Expected Error Prediction Derivation

Question

I'm confused about how a formula was found in Elements of Statistical Learning, pg. 11.

$$ EPE(f) = E(Y - f(X))^2 = \int [y - f(x)]^2 Pr(dx, dy) $$

$Pr(dx, dy)$ is the joint probability of $dx$ and $dy$ but how can we find probabilities of infinitesimals? Also, how did they find that integral?

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score 3 · Accepted Answer · answered Jan 07 '16 at 07:45

The notation $Pr(dx, dy)$ is likely meant to cover both discrete and continuous distributions. In the continuous case, read this as $$Pr(dx, dy)=f(x,y)dxdy$$ The integral follows from the definition of an expected value of a function of random variables, see e.g. here. For example, in the univariate continuous case: $$ E[g(X)]:=\int g(x)f(x)dx $$

Expected Error Prediction Derivation

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