Bayes risk bound for one-nearest-neighbour classification
= Bayes risk bound for one-nearest-neighbour classification
{c}
{title2=$R^*\leq R_\infty\leq2R^*$}
Under the hypotheses for the <one-nearest-neighbour asymptotic risk>, the limiting risk satisfies $R^*\leq R_\infty\leq2R^*$. Put $a=\min(\eta,1-\eta)$: then $a\leq2a(1-a)\leq2a$ since $0\leq a\leq1/2$. Integrate, using the definition of <Bayes risk>. Constant $\eta=1/4$ gives limiting risk $3/8$ and <Bayes risk> $1/4$, so the lower comparison is not generally equality.