5.5 Conditional Expectation


5.5 Conditional Expectation

Just as it makes sense to update beliefs in light of new information, it makes sense to update expectations in light of new information. In the case of probability, there is an obvious definition of expectation on U if μ(U) > 0:

That is, to update the expected value of X with respect to μ given the new information U, just compute the expected value of X with respect to μ|U.

It is easy to check that Eμ(XV | U) = μ(V | U) (Exercise 5.47), so conditional expectation with respect to a probability measure can be viewed as a generalization of conditional probability.

For sets of probability measures, the obvious definition of conditional lower expectation is just

where E (X | U) is undefined if | U is, that is, if *(U) = 0 (i.e., if μ(U) = 0 for all μ ). An analogous definition applies to upper expectation.

By identifying EBel with EBel, this approach immediately gives a definition of EBel( |U). Moreover, it is easy to check that EBel(X | U) = EBel|U (X) (Exercise 5.48).

These examples suggest an obvious definition for conditional expectation with respect to an arbitrary plausibility measure. Given a cps (W, , ) and an expectation domain ED, define EPl, ED(X | U) = EPl|U, ED(X) for U .

There is another approach to defining conditional expectation, which takes as its point of departure the following characterization of conditional expectation in the case of probability:

Lemma 5.5.1

start example

If μ(U) > 0, then Eμ(X | U) = α iff Eμ(X XU αXU) = 0 (where X XU αXU is the gamble Y such that Y(w) = X(w) XU(w) αXU(w)).

end example

Proof See Exercise 5.50.

Lemma 5.5.1 says that conditional expectation can be viewed as satisfying a generalization of Bayes' Rule. In the special case that X = XV, using the fact that XV XU = XUV and Eμ(XV | U) = μ(V | U), as well as the linearity of expectation, Lemma 5.5.1 says that Eμ(XUV) = αE(XU), that is, μ(U V) = μ(V | U) μ(U), so this really is a generalization of Bayes' Rule.

A characterization similar to that of Lemma 5.5.1 also holds in the case of sets of probability measures.

Lemma 5.5.2

start example

If *(U) > 0, then E(X | U) = α iff E(X XU αXU) = 0.

end example

Proof See Exercise 5.51.

Analogues of this characterization of conditional expected utility for other notions of expectation have not been considered; it may be interesting to do so.




Reasoning About Uncertainty
Reasoning about Uncertainty
ISBN: 0262582597
EAN: 2147483647
Year: 2005
Pages: 140

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