Abstract
We construct an infinite-dimensional information manifold based on exponential Orlicz spaces without using the notion of exponential convergence. We then show that convex mixtures of probability densities lie on the same connected component of this manifold, and characterize the class of densities for which this mixture can be extended to an open segment containing the extreme points. For this class, we define an infinite-dimensional analogue of the mixture parallel transport and prove that it is dual to the exponential parallel transport with respect to the Fisher information. We also define α-derivatives and prove that they are convex mixtures of the extremal (±1)-derivatives.
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References
Amari S.-I. (1985). Differential-geometrical methods in statistics. Springer, New York
Amari, S.-I., Nagaoka, H. (2000). Methods of information Geometry. Providence, RI: American Mathematical Society. Translated from the 1993 Japanese original by Daishi Harada.
Čencov, N.N. (1982). Statistical decision rules and optimal inference. Providence, RI: American Mathematical Society. Translation from the Russian edited by Lev J. Leifman.
Dawid A.P. (1975). On the concepts of sufficiency and ancillarity in the presence of nuisance parameters. Journal of the Royal Statistical Society B 37: 248–258
Efron, B. (1975). Defining the curvature of a statistical problem (with applications to second order efficiency). Annals of Statistics, 3, 1189–1242. With a discussion by C. R. Rao, Don A. Pierce, D. R. Cox, D. V. Lindley, Lucien LeCam, J. K. Ghosh, J. Pfanzagl, Neils Keiding, A. P. Dawid, Jim Reeds and with a reply by the author.
Gibilisco P. and Isola T. (1999). Connections on statistical manifolds of density operators by geometry of noncommutative L p-spaces. Infinite Dimensional Analysis Quantum Probability and Related Topics 2: 169–178
Gibilisco P. and Pistone G. (1998). Connections on non-parametric statistical manifolds by Orlicz space geometry. Infinite Dimensional Analysis Quantum Probability and Related Topics 1: 325–347
Grasselli, M.R. (2001). Classical and quantum information geometry. Ph.D. thesis, King’s College, London.
Jeffreys H. (1946). An invariant form for the prior probability in estimation problems. Proceedings of Royal Society A 186: 453–461
Kass R.E. and Vos P.W. (1997). Geometrical foundations of asymptotic inference. Wiley- Interscience, New York
Krasnosel′skiĭ M.A. and Rutickiĭ J.B. (1961). Convex functions and Orlicz spaces. P. Noordhoff, Groningen
Lang S. (1995). Differential and Riemannian manifolds (3rd ed). Springer, New York
Murray M.K. and Rice J.W. (1993). Differential geometry and statistics. Chapman & Hall, London
Pistone, G. (2001). New ideas in nonparametric estimation. In P. Sollich, et al. (Eds.), Disordered and complex systems. American Institute of Physics. AIP Conference Proceedings 553.
Pistone G. and Rogantin M.P. (1999). The exponential statistical manifold: Mean parameters, orthogonality and space transformations. Bernoulli 5: 721–760
Pistone G. and Sempi C. (1995). An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. Annals of Statistics 23: 1543–1561
Rao C.R. (1945). Information and accuracy attainable in the estimation of statistical parameters. Bulletin of the Calcutta Mathematical Society 37: 81–91
Rao M.M. and Ren Z.D. (1991). Theory of Orlicz spaces. Marcel Dekker, New York
Sollich, P., et al. (Eds.) (2001). Disordered and complex systems. American Institute of Physics. AIP Conference Proceedings 553.
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Grasselli, M.R. Dual connections in nonparametric classical information geometry. Ann Inst Stat Math 62, 873–896 (2010). https://doi.org/10.1007/s10463-008-0191-3
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DOI: https://doi.org/10.1007/s10463-008-0191-3