An Approach to Fisher-Rao Metric for Infinite Dimensional Non-Parametric Information Geometry

Non-parametric information geometry has long faced an “intractability barrier”: in the infinite-dimensional setting, the Fisher–Rao metric is a weak Riemannian metric functional that lacks a bounded inverse, rendering classical optimization and estimation techniques computationally inaccessible. This paper resolves this barrier by building the statistical manifold on the Orlicz space L0Φ(Pf) (the Pistone–Sempi manifold), which provides the necessary exponential integrability for score functions and a rigorous Fréchet differentiability for the Kullback–Leibler divergence. We introduce a novel Structural Decomposition of the Tangent Space (TfM=S⊕S⊥), where the infinite-dimensional space is split into a finite-dimensional covariate subspace (S)—representing the observable system—and its orthogonal complement (S⊥). Through this decomposition, we derive the Covariate Fisher Information Matrix (cFIM), denoted as Gf, which acts as the computable “Hilbertian slice” of the otherwise intractable metric functional. Key theoretical contributions include proving the Trace Theorem (HG(f)=Tr(Gf)) to identify G-entropy as a fundamental geometric invariant; demonstrating the Geometric Invariance of the Covariate Fisher Information Matrix (cFIM) as a covariant (0,2)-tensor under reparameterization; establishing the cFIM as the local Hessian of the KL-divergence; and characterizing the Efficiency Standard through a generalized Cramer–Rao Lower Bound for semi-parametric inference within the Orlicz manifold. Furthermore, we demonstrate that this framework provides a formal mathematical justification for the Manifold Hypothesis, as the structural decomposition naturally identifies the low-dimensional subspace where information is concentrated. By shifting the focus from the intractable global manifold to the tractable covariate geometry, this framework proves that statistical information is not a property of data alone, but an active geometric interaction between the environment (data), the system (covariate subspace), and the mechanism (Fisher–Rao connection).