Computability Limits of Sequential Hypothesis Testing

TL;DR

This paper characterizes which countable sets of real numbers admit computable sequential decision procedures with finitely many errors, clarifying limits of empirical methods under eventual correctness.

Contribution

It provides a complete necessary and sufficient characterization of subsets of rationals that allow computable finite-error sequential tests, extending Cover's theorem.

Findings

Characterization of sets admitting computable finite-error tests

Extension of results to effectively presented countable families

Clarification of limits of empirical convergence in probabilistic settings

Abstract

Sequential hypothesis testing asks for decision rules that update as data arrive. A natural goal is \emph{eventual correctness}: the rule may change its mind early on, but it should make only finitely many wrong decisions almost surely. Starting from Cover's theorem, which guarantees such behavior for membership in a countable set of candidate means, we ask a sharper question: \emph{which sets actually admit computable sequential decision procedures with finitely many errors?} We answer this optimally by giving a complete characterization both necessary and sufficient of the subsets of $\Q$ that admit a computable finite-error sequential membership test. We further extend the characterization to any \emph{effectively presented} countable family of real means, exactly the setting in which Cover's identification rule can be implemented computably. Beyond the technical boundary, the results clarify within a precise probabilistic setting what it can mean for inquiry to ``converge to the truth,'' and they formalize a limit to which empirical methods can be expected to succeed when only eventual stabilization (rather than fixed-time guarantees) is demanded. keywords: Cover's theorem, sequential decision procedures, finite error learning, limit computability, $\Delta^0_2$ sets.