Geometry of Statistical Manifolds
Paul W. Vos

TL;DR
This paper explores the geometric structure of statistical manifolds and introduces a new framework for understanding generalized estimators and their statistical properties.
Contribution
The paper introduces a Hilbert bundle HM to extend the tangent bundle TM, enabling a geometric interpretation of generalized estimators and their statistical properties.
Findings
The log likelihood and score functions are examples of generalized estimators on statistical manifolds.
The Λ-information of an estimator gθ^ is bounded by the Fisher information I.
The geometric perspective is demonstrated through the two-sample problem.
Abstract
A statistical manifold M can be defined as a Riemannian manifold each of whose points is a probability distribution on the same support. In fact, statistical manifolds possess a richer geometric structure beyond the Fisher information metric defined on the tangent bundle TM. Recognizing that points in M are distributions and not just generic points in a manifold, TM can be extended to a Hilbert bundle HM. This extension proves fundamental when we generalize the classical notion of a point estimate—a single point in M—to a function on M that characterizes the relationship between observed data and each distribution in M. The log likelihood and score functions are important examples of generalized estimators. In terms of a parameterization θ:M→Θ⊂Rk, θ^ is a distribution on Θ while its generalization gθ^=θ^−Eθ^ as an estimate is a function over Θ that indicates inconsistency between the…
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Taxonomy
TopicsStatistical Mechanics and Entropy · Morphological variations and asymmetry · Advanced Statistical Methods and Models
