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Optimizing Conditional Expectation

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  1. Sep 12, 2015 #1

    WWGD

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    Hi all,
    Let X be a random EDIT variable with (infinite) sample space S. Are there some results dealing with how to maximize

    E(X|s ) (conditional expectation of X given s ) for s in S ?

    Thanks.
     
    Last edited: Sep 12, 2015
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  3. Sep 12, 2015 #2

    mfb

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    If you know nothing about X or S, I don't see how a general strategy would work. The different expectation values can be completely independent.
    If you know something about X or S, there can be nice ways to find the optimal s.
     
  4. Sep 12, 2015 #3

    WWGD

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    Couldn't we see this as a sort of an infinite-dimensional variant of Factor Analysis/ PCA (where the elements of S are the infinite factors)? EDIT : Maybe we can use some functional-analytic techniques dealing with infinite-dimensional linear operators? I know there are generalizations to the infinite-dimensional case of , e.g., determinants, maybe there are generalizations of other aspects?
     
    Last edited: Sep 12, 2015
  5. Sep 13, 2015 #4

    gill1109

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    What is S supposed to be: the set of values which the random variable X can take, or the probability space on which it is defined?

    In the latter case, X is a function from S to the real numbers R. When you condition on a particular outcome s in S, you fix the value of X; it's X(s). According to any sensible definition of conditional expectation, we should take E(X | s) to be X(s).

    And now you want to find the maximal value which this can be? This is no longer a probability question.
     
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