Proposition: Approximate accessible-input limitation

If all accessible inputs obey $d_{\mathrm{tr}}({\rho }_x,{\rho }_{\ast })\le r$, and the apparatus is fixed, then every estimator has success probability at most $\min \{1,{\max }_x{\pi }_x+r\}$.

Proof. Contractivity gives output law distance at most $r$ from the law for ${\rho }_{\ast }$. The probability of the estimator's correct decision changes by at most $r$ for each $x$. Average over the priors and apply Theorem thm:nolabel.