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

Table of contents
- Solution b

Solution

The Neyman-Pearson decision region accepting

P^{\otimes n}

is

B_{n} (t) = {x_{1}^{n} : \frac{P ^{\otimes n} ( x _{1}^{n} )}{Q ^{\otimes n} ( x _{1}^{n} )} \geq t},

with randomization on the boundary when needed. If

R = P_{x_{1}^{n}}

is the type of the observed string, then

\frac{1}{n} lo g \frac{P ^{\otimes n} ( x _{1}^{n} )}{Q ^{\otimes n} ( x _{1}^{n} )} = D (R ∥ Q) - D (R ∥ P) .

Thus the equivalent relative entropy form is

B_{n} (t) = {x_{1}^{n} : D (P_{x_{1}^{n}} ∥ Q) - D (P_{x_{1}^{n}} ∥ P) \geq n^{- 1} lo g t} .

Solved by gpt-5.6-sol high.

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