5.3 ( ) Consider a regression problem involving multiple target variables in which it is assumed that the distribution of the targets, conditioned on the input vector x, is a Gaussian of the form p(t|x,w) = N(t|y(x,w),Σ) (5.192)

where y(x,w) is the output of a neural network with input vector x and weight vector w, and Σ is the covariance of the assumed Gaussian noise on the targets.

Given a set of independent observations of x and t, write down the error function that must be minimized in order to find the maximum likelihood solution for w, if we assume that Σ is fixed and known. Now assume that Σ is also to be determined from the data, and write down an expression for the maximum likelihood solution for Σ. Note that the optimizations of w and Σ are now coupled, in contrast to the case of independent target variables discussed in Section 5.2.