🤖 AI Summary
This work addresses the limitations of classical Fisher–Rao information geometry in handling non-regular statistical models—such as those lacking densities, featuring parameter-dependent support, or exhibiting biased score functions—by developing a unified weak information geometry framework. Within the setting of tempered distributions, the authors introduce weakly regular inference functions and Stein representations as tools for information extraction, constructing a Riemannian metric induced by Godambe information that transcends Fisher–Rao constraints. This approach yields a coherent geometric theory for five classes of non-regular settings, including likelihood-free models, varying-support families, transformation-based inference, and mixture models, while revealing a geometric correspondence between inferential non-identifiability and block-diagonal metric structure. Leveraging tempered distribution theory, Schwartz kernels, quadratic Stein discrepancies, and reproducing kernel Hilbert spaces, the study derives closed-form weak Godambe metrics for models such as Cantor location families, uniform scale families, shifted exponentials, hierarchical mixtures, and α-stable noise-driven lattice heat equations, demonstrating their geometric validity and stability.
📝 Abstract
The class of parametric statistical models that can be treated as Riemannian manifolds is considerably larger than the classical Fisher-Rao setting allows, once one works in the space of tempered distributions. A law is represented by a tempered distribution T in S'(R^k), while an instrument - a positive Schwartz kernel, a weak regular inference function, or a weak Stein representation - extracts information from the law without being part of it. Any instrument with full-rank sensitivity and positive-definite variability induces the Godambe information G = S^T V^{-1} S, a Riemannian metric on the parameter space; the Fisher-Rao manifold is recovered exactly when the score is an admissible instrument, and every Godambe metric is dominated by the Fisher metric in the Loewner order whenever the latter exists. Four examples lie outside the Fisher-Rao class for four different reasons: a location model built on the Cantor distribution (an undominated family - no likelihood, no score, and no Fisher information exist at all), the uniform scale model (parameter-dependent support), the shifted exponential model (transform-based inference), and a stratified finite mixture (a provably biased score in a dominated model); a lattice stochastic heat equation driven by alpha-stable noise provides a fifth, dynamical example, whose closed-form weak Godambe information stabilises at a rate governed by the spectral gap of the discrete Laplacian. Quadratic Stein discrepancies induce the same local geometry, and reproducing-kernel constructions generate a hierarchy of geometries. Because there is no canonical instrument, the model carries a family of Godambe metrics; we discuss the inferential, diagnostic, geometric, and computational roles of its members, and show that weak inferential separation (nonformation) appears geometrically as block-diagonality of the Godambe metric.