Topological Criteria for Hypothesis Testing with Finite-Precision Measurements

Abstract

We establish topological necessary and sufficient conditions under which a pair of statistical hypotheses can be consistently distinguished when i.i.d. observations are recorded only to finite precision. To accommodate finite-precision data, we introduce finite-precision tests: tests whose decision regions are open in the sample-space topology. We first show that, both for classical and finite-precision tests, the existence of such tests with finite-sample error control, asymptotic error control, or uniform convergence of the errors are all equivalent. A pair of null- and alternative hypotheses $H_0$ and $H_1$ admits a consistent finite-precision test if and only if both are $F_\sigma$ in the weak topology on the space of probability measures $W := H_0\cup H_1$. The hypotheses admit uniform error control under $H_i$ if and only if $H_i$ is closed in $W$, and admit uniformly consistent testing with bounded precision under metric separation of $H_0$ and $H_1$. These criteria imply that, without regularity assumptions, conditional independence is not consistently testable from finite-precision data when the conditioning space has no isolated points - strengthening existing impossibility results to Polish sample spaces and showing that even pointwise consistency cannot be obtained. We introduce an equicontinuity assumption on the family of conditional distributions under which we recover consistent finite-precision testability of conditional independence with uniform error control under the null, provided sample spaces are Polish and the conditioning space is locally compact. The equicontinuity assumption is itself a finite-precision-testable hypothesis, so the resulting test for conditional independence is, in a precise sense, assumption-free.