Kullback-Leibler Information. Given the random vector y, we define the information for discriminating between two densities in the same family, indexed by a parameter θ, say f(y; θ1) and f(y; θ2), as I(θ1; θ2) = 1 n

E1 log f(y; θ1)

f(y; θ2)

, (2.57)

where E1 denotes expectation with respect to the density determined by θ1.

For the Gaussian regression model, the parameters are θ = (β0

, σ2)0

. Show that we obtain I(θ1; θ2) = 1 2

σ2 1

σ2 2

− log σ2 1

σ2 2

− 1



+

1 2

(β1 − β2)0 Z0 Z(β1 − β2)

nσ2 2

(2.58)

in that case.