[Designing the optimal predictor for continuous output spaces] We studied in class that the "Bayes Classifier"

3 [Designing the optimal predictor for continuous output spaces] We studied in class that the "Bayes Classifier" (f := arg max, P[Y|X] ) is optimal in the sense that it minimizes generalization error over the underlying distribution, that is, it maximizes Ex,y [1 [g(x) = y]]. But what can we say when the output space ) is continuous? Consider predictors of the kind g : X' -> R that predict a real-valued output for a given input x E X'. One intuitive way to define the quality of of such a predictor g is as Q(g) := Ex,yl(g(x) - y)2]. Observe that one would want a predictor g with the lowest Q(g). (i) Show that if one defines the predictor as f (x) := E[Y|X = x], then Q(f)