3.4 () www Consider a linear model of the form y(x,w) = w0 +

D i=1 wixi (3.105)

together with a sum-of-squares error function of the form ED(w) =

1 2

N n=1

{y(xn,w) − tn}2 . (3.106)

Now suppose that Gaussian noise i with zero mean and variance σ2 is added independently to each of the input variables xi. By making use of E[i] = 0 and E[ij] = δijσ2, show that minimizing ED averaged over the noise distribution is equivalent to minimizing the sum-of-squares error for noise-free input variables with the addition of a weight-decay regularization term, in which the bias parameter w0 is omitted from the regularizer.