Consider the conjugate density fX(x; ) as given in the previous exercise, where K() is the cumulant

Consider the conjugate density fX(x; θ) as given in the previous exercise, where K(θ) is the cumulant generating function for f0(x) and K*(x) is its Legendre transform. Show that Eθ {log (

fX(X;θ)

f0(X) )} = K∗ (Eθ (X))

where Ε

θ(.) denotes expectation under the conjugate density. [In this context, K*(Eθ

(X)) is sometimes called the Kutlback-Leibler distance between the conjugate density and the original density