In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs. For example, if A and B are two events that individually increase the probability of a third event C, and do not directly affect each other, then initially But suppose that now C is observed to occur. If event B occurs the probability of occurrence of the event A will decrease because its positive relation to C is less necessary as an explanation for the occurrence of C. Hence, now the two events A and B are conditionallynegatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs. We have Conditional dependence is different from conditional independence. In conditional independence two events become independent given the occurrence of a third event.
Example
In essence probability is influenced by a person's information about the possible occurrence of an event. For example, let the event A be 'I have a new phone'; event B be 'I have a new watch'; and event C be 'I am happy'; and suppose that having either a new phone or a new watch increases the probability of my being happy. Let us assume that the event C has occurred – meaning 'I am happy'. Now if another person sees my new watch, he/she will reason that my likelihood of being happy was increased by my new watch, so there is less need to attribute my happiness to a new phone. To make the example more numerically specific, suppose that there are four possible states, given in the four columns of the following table, in which the occurrence of event A is signified by a 1 in rowA and its non-occurrence is signified by a 0 :
probability
1/4
1/4
1/4
1/4
A
0
1
0
1
B
0
0
1
1
C
0
1
1
1
In this example, C occurs if and only if at least one of A, B occurs. Unconditionally, A and B are independent of each other because P—the sum of the probabilities associated with a 1 in row A—is while P = P / P = = P. But conditional on C having occurred, we have P = P / P = while P = P / P = < P. Since in the presence of C the probability of A is affected by the presence or absence of B, A and B are mutually dependent conditional on C.
