弱信息几何：基于分布推断函数和斯坦因差异的黎曼结构
Weak Information Geometry: Riemannian Structures from Distributional Inference Functions and Stein Discrepancies
AI总结：
研究在缓增广义函数空间中可视为黎曼流形的参数统计模型，通过具有满秩灵敏度和正定变异性的工具诱导戈达姆贝信息及黎曼度量，给出多个超出费希尔 - 拉奥类别的例子，探讨戈达姆贝度量相关作用及弱推断分离的几何表现。
AI中文摘要：
当在缓增广义函数空间中工作时，可被视为黎曼流形的参数统计模型类别比经典的费希尔 - 拉奥设定所允许的要大得多。一个概率分布由\(S'(R^k)\)中的缓增广义函数\(T\)表示，而一个工具（正施瓦茨核、弱正则推断函数或弱斯坦因表示）从该分布中提取信息且不属于该分布本身。任何具有满秩灵敏度和正定变异性的工具都会诱导出戈达姆贝信息\(G = S^T V^{-1} S\)，这是参数空间上的黎曼度量；当得分是一个可允许的工具时，能精确恢复费希尔 - 拉奥流形，并且只要后者存在，每个戈达姆贝度量在洛纳偏序中都被费希尔度量所支配。给出了四个超出费希尔 - 拉奥类别的例子及原因，还有一个由\(\alpha\) - 稳定噪声驱动的格点随机热方程的动态例子，其封闭形式的弱戈达姆贝信息以由离散拉普拉斯算子的谱隙所控制的速率稳定下来。二次斯坦因差异诱导相同的局部几何，再生核构造产生几何层次结构。由于没有规范工具，模型带有一族戈达姆贝度量，讨论了其成员的推断、诊断、几何和计算作用，并表明弱推断分离（非形成）在几何上表现为戈达姆贝度量的块对角性。
英文摘要：
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.
