By the <Cauchy-Schwarz inequality>, part (b), and part (e),
$$
\mathbb E|\widehat p_1(x)-p_1(x)|
\leq\frac1{\sqrt L}
+C\left\{
\left(\frac{2L}{nM}\right)^{1/d}+\frac2L
\right\}.
$$
This bound tends uniformly to zero if
$$
L\longrightarrow\infty,
\qquad \frac Ln\longrightarrow0.
$$
The <plug-in classifier excess-risk bound> then shows that the misclassification risk of the nearest-neighbour classifier converges to the <Bayes risk>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 2 / f / Solution

For binary classification with posterior estimate $\widehat p_1$, the plug-in classifier satisfies
$$
R(\widehat C)-R(C^*)\leq2\mathbb E|\widehat p_1(X)-p_1(X)|.
$$


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