Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topological Criteria for Hypothesis Testing with Finite-Precision Measurements

Philip Boeken, Eduardo Skapinakis, Konstantin Genin, Joris M. Mooij

TL;DR

This work provides a unified topological framework for hypothesis testing with finite-precision measurements by characterizing FP-testability in terms of the weak topology on probability measures. It shows that a pair of disjoint null and alternative hypotheses admits a consistently testable FP-test if and only if they are $F_\sigma$ in the weak topology, with uniform error control corresponding to closedness and, under compactness, to disjoint closures. The paper then applies these results to the hardness of unconditional conditional independence testing and identifies Lipschitz continuity of conditional distributions as a key regularity condition that restores testability with uniform error control. Additional constructive results detail when Lipschitz Markov kernels and related density-regularity conditions ensure weak closedness and testability for CI. The findings offer a rigorous, general criterion for when statistical hypotheses are testable under finite-precision constraints and highlight the fundamental non-testability of CI without regularity, guiding practical methodology in causal inference and related areas.

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.

Topological Criteria for Hypothesis Testing with Finite-Precision Measurements

TL;DR

This work provides a unified topological framework for hypothesis testing with finite-precision measurements by characterizing FP-testability in terms of the weak topology on probability measures. It shows that a pair of disjoint null and alternative hypotheses admits a consistently testable FP-test if and only if they are in the weak topology, with uniform error control corresponding to closedness and, under compactness, to disjoint closures. The paper then applies these results to the hardness of unconditional conditional independence testing and identifies Lipschitz continuity of conditional distributions as a key regularity condition that restores testability with uniform error control. Additional constructive results detail when Lipschitz Markov kernels and related density-regularity conditions ensure weak closedness and testability for CI. The findings offer a rigorous, general criterion for when statistical hypotheses are testable under finite-precision constraints and highlight the fundamental non-testability of CI without regularity, guiding practical methodology in causal inference and related areas.

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 and admits a consistent test if and only if they are in the weak topology on the space of probability measures . Additionally, the hypotheses admit uniform error control under and/or if and only if and/or are closed in . Under compactness assumptions, uniform consistency is characterised by and 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.
Paper Structure (20 sections, 23 theorems, 45 equations)

This paper contains 20 sections, 23 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Accommodating finite-precision measurements
  3. Consistency, uniform consistency, and error control
  4. Related literature
  5. The weak topology
  6. Main results
  7. Proofs
  8. Proof of Theorem \ref{['thm:testable_iff_weakly_closed']}
  9. Proof of Theorem \ref{['thm:discernible_iff_weakly_fsigma']}
  10. Proof of Theorem \ref{['thm:testable_iff_weakly_clopen']}
  11. Proof of Theorem \ref{['thm:testable_iff_weakly_separated']}
  12. The hardness of conditional independence testing
  13. Weak closedness of conditional independence
  14. Sufficient conditions for Lipschitz Markov kernels
  15. Weak closedness of conditional independence via total variation
  16. ...and 5 more sections

Key Result

Theorem 1

Given a pair of hypotheses $H_0, H_1\subseteq \mathcal{P}(\mathcal{X})$, there exists a weakly consistent FP-test if and only if there exists a strongly consistent FP-test.

Theorems & Definitions (45)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 1
  • ...and 35 more