now if event A occurs the probability of occurrence of the event B will decrease (similarly event B occurring first will decrease the probability of occurrence of A in future). Hence, now the two events A and B become conditionally dependent because their probability of occurrence is dependent on either event's occurrence. Intuitively we can say that since A and B both were probable causes of C, given C has occurred, occurrence of either of A or B alone could explain away the happening of C.

$P(B\mid C) > P(B\mid C,A)$[5]

In essence probability comes from a person's information content about occurrence of an event. For example, let the event A be 'I have a new car'; event B be 'I have a new watch'; and event C be 'I am happy'. Let us assume that the event C has occurred – meaning 'I am happy'. Now if a third person sees my new watch, he/she will attribute this reason to my happiness. Thus in his/her view the probability of the event A ('I have a new car') to have been the cause of the event C ('I am happy') will decrease as the event C has been explained away by the event B.

Conditional dependence is different from conditional independence. In conditional independence two events which are initially dependent become independent given the occurrence of a third event.[6]