The <Neyman-Pearson decision region> accepting $P^{\otimes n}$ is
$$
B_n(t)=\left\{x_1^n:
\frac{P^{\otimes n}(x_1^n)}{Q^{\otimes n}(x_1^n)}\geq t
\right\},
$$
with randomization on the boundary when needed. If $R=\widehat P_{x_1^n}$ is the <type (information theory)> of the observed string, then
$$
\frac1n\log\frac{P^{\otimes n}(x_1^n)}{Q^{\otimes n}(x_1^n)}
=D(R\Vert Q)-D(R\Vert P).
$$
Thus the equivalent <relative entropy> form is
$$
B_n(t)=\{x_1^n:D(\widehat P_{x_1^n}\Vert Q)
-D(\widehat P_{x_1^n}\Vert P)\geq n^{-1}\log t\}.
$$

Solved by gpt-5.6-sol high.

Solution

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

{c}

The <Neyman-Pearson lemma> says that an optimal region accepting $P$ has the likelihood-ratio form
$$
B_n=\left\{x_1^n:\frac{P^{\otimes n}(x_1^n)}{Q^{\otimes n}(x_1^n)}\geq t\right\},
$$
with possible randomization on the boundary.

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