Conditional expectation
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Loss function ▪ Definition
Consider two random variables, X and Y. Assume: ▪ X is Rⁿ-value ▪ Y is real valued The idea is that X is observable and Y has to be “predicted” on the basis of X We consider the following “Best prediction problem” find a function Ϥ: Rⁿ –> R such that the “loss function” is minimized L(Ϥ) = E[(Y-Ϥ(X))²] –> least squared loss function
Conditional density ▪ Definition
We assume (X,Y) is a discrete random vector, with joint density pₓ,y p y|x (y|x) = P(Y=y|X=x) p y|x (y|x) = (P(Y=y|X=x)) / P(X=x) p y|x (y|x) = pₓ,y (x,y) / pₓ(x)
Conditional expectation ▪ Definition
We define the conditional expectation of Y given X=x as follows E(Y|X=x) = Σ y py|ₓ (y|x) More generally E(g(x,y)|X=x) = Σ g(x, y) py|ₓ (y|x)
Conditional expectation ▪ Properties
1) E[aY+bZ|X=x] = aE(Y|X=x) + bE(Z|X=x) 2) E[f(x)|X=x] = f(x) 3) E[f(x)Y|X=x] = f(x) * E[Y|X=x] 4) E[ E(Y|X) ] = E(Y)
Conditional expectation ▪ Theorem
The function Ϥ(x) = E(Y|X=x) solves the best prediction problem In other words, for any g:Rⁿ–>R L(g) ⩾ L(Ϥ)