8.1. One approach to Bayesian variable selection for linear regression models is described in Section 8.2.1 and further examined in Example 8.3. For a Bayesian analysis for the model in Equation (8.20), we might adopt the normal–gamma conjugate class of priors

β|mk ∼ N(αmk , σ2Vmk ) and νλ/σ2 ∼ χ2

ν. Show that the marginal density of Y|mk is given by

 ((ν + n)/2) (νλ)

ν/2

πn/2(ν/2)|I + XmkVmkXT mk

|

1/2

×

%

λν +

Y − Xmkαmk T

I + XmkVmkXT mk

−1 Y − Xmkαmk

&−(ν+n)/2

, where Xmk is the design matrix, αmk is the mean vector, and Vmk is the covariance matrix for βmk for the model mk.