Conditional Probability Measures and Conditional Independence

The notion of a probability measure relative to a sub-a – algebra can be defined as in Definition 3.1 using the conditional expectation of an indicator function:

Definition 3.2: Let {2, X, P} be a probability space, and let X0 C X be a a-algebra. Then for any set A in IX, P(A|X0) = E[IA|X0], where IA(ш) = I(ш e A).

In the sequel I will use the shorthand notation P(Y e B|X) to indicate the conditional probability P({ш e 2 : Y(ш) e B}|XX), where B is a Borel set and XX is the a – algebra generated by X, and P (Y e B|X0) to indicate P ({ш e 2 : Y(ш) e B}|X0) for any sub-a-algebra X0 of X. The event Y e B involved may be replaced by any equivalent expression.

Recalling the notion of independence of sets and random variables, vectors, or both (see Chapter 1), we can now define conditional independence:

Definition 3.3: A sequence of sets Aj e X is conditional independent rela­tive to a sub-a-algebra X0 of X if for any subsequence jn, P(m Ajn |X0) = nn P (Ajn |X0). Moreover, asequence Yj of random variables or vectors defined on a common probability space {2, X, P} is conditional independent relative to a sub-a-algebra X0 of X if for any sequence Bj of conformable Borel sets the sets Aj = {ш e 2 : Yj(ш) e Bj} are conditional independent relative to X0.

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