It follows immediately from Grenander's result that and hence λ opt (x, θ * ) ≤ µ x /τ for any θ * , hence Λ(x; θ I ) ≤ µ x /τ .
if we choose θ 0 = θ I , we have that p(z | θ 0 , x) > µ x for some z , hence, for θ * such that it assigns probability τ on z , we have that and hence λ opt (x, θ * ; θ ) > µ x /τ , so Λ(x; θ ) > µ x /τ .