Cartan meets Cram'er-Rao

2025-11-19

AI Summary

The Cramér–Rao bound (CRB) exhibits curvature-induced bias on non-Euclidean statistical manifolds, limiting its accuracy in geometric statistical inference. Method: We introduce a novel theoretical framework based on sprays and Cartan geometry, pioneering the application of Cartan prolongation to statistical inference. Estimation efficiency is characterized via integral curves on the spray manifold, and a fundamental connection is established between CRB curvature corrections and the non-integrability and torsion of Cartan distributions. By integrating square-root embeddings, Ehresmann connections, and contact structures, higher-order information inequalities are recast as problems of differential equation integrability. Contribution: Our work unifies the CRB with Bhattacharyya-type bounds, providing the first differential-geometric interpretation of higher-order statistical lower bounds grounded in affine connections and integrability conditions. It reveals a deep correspondence between variance lower bounds and the intrinsic geometric structure—particularly curvature, torsion, and distributional non-integrability—of statistical models.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

This paper develops a jet bundle and Cartan geometric foundation for the curvature-aware refinements of the Cram'er-Rao bound (CRB) introduced in our earlier work. We show that the extrinsic corrections to variance bounds, previously derived from the second fundamental form of the square root embedding $s_ heta=sqrt{f(cdot; heta)}in L^2(mu)$ for model density $f(cdot; heta)$ with scalar parameter $ heta$, admit an intrinsic formulation within the Cartan prolongation framework. Starting from the canonical contact forms and total derivative on the finite jet bundle $J^m(mathbb{R} imes mathbb{R})$, we construct the Cartan distribution and the associated Ehresmann connection, whose non-integrability and torsion encode the geometric source of curvature corrections in statistical estimation. In the statistical jet bundle $E=mathbb{R} imes L^2(mu)$, we point out that the condition for an estimator error to lie in the span of derivatives of $s_ heta$ up to order $m$ is equivalent to the square root map satisfying a linear differential equation of order~$m$. The corresponding submanifold of $J^m(E)$ defined by this equation represents the locus of $m$-th order efficient models, and the prolonged section must form an integral curve of the restricted Cartan vector field. This establishes a one-to-one correspondence between algebraic projection conditions underlying CRB and Bhattacharyya-type bounds and geometric integrability conditions for the statistical section in the jet bundle hierarchy. The resulting framework links variance bounds, curvature, and estimator efficiency through the geometry of Cartan distributions, offering a new differential equation and connection-theoretic interpretation of higher-order information inequalities.

Problem

• Develops geometric foundation for curvature corrections in Cramér-Rao bound
• Links variance bounds and estimator efficiency through Cartan geometry
• Provides connection-theoretic interpretation of higher-order information inequalities

Innovation

• Jet bundle framework for curvature-aware Cramér-Rao bound refinements
• Cartan geometry encoding curvature corrections in statistical estimation
• Differential equation linking estimator efficiency to jet bundle geometry