The uniqueness of the Fisher metric as information metric

Article

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 

References

  1. Adams, R.A., Fournier, J.J.F. (2006). Sobolev spaces. Amsterdam: Elsevier/Academic Press.Google Scholar
  2. Amari, S. (1987). Differential geometrical theory of statistics. In: Differential geometry in statistical inference. Lecture note-monograph series, (Vol. 10). California: Institute of Mathematical Statistics.Google Scholar
  3. Amari, S., Nagaoka, H. (2000). Methods of information geometry. Translations of mathematical monographs (Vol. 191). Providence/Oxford: American Mathematical Society/Oxford University Press.Google Scholar
  4. Ay, N., Jost, J., Lê, H. V., Schwachhöfer, L. (2015). Information geometry and sufficient statistics. Probability Theory and related Fields, 162, 327–364. arXiv:1207.6736.
  5. Ay, N., Jost, J., Lê, H. V. and Schwachhöfer, L., Information geometry (book in preparation).Google Scholar
  6. Ay, N., Olbrich, E., Bertschinger, N., Jost, J. (2011). A geometric approach to complexity. Chaos, 21, 37–103.Google Scholar
  7. Billingsley, P. (1999). Convergence of probability measures. New York: Wiley.CrossRefMATHGoogle Scholar
  8. Bogachev, V.I. (2007). Measure Theory (Vol. I, II). Berlin: Springer.Google Scholar
  9. Campbell, L. L. (1986). An extended Chentsov characterization of a Riemannian metric. Proceedings of the American Mathematical Society, 98, 135–141.MathSciNetMATHGoogle Scholar
  10. Chentsov, N. (1978). Algebraic foundation of mathematical statistics. Mathematische Operationsforschung und Statistik Serie Statistics, 9, 267–276.MathSciNetMATHGoogle Scholar
  11. Chentsov, N. (1982). Statistical decision rules and optimal inference. Translation of mathematical monographs (Vol. 53). Providence: American Mathematical Society.Google Scholar
  12. Hamilton, R. (1982). The inverse function theorem of Nash and Moser. Bulletin of the American Mathematical Society, 7, 65–222.MathSciNetCrossRefMATHGoogle Scholar
  13. Jost, J. (2005). Postmodern analysis. Berlin: Springer.MATHGoogle Scholar
  14. Morozova, E., Chentsov, N. (1991). Natural geometry of families of probability laws, Itogi Nauki i Techniki, Current problems of mathematics, Fundamental directions 83 (pp. 133–265). Moscow.Google Scholar
  15. Neveu, J. (1965). Mathematical foundations of the calculus of probability. San Francisco: Holden-Day Inc.MATHGoogle Scholar
  16. Shahshahani, S. (1979). A new mathematical framework for the study of linkage and selection. Memoirs of the American Mathematical Society, volume 17, Nr. 211.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2016

Authors and Affiliations

  1. 1.Institute of Mathematics of ASCRPrague 1Czech Republic

Personalised recommendations