The Topology of Statistical Verifiability

TL;DR

This paper develops a topological framework for understanding statistical verifiability and learnability, connecting formal hypothesis verification with statistical data analysis through a unique topology on probability measures.

Contribution

It introduces a unique topology on probability measures where open sets correspond to statistically verifiable hypotheses, bridging propositional and statistical data analysis.

Findings

Identifies a unique topology on probability measures for statistical verifiability

Characterizes learnability in the limit using topological methods

Bridges the gap between formal hypothesis verification and statistical data analysis

Abstract

Topological models of empirical and formal inquiry are increasingly prevalent. They have emerged in such diverse fields as domain theory [1, 16], formal learning theory [18], epistemology and philosophy of science [10, 15, 8, 9, 2], statistics [6, 7] and modal logic [17, 4]. In those applications, open sets are typically interpreted as hypotheses deductively verifiable by true propositional information that rules out relevant possibilities. However, in statistical data analysis, one routinely receives random samples logically compatible with every statistical hypothesis. We bridge the gap between propositional and statistical data by solving for the unique topology on probability measures in which the open sets are exactly the statistically verifiable hypotheses. Furthermore, we extend that result to a topological characterization of learnability in the limit from statistical data.