[Paper Review] Topological Criteria for Hypothesis Testing with Finite-Precision Measurements

Philip Boeken, Eduardo Skapinakis+2 more|arXiv (Cornell University)|Jan 20, 2026

Statistical Methods and Inference0 citations

TL;DR

The paper provides topological necessary and sufficient conditions for when two statistical hypothesis sets can be consistently distinguished under finite-precision observations, using open decision regions in the sample space and weak topology on probability measures.

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. Requiring the test's decision regions to be open in the sample-space topology to accommodate finite-precision data, we show that a pair of null- and alternative hypotheses $H_0$ and $H_1$ admits a consistent test if and only if they are $F_σ$ in the weak topology on the space of probability measures $W := H_0\cup H_1$. Additionally, the hypotheses admit uniform error control under $H_0$ and/or $H_1$ if and only if $H_0$ and/or $H_1$ are closed in $W$. Under compactness assumptions, uniform consistency is characterised by $H_0$ and $H_1$ having disjoint closures in the ambient space of probability measures. These criteria imply that - without regularity assumptions - conditional independence is not consistently testable. We introduce a Lipschitz-continuity assumption on the family of conditional distributions under which we recover testability of conditional independence with uniform error control under the null, with testable smoothness constraints.

Motivation & Objective

• Motivate the question of testability under finite-precision measurements across sciences and identify gaps in existing criteria.
• Characterize when two hypothesis sets are consistently testable using openness of decision regions in FP-tests.
• Link topological properties (F_sigma, closed, clopen) to various modes of consistency and uniform error control.
• Apply the developed framework to conditional independence testing and discuss the implications for regularity assumptions.

Proposed method

• Model hypothesis testing with i.i.d. samples and finite-precision decision regions (FP-tests) that are open for binary outcomes.
• Use the weak topology on the space of probability measures and analyze H0, H1 as subsets of P(X) within W=H0∪H1.
• Prove equivalences: existence of consistent FP-tests iff H0 and H1 are F_sigma in the weak topology; and conditions for uniform error control.
• Show implications for conditional independence by proving density of conditional independence and conditional dependence under broad settings, and propose Lipschitz-regularity conditions for recoverability of testable conditional independence.

Experimental results

Research questions

• RQ1 Under what topological conditions on H0 and H1 does a consistently FP-test exist?
• RQ2 When can FP-tests achieve uniform error control under H0 and/or H1?
• RQ3 What are the implications of these topological criteria for conditional independence testing?
• RQ4 How do finite-precision considerations influence the design of decision regions in hypothesis testing?
• RQ5 What regularity assumptions (e.g., Lipschitz continuity) restore testability for conditional independence?

Key findings

• There exists a consistent FP-test if and only if H0 and H1 are F_sigma in the weak topology on W.
• Uniform error control under H0 and/or H1 is possible if and only if H0 and/or H1 are closed in W, with disjoint closures implying uniform consistency under compactness.
• If both hypotheses are relatively compact with disjoint closures in the ambient space, a uniformly consistent FP-test exists.
• Conditional independence testing is not consistently FP-testable in full generality; it becomes testable under Lipschitz-continuity assumptions on conditional distributions.
• The paper connects testability to topological properties, showing that regularity requirements (e.g., open critical regions) resolve issues highlighted in prior work.