Consider a linear basis function regression model for a multivariate target variable $t$ having a Gaussian distribution of the form

$(W,|)(y(x,W),|)$

where

$y(x,W)=W^T(x)$

together with a training data set comprising input basis vectors $(x_n)$ and corresponding target vectors $t_n,$ with $n=1,$dots,$N.$ Show that the maximum likelihood solution $W_{ML}$ for the parameter matrix $W$ has the property that each column is given by an expression of the form $(3\mathrm{.}15),$ which was the solution for an isotropic noise distribution. Note that this is independent of the covariance matrix $.$ Show that the maximum likelihood solution for $$ is given by

$=\frac{1}{N}{\displaystyle \underset{n=1}{\overset{N}{}}}(t_n-W_{ML}^T(x_n))(t_n-W_{ML}^T(x_n){)}^T$