Consider a linear basis function regression model for a multivariate target variable t having a Gaussian distribution of the form p(t IW, E) = ly(x, W), E XXXXXXXXXXwhere y(x, W) = WIT/5(x...

Consider a linear basis function regression model for a multivariate target variable t having a Gaussian distribution of the form p(t IW, E) = ly(x, W), E) (3.107)
where
y(x, W) = WIT/5(x) (3.108)
Exercises 175
together with a training data set comprising input basis vectors 0(x„) and corre-sponding target vectors t„, with n = 1,... , N. Show that the maximum likelihood solution Wm', for the parameter matrix W has the property that each column is given by an expression of the form (3.15), which was the solution for an isotropic noise distribution. Note that this is independent of the covariance matrix E. Show that the maximum likelihood solution for E is given by
1 tr E = E _wq; cxx„))0„_wi,;,,40,0) Ty n h n=1
(3.109)