Overview
This study investigates the limitations of statistical inference when pointwise consistency—convergence to the truth in every possible state of the world—is unattainable. Rather than altering the inferential target or strengthening background assumptions, the research explores maintaining the inference problem while identifying the highest achievable standard. A hierarchy of consistency standards, defined in topological terms and weaker than pointwise consistency, is introduced. These standards require convergence to the truth not ubiquitously, but on a “large” set of probability measures.
Research Context
The standard of pointwise consistency in statistical inference dictates that a statistical procedure should converge to the true state of the world across all possible scenarios. However, this ideal is sometimes provably unachievable. Existing responses to this challenge typically involve either modifying the inferential goal or imposing more stringent background assumptions on the problem. This paper proposes an alternative approach: retaining the original inference problem and determining the maximal standard of success that remains feasible. The work complements analysis by Boeken et al. (2026) regarding pointwise consistency by providing characterizations of when these weaker topological standards are achievable.
Approach
The methodology involves defining a hierarchy of consistency standards that are weaker than pointwise consistency. These standards are framed using topological concepts, specifically requiring convergence to the truth over a “large” subset of probability measures, rather than across all measures. The study then applies this framework to analyze the behavior of finite-precision tests within hypothesis testing scenarios.
Findings
- For finite-precision tests, convergence to the truth densely within each hypothesis results in inconsistency over a comeager set of measures. This phenomenon occurs when the two hypotheses are dense within their combined union.
- The concept of "comeager" is used to denote "topologically almost all" sets of measures, indicating a widespread inconsistency under specific conditions.
- Distribution-free testing of conditional independence serves as an example where the conditions leading to this impossibility theorem are met.
- Two additional theorems provide a characterization, exclusively in topological terminology, of the precise conditions under which each of the introduced weaker consistency standards can be achieved.