Abstract
We propose a dimensionality reduction method for infinite—dimensional measure—valued evolution equations such as the Fokker–Planck partial differential equation or the Kushner–Stratonovich resp. Duncan–Mortensen–Zakai stochastic partial differential equations of nonlinear filtering, with potential applications to signal processing, quantitative finance, heat flows and quantum theory among many other areas. Our method is based on the projection coming from a duality argument built in the exponential statistical manifold structure developed by G. Pistone and co-authors. The choice of the finite dimensional manifold on which one should project the infinite dimensional equation is crucial, and we propose finite dimensional exponential and mixture families. This same problem had been studied, especially in the context of nonlinear filtering, by D. Brigo and co-authors but the \(L^2\) structure on the space of square roots of densities or of densities themselves was used, without taking an infinite dimensional manifold environment space for the equation to be projected. Here we re-examine such works from the exponential statistical manifold point of view, which allows for a deeper geometric understanding of the manifold structures at play. We also show that the projection in the exponential manifold structure is consistent with the Fisher Rao metric and, in case of finite dimensional exponential families, with the assumed density approximation. Further, we show that if the sufficient statistics of the finite dimensional exponential family are chosen among the eigenfunctions of the backward diffusion operator then the statistical-manifold or Fisher–Rao projection provides the maximum likelihood estimator for the Fokker Planck equation solution. We finally try to clarify how the finite dimensional and infinite dimensional terminology for exponential and mixture spaces are related.
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Abraham, R., Marsden, J. E., & Ratiu, T. (1988). Manifolds, tensor analysis, and applications. Applied mathematical sciences (2nd ed., Vol. 75). New York: Springer.
Amari, S. (1987). Dual connections on the Hilbert bundles of statistical models. Geometrization of statistical theory (Lancaster, 1987) (pp. 123–151). Lancaster: ULDM Publ.
Amari, S., & Nagaoka, H. (2000). Methods of information geometry. Providence: American Mathematical Society. (Translated from the 1993 Japanese original by Daishi Harada).
Armstrong, J., & Brigo, D. (2015). Extrinsic projection of Itô SDEs on submanifolds with applications to non-linear filtering. To appear in the same volume of this paper.
Armstrong, J., & Brigo, D. (2016). Nonlinear filtering via stochastic PDE projection on mixture manifolds in \(L^2\) direct metric. Mathematics of Control, Signals and Systems, 28(1), Art.5, p. 33.
Ay, N., Jost, J., Lê, H.V., & Schwachhöfer, L. (2016). Parametrized measure models. arXiv:1510.07305.
Brezis, H. (2011). Functional analysis, Sobolev spaces and partial differential equations, Universitext. New York: Springer.
Brigo, D. (1997). On nonlinear SDEs whose densities evolve in a finite–dimensional family. Stochastic differential and difference equations, Progress in systems and control theory (Vol. 23, pp. 11–19). Boston: Birkhäuser.
Brigo, D. (1998). On some filtering problems arising in mathematical finance. Insurance: Mathematics and Economics, 22(1), 53–64.
Brigo, D. (1999). Diffusion processes, manifolds of exponential densities, and nonlinear filtering. In O. E. Barndorff-Nielsen, et al. (Eds.), Geometry in present day science. Proceedings of the Conference, Aarhus, Denmark, January 16–18, 1997 (pp. 75–96). Singapore: World Scientific.
Brigo, D. (2000). On SDEs with marginal laws evolving in finite-dimensional exponential families. Statistics & Probability Letters, 49(2), 127–134.
Brigo, D. (2011). The direct L2 geometric structure on a manifold of probability densities with applications to Filtering. arXiv:1111.6801.
Brigo, D., & Pistone, G. (1996). Projecting the Fokker-Planck equation onto a finite dimensional exponential family. Preprint 4/1996, Department of Mathematics, University of Padua, posted in 2009 on arXiv:0901.1308.
Brigo, D., Hanzon, B., & Le Gland, F. (1998). A differential geometric approach to nonlinear filtering: the projection filter. IEEE Transactions on Automatic Control, 43(2), 247–252.
Brigo, D., Hanzon, B., & Le Gland, F. (1999). Approximate nonlinear filtering by projection on exponential manifolds of densities. Bernoulli, 5(3), 495–534.
Brown, L. D. (1986). Fundamentals of statistical exponential families with applications in statistical decision theory. IMS Lecture Notes–Monograph Series (Vol. 9). Hayward, CA: IMS.
