Manifolds of differentiable densities

Newton, Nigel J.

ESAIM: Probability and Statistics, Tome 22 (2018), pp. 19-34.

We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class $C_b^k$ with respect to appropriate reference measures. The case $k=\infty$, in which the manifolds are modelled on Fréchet spaces, is included. The manifolds admit the Fisher-Rao metric and, unusually for the non-parametric setting, Amari’s $\alpha$-covariant derivatives for all $\alpha \in ℝ$. By construction, they are $C^{\infty }$-embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually -embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually (α = ±1) flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the α-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the α-divergences are of class C$(\alpha =\pm 1)$ flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the $\alpha$-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the $\alpha$-divergences are of class $C^{\infty }$.

[1] S.-I. Amari and H. Nagaoka, Methods of Information Geometry. Vol. 191 of Translations of Mathematical Monographs. American Mathematical Society, Providence (2000). | MR | Zbl

[2] N. Ay, J. Jost, H.V. Lê and L. Schwachhofer, Information geometry and sufficient statistics. Probab. Theory Relat. Fields 162 (2015) 327–364. | DOI | MR | Zbl

[3] N. Ay, J. Jost, H.V. Lê and L. Schwachhofer, Paramerized Measure Models. Bernoulli 24 (2018) 1692–1725. | MR | Zbl

[4] O.E. Barndorff-Nielsen, Information and Exponential Families in Statistical Theory. Wiley (1978). | MR | Zbl

[5] M. Bauer, M. Bruveris and P.W. Michor, Uniqueness of the Fisher-Rao metric on the space of smooth densities. Bull. Lond. Math. Soc. 48 (2016) 499–506. | DOI | MR | Zbl

[6] D. Brigo and G. Pistone, Dimensionality reduction for measure valued evolution equations in statistical manifolds, in Computational Information Geometry for Image and Signal Processing, edited by F. Nielsen, F. Critchley and C.T.J. Dodson. Springer (2017) 217–265. | DOI | MR

[7] M. Bruveris and P.W. Michor, Geometry of the Fisher-Rao Metric on the Space of Smooth Densities on a Compact Manifold. Preprint (2016). | arXiv | MR

[8] A. Cena and G. Pistone, Exponential statistical manifold. Ann. Inst. Stat. Math. 59 (2007) 27–56. | DOI | MR | Zbl

[9] N.N. Chentsov, Statistical Decision Rules and Optimal Inference. Vol. 53 of Translations of Mathematical Monographs. American Mathematical Society, Providence (1982). | Zbl

[10] C.T.J. Dodson, G. Galanis and E. Vassiliou, Geometry in a Fréchet Context: A Projective Limit Approach. Vol. 428 of London Mathematical Society Lecture Note Series. Cambridge University Press (2016). | MR

[11] K. Fukumizu, Exponential manifold by reproducing kernel Hilbert spaces, in Algebraic and Geometric Methods in Statistics, edited by P. Gibilisco, E. Riccomagno, M.P. Rogantin and H. Winn. Cambridge University Press (2009) 291–306. | MR

[12] P. Gibilisco and G. Pistone, Connections on non-parametric statistical manifolds by Orlicz space geometry. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998) 325–347. | DOI | MR | Zbl

[13] S.L. Lauritzen, Statistical Manifolds. Vol. 10 of IMS Lecture Notes Series. Institute of Mathematical Statistics (1987).

[14] J.A. Leslie, On a differential structure for the group of diffeomorphisms. Topology 46 (1967) 263–271. | DOI | MR | Zbl

[15] R.S. Liptser and A.N. Shiryayev, Statistics of Random Processes I – General Theory. Springer (2001). | MR

[16] G. Loaiza and H.R. Quiceno, A q-exponential statistical Banach manifold. J. Math. Ann. Appl. 398 (2013) 466–476. | DOI | MR | Zbl

[17] B. Lods and G. Pistone, Information geometry formalism for the spatially homogeneous Boltzmann equation. Entropy 17 (2015) 4323–4363. | DOI | MR | Zbl

[18] P.A. Meyer, Quantum Probability for Probabilists. Vol. 1538 of Lecture Notes in Mathematics. Springer (1995). | Zbl

[19] M.K. Murray and J.W. Rice, Differential Geometry and Statistics. Vol. 48 of Monographs in Statistics and Applied Probability. Chapman Hall (1993). | MR | Zbl

[20] J. Naudts, Generalised Thermostatistics. Springer, London (2011). | DOI | MR | Zbl

[21] N.J. Newton, An infinite-dimensional statistical manifold modelled on Hilbert space. J. Funct. Anal. 263 (2012) 1661–1681. | DOI | MR | Zbl

[22] N.J. Newton, Information geometric nonlinear filtering. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18 (2015) 1550014. | DOI | MR | Zbl

[23] N.J. Newton, Infinite-dimensional statistical manifolds based on a balanced chart. Bernoulli 22 (2016) 711–731. | DOI | MR | Zbl

[24] F. Nielsen and F. Barbaresco, Proceedings of GSI 2013 Conference. Vol. 8085 of Lecture Notes in Computer Science. Springer, Berlin (2013). | Zbl

[25] F. Nielsen and F. Barbaresco, Proceedings of GSI 2015 Conference. Vol. 9389 of Lecture Notes in Computer Science. Springer, Berlin (2015). | DOI | Zbl

[26] G. Pistone and M.P. Rogantin, The exponential statistical manifold: mean parameters, orthogonality and space transformations. Bernoulli 5 (1999) 721–760. | DOI | MR | Zbl

[27] G. Pistone and C. Sempi, An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Stat. 23 (1995) 1543–1561. | DOI | MR | Zbl

[28] C.R. Rao, Information and accuracy obtainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37 (1945) 81–91. | MR | Zbl