🤖 AI Summary
This paper addresses the limitation in information geometry that the Fisher metric is defined only on the tangent bundle, hindering its direct connection to statistical quantities such as variance. It introduces, for the first time, an intrinsic definition of the Fisher cometric—the dual of the Fisher metric on the cotangent bundle—without relying on the original tangent-bundle Fisher metric. By establishing a natural correspondence between cotangent vectors and random variables, the variance/covariance structure is directly embedded into the cotangent space. Methodologically, the work integrates information geometry, statistical manifold theory, and invariance analysis under Markov morphisms. Key contributions include: (1) a cotangent-space analogue of the Čencov characterization theorem, simultaneously characterizing variance and covariance; (2) rendering the Cramér–Rao inequality a trivial consequence of the cometric structure; and (3) proving the invariance of the Fisher cometric under sufficient statistics and data processing, thereby providing a more fundamental geometric foundation for statistical inference.
📝 Abstract
The Fisher metric on a manifold of probability distributions is usually treated as a metric on the tangent bundle. In this paper, we focus on the metric on the cotangent bundle induced from the Fisher metric with calling it the Fisher co-metric. We show that the Fisher co-metric can be defined directly without going through the Fisher metric by establishing a natural correspondence between cotangent vectors and random variables. This definition clarifies a close relation between the Fisher co-metric and the variance/covariance of random variables, whereby the Cramér-Rao inequality is trivialized. We also discuss the monotonicity and the invariance of the Fisher co-metric with respect to Markov maps, and present a theorem characterizing the co-metric by the invariance, which can be regarded as a cotangent version of Čencov’s characterization theorem for the Fisher metric. The obtained theorem can also be viewed as giving a characterization of the variance/covariance.