4.10 ( ) Consider the classification model of Exercise 4.9 and now suppose that the class-conditional densities are given by Gaussian distributions with a shared covariance matrix, so that p(φ|Ck) = N(φ|μk,Σ). (4.160)

Show that the maximum likelihood solution for the mean of the Gaussian distribution for class Ck is given by

μk =

1 Nk

N n=1 tnkφn (4.161)

which represents the mean of those feature vectors assigned to class Ck. Similarly, show that the maximum likelihood solution for the shared covariance matrix is given by Σ = K k=1 Nk N Sk (4.162)
where Sk = 1 Nk N n=1 tnk(φn − μk)(φn − μk)T. (4.163)
Thus Σ is given by a weighted average of the covariances of the data associated with each class, in which the weighting coefficients are given by the prior probabilities of the classes.