Notes on Transversality and Statistical Degeneracies in Distributional Models

TL;DR

This paper introduces transversality theory as a geometric framework to analyze statistical degeneracies in distributional models, providing insights into model regularity and identifiability issues.

Contribution

It offers a pedagogical, geometric perspective on statistical pathologies, connecting them to transversality conditions and expanding on prior research with examples and exercises.

Findings

Degeneracies correspond to geometric non-transversality.

Transversality theory clarifies when statistical models are well-behaved.

The geometric approach unifies various statistical pathologies.

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.