Optimizing Conditional Expectation

[WWGD]

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.

 

[mfb]

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.

 

[WWGD]

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?

 

[gill1109]

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.

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.