 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 224

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

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

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

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

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)