Conditional independence

The idea of stochastic (probabilistic) independence is explored in the unit Independence of Events . The concept is approached as lack of conditioning: $P\left(A\right|B)=P(A)$ . This is equivalent to the product rule $P\left(AB\right)=P\left(A\right)P\left(B\right)$ . We consider an extension to conditional independence.

The concept

Examination of the independence concept reveals two important mathematical facts:

• Independence of a class of non mutually exclusive events depends upon the probability measure, and not on the relationship between the events. Independence cannot be displayedon a Venn diagram, unless probabilities are indicated. For one probability measure a pair may be independent while for another probability measure the pair may not beindependent.
• Conditional probability is a probability measure, since it has the three defining properties and all those properties derived therefrom.

This raises the question: is there a useful conditional independence—i.e., independence with respect to a conditional probability measure? In this chapter we explore thatquestion in a fruitful way.

Among the simple examples of “operational independence" in the unit on independence of events, which leadnaturally to an assumption of “probabilistic independence” are the following:

• If customers come into a well stocked shop at different times, each unaware of the choice made by the other, the the item purchased by one should not be affected bythe choice made by the other.
• If two students are taking exams in different courses, the grade one makes should not affect the grade made by the other.

Buying umbrellas and the weather

A department store has a nice stock of umbrellas. Two customers come into the store “independently.” Let A be the event the first buys an umbrella and B the event the second buys an umbrella. Normally, we should think the events $\{A,B\}$ form an independent pair. But consider the effect of weather on the purchases. Let C be the event the weather is rainy (i.e., is raining or threatening to rain). Now we should think $P\left(A\right|C)>P\left(A\right|C^c)$ and $P\left(B\right|C)>P\left(B\right|C^c)$ . The weather has a decided effect on the likelihood of buying an umbrella. But given the fact the weather is rainy(event C has occurred), it would seem reasonable that purchase of an umbrella by one should not affect the likelihood of such a purchase by the other. Thus,it may be reasonable to suppose