信息几何与功能方差分析的结合：一种精确的费舍尔信息分解及其在射电光度函数中的应用
Information Geometry meets Functional ANOVA: An Exact Fisher-Information Decomposition, with an Application to Radio Luminosity Functions
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中文总结 AI 辅助
该研究将信息几何与功能方差分析结合，提出费舍尔信息的精确分解方法，并将其应用于射电光度函数模型的参数信息分解，还以植物CO2吸收实验作为示例验证方法。
中文摘要 AI 辅助
信息几何将拟合模型表示为一个流形，其度量为费舍尔信息。我们证明，对于非线性回归模型，该度量可在其协变量的Hoeffding-Sobol'（功能方差分析）通道上精确分解。将主效应和交互作用的得分贡献视为希尔伯特空间中的向量，它们的格拉姆矩阵精确再现了（高斯-牛顿）费舍尔信息度量，且该恒等式可扩展为费舍尔信息矩阵的精确加性分解，分解为主效应、交互作用和跨通道项：这是一种Sobol'式的参数信息而非输出方差的核算，等价于沿模型嵌入的环境费舍尔度量的通道分辨拉回。我们给出了带简短证明的精确分解定理（推导了中心化和非中心化费舍尔信息），定义了Sobol'指数的信息论类似物，表征了跨通道项的符号和有限样本行为，并描述了两种实用估计量。作为天体物理应用，我们分解了涵盖活动星系核（AGN）和恒星形成星系（SFG）的组合1.4 GHz射电光度函数模型的参数信息，该模型拟合于6dFGS-NVSS和VLA-COSMOS 3 GHz数据：该分解确定了每个星系种群和巡天在哪个光度处约束拟合，并揭示了光度函数 turnover附近的补偿性（“ sloppy”）通道组合。植物CO2吸收实验作为一个小型的全分类伴随示例。
英文摘要
Information geometry represents a fitted model as a manifold whose metric is the Fisher information. We show that, for a nonlinear regression model, this metric decomposes exactly across the Hoeffding-Sobol' (functional ANOVA) channels of its covariates. Treating the score contributions of main effects and interactions as vectors in a Hilbert space, their Gram matrix reproduces the (Gauss-Newton) Fisher information metric exactly, and this identity extends to an exact, additive decomposition of the Fisher information matrix into main-effect, interaction, and cross-channel terms: a Sobol'-type accounting for parameter information rather than output variance, and equivalently a channel-resolved pullback of the ambient Fisher metric along the model embedding. We give the exact decomposition theorem with a short proof (deriving both the centered and uncentered Fisher information), define information-theoretic analogues of Sobol' indices, characterize the sign and finite-sample behaviour of the cross-channel terms, and describe two practical estimators. As an astrophysical application we decompose the parameter information of a combined 1.4 GHz radio luminosity-function model spanning active galactic nuclei (AGN) and star-forming galaxies (SFG), fit to 6dFGS-NVSS and VLA-COSMOS 3 GHz data: the decomposition localizes where in luminosity each population and survey constrains the fit, and exposes the compensating ("sloppy") channel combinations near the luminosity-function turnover. A plant CO2-uptake experiment serves as a small, fully categorical companion example.
