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The notion of conditional independence is extended to random variables by considering appropriate indicator functions, then extending to more general random vectors. A variety of equivalent conditions provide a basis for practical analysis and interpretation. An important application is the Bayesian approach to statistics. The essential idea is that an unknown parameter about which there is uncertainty is modeled as the value of a random variable. The name Bayesian comes from the role of Bayesian reversal in the analysis. The Bayesian estimate often seems preferable for small samples, and it has the advantage that prior information may be utilized. The sampling procedure upgrades the prior distribution.

In the unit on Conditional Independence , the concept of conditional independence of events is examined andused to model a variety of common situations. In this unit, we investigate a more general concept of conditional independence, based on the theory of conditionalexpectation. This concept lies at the foundations of Bayesian statistics, of many topics in decision theory, and of the theory of Markov systems. We examine in this unit, verybriefly, the first of these. In the unit on Markov Sequences , we provide an introduction to the third.

The concept

The definition of conditional independence of events is based on a product rule which may be expressed in terms of conditional expectation, given an event. The pair { A , B } is conditionally independent, given C , iff

E [ I A I B | C ] = P ( A B | C ) = P ( A | C ) P ( B | C ) = E [ I A | C ] E [ I B | C ]

If we let A = X - 1 ( M ) and B = Y - 1 ( N ) , then I A = I M ( X ) and I B = I N ( Y ) . It would be reasonable to consider the pair { X , Y } conditionally independent, given event C , iff the product rule

E [ I M ( X ) I N ( Y ) | C ] = E [ I M ( X ) | C ] E [ I N ( Y ) | C ]

holds for all reasonable M and N (technically, all Borel M and N ). This suggests a possible extension to conditional expectation, given a random vector.We examine the following concept.

Definition . The pair { X , Y } is conditionally independent, given Z , designated { X , Y } ci | Z , iff

E [ I M ( X ) I N ( Y ) | Z ] = E [ I M ( X ) | Z ] E [ I N ( Y ) | Z ] for all Borel M . N

Remark . Since it is not necessary that X , Y , or Z be real valued, we understand that the sets M and N are on the codomains for X and Y , respectively. For example, if X is a three dimensional random vector, then M is a subset of R 3 .

As in the case of other concepts, it is useful to identify some key properties, which we refer to by the numbers used in the table in Appendix G. We note twokinds of equivalences. For example, the following are equivalent.

(CI1) E [ I M ( X ) I N ( Y ) | Z ] = E [ I M ( X ) | Z ] E [ I N ( Y ) | Z ] a . s . for all Borel sets M , N

(CI5) E [ g ( X , Z ) h ( Y , Z ) | Z ] = E [ g ( X , Z ) | Z ] E [ h ( Y , Z ) | Z ] a . s . for all Borel functions g , h

Because the indicator functions are special Borel functions, (CI1) is a special case of (CI5) . To show that (CI1) implies (CI5) , we need to use linearity, monotonicity, and monotone convergence in a manner similar to that used in extending properties (CE1) to (CE6) for conditional expectation.A second kind of equivalence involves various patterns. The properties (CI1) , (CI2) , (CI3) , and (CI4) are equivalent, with (CI1) being the defining condition for { X , Y } ci | Z .

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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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