5.17 () Consider a squared loss function of the form E =

1 2



{y(x,w) − t}2 p(x, t) dx dt (5.193)

where y(x,w) is a parametric function such as a neural network. The result (1.89)

shows that the function y(x,w) that minimizes this error is given by the conditional expectation of t given x. Use this result to show that the second derivative of E with respect to two elements wr and ws of the vector w, is given by

∂2E

∂wr∂ws

=



∂y

∂wr

∂y

∂ws p(x) dx. (5.194)

Note that, for a finite sample from p(x), we obtain (5.84).