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The Fisher metric as a metric on the cotangent bundle

Hiroshi Nagaoka

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 viewed as giving a characterization of the variance/covariance.

The Fisher metric as a metric on the cotangent bundle

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 viewed as giving a characterization of the variance/covariance.
Paper Structure (12 sections, 14 theorems, 103 equations)

This paper contains 12 sections, 14 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. The Fisher co-metric
  3. The correspondence between a metric and a co-metric
  4. The Fisher co-metric on a submanifold and the Cramér-Rao inequality
  5. On the e, m-connections
  6. Monotonicity
  7. Invariance
  8. Strong invariance
  9. Weak invariance for affine connections
  10. Concluding remarks
  11. Proof of Theorem \ref{['theorem_invariance_covariance']}
  12. An exact form of Proposition \ref{['proposition_characterization_co-metric_strong_invariance']}

Key Result

Proposition 2.1

For every $p\in \mathcal{P}$, the linear map $\delta_p : \mathbb{R}^\Omega \rightarrow T^*_{p} (\mathcal{P})$ is surjective with $\mathrm{Ker}\, \delta_p =\mathbb{R}$, where $\mathbb{R}$ is regarded as a subspace of $\mathbb{R}^\Omega$ by identifying a constant $c\in\mathbb{R}$ with the constant fun

Theorems & Definitions (28)

  • Remark 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • Corollary 4.2: The Cramér-Rao inequality
  • proof
  • Proposition 6.1
  • ...and 18 more