The Fisher metric as a metric on the cotangent bundle
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916988586033152 |
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| author | Nagaoka, Hiroshi |
| author_facet | Nagaoka, Hiroshi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13237 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Fisher metric as a metric on the cotangent bundle Nagaoka, Hiroshi Information Theory Differential Geometry Probability Statistics Theory 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. |
| title | The Fisher metric as a metric on the cotangent bundle |
| topic | Information Theory Differential Geometry Probability Statistics Theory |
| url | https://arxiv.org/abs/2310.13237 |