3.6 () www Consider a linear basis function regression model for a multivariate target variable t having a Gaussian distribution of the form p(t|W,Σ) = N(t|y(x,W),Σ) (3.107)

where y(x,W) =WTφ(x) (3.108)

together with a training data set comprising input basis vectors φ(xn) and corresponding target vectors tn, with n = 1, . . . , N. Show that the maximum likelihood solution WML 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 Σ. Show that the maximum likelihood solution for Σ is given by Σ = 1 N N n=1 tn −WT MLφ(xn)
tn −WT MLφ(xn)
T . (3.109)