For $Z=\mathbb E_Qe^g$, define the exponentially tilted mass function $Q_g(x)=Q(x)e^{g(x)}/Z$. <Gibbs inequality> gives
$$
0\leq D_e(P\Vert Q_g)
=D_e(P\Vert Q)-\mathbb E_Pg+\log Z.
$$
Rearranging proves
$$
D_e(P\Vert Q)\geq\mathbb E_Pg-log\mathbb E_Qe^g.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 224 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-224/1/c/solution