# The uniqueness of the Fisher metric as information metric

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## Abstract

We define a mixed topology on the fiber space \(\cup _\mu \oplus ^n L^n(\mu )\) over the space \({\mathcal M}({\Omega })\) of all finite non-negative measures \(\mu \) on a separable metric space \({\Omega }\) provided with Borel \(\sigma \)-algebra. We define a notion of strong continuity of a covariant *n*-tensor field on \({\mathcal M}({\Omega })\). Under the assumption of strong continuity of an information metric, we prove the uniqueness of the Fisher metric as information metric on statistical models associated with \({\Omega }\). Our proof realizes a suggestion due to Amari and Nagaoka to derive the uniqueness of the Fisher metric from the special case proved by Chentsov by using a special kind of limiting procedure. The obtained result extends the monotonicity characterization of the Fisher metric on statistical models associated with finite sample spaces and complement the uniqueness theorem by Ay–Jost–Lê–Schwachhöfer that characterizes the Fisher metric by its invariance under sufficient statistics.

## Keywords

Monotonicity of the Fisher metric Chentsov’s theorem Mixed topology## Notes

### Acknowledgments

The author thanks Shun-ichi Amari, Nihat Ay, Lorenz Schwachhöfer and Alesha Tuzhilin for valuable conversations. She is grateful to Vladimir Bogachev and Jürgen Jost for their helpful comments and suggestions. The final version of this manuscript is greatly improved thanks to critical helpful suggestions of the anonymous referees. She acknowledges the VNU for Sciences in Hanoi for excellent working conditions and financial support during her visit when a part of this note has been done.

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