2.32 ( ) www This exercise and the next provide practice at manipulating the quadratic forms that arise in linear-Gaussian models, as well as giving an independent check of results derived in the main text. Consider a joint distribution p(x, y)

defined by the marginal and conditional distributions given by (2.99) and (2.100).

By examining the quadratic form in the exponent of the joint distribution, and using the technique of ‘completing the square’ discussed in Section 2.3, find expressions for the mean and covariance of the marginal distribution p(y) in which the variable x has been integrated out. To do this, make use of the Woodbury matrix inversion formula (2.289). Verify that these results agree with (2.109) and (2.110) obtained using the results of Chapter 2.