🤖 AI Summary
This study addresses statistical ill-posedness in distributional models—such as non-identifiability, singular Fisher information, and moment indeterminacy—arising from geometric degeneracies. It introduces transversality theory from differential topology to systematically characterize well-behaved models through the geometric properties of kernel-induced feature maps. Leveraging tools including Sard’s theorem, smooth mappings, jet spaces, and parametric transversality, the work unifies the explanation of six classical pathological phenomena, including representation collapse and the Behrens–Fisher problem, demonstrating that these issues are non-generic under typical conditions. The analysis provides a geometric perspective and theoretical foundation for robust statistical modeling, while presenting the underlying topological machinery in an accessible, pedagogically oriented manner.
📝 Abstract
These notes provide a pedagogical introduction to the role of transversality theory in the analysis of statistical degeneracies within the framework of distributional statistical models. The classical question of when a statistical model is well-behaved - in the sense of being identifiable, having non-singular Fisher information, and admitting robust estimation - is reformulated as a question about the geometry of a kernel-induced feature map. Statistical pathologies correspond to geometric degeneracies of this map, and transversality theory provides a precise language for understanding when and why such degeneracies are non-generic.
The exposition is organised in three parts. Part I surveys the statistical phenomena that motivate the geometric treatment: representation failure, non-identifiability, moment indeterminacy, singular information, nuisance parameters, and the Behrens-Fisher problem. Part II develops the necessary geometric toolkit - smooth maps, Sard's theorem, transversality, jets, stratifications, and the parametric transversality theorem - at a level accessible to students with a background in analysis and linear algebra but no prior exposure to differential topology. Part~III returns to the statistical problems of Part~I and shows how each one admits a unified geometric interpretation as a transversality condition on the feature map.
These notes are a pedagogical companion to the research paper Labouriau (2026) "Transversality and Geometric Regularisation in Distributional Statistical Models" (arXiv:2605.04536 [math.ST]), expanding its arguments with motivating examples, geometric intuition, and exercises aimed at advanced Master's and PhD students with a background in mathematical statistics and measure theory. They are designed to support seminars or reading groups.