The Topology of Statistical Verifiability
This work addresses a foundational problem in statistics and machine learning by bridging theoretical models with practical data analysis, though it appears incremental in extending existing topological frameworks.
The paper tackles the gap between propositional and statistical data by solving for the unique topology on probability measures where open sets correspond to statistically verifiable hypotheses, and extends this to characterize learnability from statistical data.
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.