#1




Problem 4.4: Interpretation
Can someone confirm if this interpretation of "experiment" in part (d) is correct? I think we have nested loops:
for each choice of Q sub f, N, and sigma { Define the normalizing constant c sup 2 = E sub a, x (f sup 2). for each choice of coefficients {a sub q: q = 1,..., Q} from standard normal distributions { f(x) = sum from 1 to Q (a sub q L sub q (x)) / c for n = 1 to N y sub n = f(x sub n) + sigma epsilon sub n Find E sub out (g sub 2) and E sub out (g sub 10) } } The reason for all this detail is that I was unclear about what the conditional distribution p(yx) might be. I think now that in each iteration of the second nested loop, we fix a choice of {a sub q}; given these coefficients, f becomes a deterministic function of x and the only randomness in y sub n is due to epsilon sub n. As a result, we have a different joint distribution P(x, y) for each choice of {a sub q}. We also have a different target function for each {a sub q}. The repeated experiments in (d), with fixed Q, N, and sigma, lead to one E sub out (g) for each set of coefficients {a sub q}. E sub out (g) is a function of the normal random vector (a sub 1, ..., a sub Q). The average of the outofsample errors is empirical mean of the distribution of E sub out. 
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joint distribution 
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