Abstract
It is well-known that a contrast function defined on a product manifold \(M \times M\) induces a Riemannian metric and a pair of dual torsion-free affine connections on the manifold M. This geometrical structure is called a statistical manifold and plays a central role in information geometry. Recently, the notion of pre-contrast function has been introduced and shown to induce a similar differential geometrical structure on M, but one of the two dual affine connections is not necessarily torsion-free. This structure is called a statistical manifold admitting torsion. This paper summarizes such previous results including the fact that an estimating function on a parametric statistical model naturally defines a pre-contrast function to induce a statistical manifold admitting torsion and provides some new insights on this geometrical structure. That is, we show that the canonical pre-contrast function can be defined on a partially flat space, which is a flat manifold with respect to only one of the dual connections, and discuss a generalized projection theorem in terms of the canonical pre-contrast function.
M. Henmi—This work was supported by JSPS KAKENHI Grant Number 15K00064.
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References
Eguchi, S.: Geometry of minimum contrast. Hiroshima Math. J. 22, 631–647 (1992)
Matsuzoe, H.: Geometry of contrast functions and conformal geometry. Hiroshima Math. J. 29, 175–191 (1999)
Amari, S., Nagaoka, H.: Method of Information Geometry. Amer. Math. Soc., Providence, Oxford University Press, Oxford (2000)
Amari, S.: Information Geometry and Its Applications. AMS, vol. 194. Springer, Tokyo (2016). doi:10.1007/978-4-431-55978-8
Kurose, T.: Statistical manifolds admitting torsion. Geometry and Something, Fukuoka University (2007)
Matsuzoe, H.: Statistical manifolds admitting torsion and pre-contrast functions. Information Geometry and Its Related Fields, Osaka City University (2010)
Henmi, M., Matsuzoe, H.: Geometry of pre-contrast functions and non-conservative estimating functions. In: AIP Conference Proceedings, vol. 1340, pp. 32–41 (2011)
Kurose, T.: On the divergences of 1-conformally flat statistical manifolds. Tohoku Math. J. 46, 427–433 (1994)
Henmi, M., Kobayashi, R.: Hooke’s law in statistical manifolds and divergences. Nagoya Math. J. 159, 1–24 (2000)
Ay, N., Amari, S.: A novel approach to canonical divergences within information geometry. Entropy 17, 8111–8129 (2015)
van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (2000)
Heyde, C.C.: Quasi-Likelihood and Its Application. Springer, New York (1997). doi:10.1007/b98823
Godambe, V.: An optimum property of regular maximum likelihood estimation. Ann. Math. Statist. 31, 1208–1211 (1960)
McCullagh, P., Nelder, J.A.: Generalized Linear Models, 2nd edn. Chapman and Hall, Boca Raton (1989)
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Henmi, M. (2017). Statistical Manifolds Admitting Torsion, Pre-contrast Functions and Estimating Functions. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_18
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DOI: https://doi.org/10.1007/978-3-319-68445-1_18
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