Let
$$
\mathcal C_\delta=\{R:D(R\Vert P)\leq\delta\},
\qquad
D(\delta)=\min_{R\in\mathcal C_\delta}D(R\Vert Q).
$$
The minimum exists because the probability simplex is compact. Let $R^*$ be the <information projection> of $Q$ onto the closed <convex set> $\mathcal C_\delta$. Its Pythagorean inequality says that every $R\in\mathcal C_\delta$ satisfies
$$
D(R\Vert Q)-D(R\Vert R^*)\geq D(R^*\Vert Q)=D(\delta).
$$
For a string $x_1^n$ of <type (information theory)> $R\in\mathcal C_\delta$, this gives
$$
\frac{Q^{\otimes n}(x_1^n)}{(R^*)^{\otimes n}(x_1^n)}
=2^{-n[D(R\Vert Q)-D(R\Vert R^*)]}
\leq2^{-nD(\delta)}.
$$
Summing over the decision region $B_n$ proves the exact bound
$$
e_1^{(n)}=Q^{\otimes n}(B_n)
\leq2^{-nD(\delta)}(R^*)^{\otimes n}(B_n)
\leq2^{-nD(\delta)}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 224 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-224/2/c/solution