🤖 AI Summary
Hypothesis testing in singular statistical models is often deemed infeasible due to non-identifiable parameters and degenerate Fisher information. This work circumvents these issues by reframing hypotheses in terms of identifiable functionals of the observable distribution rather than unidentifiable parameter functions, thereby recasting the problem as a classical testing problem in the space of probability distributions. The study introduces the novel concept of an “overlap barrier,” which reveals that hypotheses involving non-identifiable quantities inevitably lead to testing impossibility. A Hellinger-distance-based criterion for testability is established, enabling a structural classification of hypotheses in singular models. By integrating distribution-space analysis, posterior contraction theory, and test-driven Bayesian arguments, the framework is validated in Gaussian mixture models and reduced-rank regression, rigorously delineating the boundary between testable and non-testable hypotheses and clarifying the limits of valid statistical inference in singular settings.
📝 Abstract
Hypothesis testing in singular statistical models is often regarded as inherently problematic due to non-identifiability and degeneracy of the Fisher information. We show that the fundamental obstruction to testing in such models is not singularity itself, but the formulation of hypotheses on non-identifiable parameter quantities. Testing is inherently a problem in distribution space: if two hypotheses induce overlapping subsets of the model class, then no uniformly consistent test exists. We formalize this overlap obstruction and show that hypotheses depending on non-identifiable parameter functions necessarily fail in this sense. In contrast, hypotheses formulated over identifiable observables-quantities that are determined by the induced distribution-reduce entirely to classical testing theory. When the corresponding distributional regimes are separated in Hellinger distance, uniformly consistent tests exist and posterior contraction follows from standard testing-based arguments. Near singular boundaries, separation may collapse locally, leading to scale-dependent detectability governed jointly by sample size and distance to the singular stratum. We illustrate these phenomena in Gaussian mixture models and reduced-rank regression, exhibiting both untestable non-identifiable hypotheses and classically testable identifiable ones. The results provide a structural classification of which hypotheses in singular models are statistically meaningful.