Cena, A., & Pistone, G. (2007). Exponential statistical manifold. Annals of the Institute of Statistical Mathematics, 59(1), 27–56.
Csiszár, I. (1975). \(I\)-divergence geometry of probability distributions and minimization problems. Annals of Probability, 3, 146–158.
Friedman, A. (1975). Stochastic differential equations and applications (Vol. I). New York: Academic Press.
Gibilisco, P., & Pistone, G. (1998). Connections on non-parametric statistical manifolds by Orlicz space geometry. IDAQP, 1(2), 325–347.
Hanzon, B. (1987). A dierential-geometric approach to approximate nonlinear ltering. In C. Dodson (Ed.), Geometrization of statistical theory (pp. 219–233). Lancaster: ULMD Publ.
Hazewinkel, M., Marcus, S., & Sussmann, H. (1983). Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem. Systems & Control Letters, 3(6), 331–340.
Lang, S. (1995). Differential and Riemannian manifolds. Graduate texts in mathematics (2nd ed., Vol. 160). New York: Springer.
Lods, B., & Pistone, G. (2015). Information geometry formalism for the spatially homogeneous Boltzmann equation. Entropy, 17(6), 4323–4363.
Mitter, S. K. (1979). On the analogy between mathematical problems of non-linear filtering theory and quantum physics. Ricerche di Automatica, 10(2), 163–216.
Musielak, J. (1983). Orlicz spaces and modular spaces, Lecture Notes in Mathematics (Vol. 1034). Berlin: Springer.
Naudts, J. (2011). Generalised thermostatistics. London: Springer London Ltd.
Newton, N. J. (2012). An infinite-dimensional statistical manifold modelled on Hilbert space. Journal of Functional Analysis, 263(6), 1661–1681.
Newton, N. J. (2013). Infinite-dimensional manifolds of finite-entropy probability measures. In F. Barbaresco & F. Nielsen (Eds.), Geometric science of information, Springer LNCS (Vol. 8085, pp. 713–720). Berlin: Springer.
Newton, N. J. (2015). Information geometric nonlinear filtering. Infinite Dimensional Analysis Quantum Probability And Related Topics, 18(2), 1550014, 24.
Pavliotis, G. A. (2014). Stochastic processes and applications: Diffusion processes, the Fokker-Planck and Langevin equations. New York: Springer.
Pistone, G. (2013). Examples of the application of nonparametric information geometry to statistical physics. Entropy, 15(10), 4042–4065.
Pistone, G. (2014). A version of the geometry of the multivariate Gaussian model, with applications. In XLVII Scientific Meeting SIS June 11–13. Cagliari: Società Italiana di Statistica.
Pistone, G., & Rogantin, M. (1999). The exponential statistical manifold: mean parameters, orthogonality and space transformations. Bernoulli, 5(4), 721–760.
Pistone, G., & 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(5), 1543–1561.
Santacroce, M., Siri, P., & Trivellato, B. (2015). New results on mixture and exponential models by Orlicz spaces. Bernoulli, 22(3), 1431–1447.
Schwachhöfer, L., Ay, N., Jost, J., & Lê, H. V. (2015). Invariant geometric structures in statistical models. In F. Barbaresco & F. Nielsen (Eds.), Geometric science of information, Springer LNCS (Vol. 8085, pp. 713–720). Berlin: Springer.
Shima, H. (2007). The geometry of Hessian structures. Hackensack: World Scientific Publishing Co. Pte. Ltd.
Stroock, D. W., & Varadhan, S. R. S. (1979). Multidimensional diffusion processes. Berlin-New York: Springer.
van Handel, R., & Mabuchi, H. (2005). Quantum projection filter for a highly nonlinear model in cavity qed. Journal of Optics B: Quantum and Semiclassical Optics, 7(10), S226.
Acknowledgements
The authors are grateful to the organizers and participants of the conference Computational information geometry for image and signal processing, held at the ICMS in Edinburgh on September 21–25, 2015. They are also grateful to Frank Nielsen for feedback on this preprint and to an anonynous referee for suggesting investigating the approximation error, as this prompted us to derive the MLE theorem. G. Pistone is supported by deCastro Statistics, Collegio Carlo Alberto, Moncalieri, and he is a member of GNAFA-INDAM.
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Brigo, D., Pistone, G. (2017). Dimensionality Reduction for Measure Valued Evolution Equations in Statistical Manifolds. In: Nielsen, F., Critchley, F., Dodson, C. (eds) Computational Information Geometry. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-47058-0_10
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