Information geometric nonlinear filtering

Newton, Nigel J.

doi:10.1142/s0219025715500149

Cited by 11 publications

(10 citation statements)

References 22 publications

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“…The choice φ(u) = u q , q = 1 reproduces the q-deformed logarithm and exponential, as introduced by Tsallis [18] and mentioned in the introduction. Taking φ(u) = u/(1+ u) leads to (see, [10,16]) log φ (u) = u − 1 + log(u). Taking φ(u) = u(1 + u) leads to (see, [25])…”

Section: Goals Organization and Notationsmentioning

confidence: 99%

Rho–tau embedding and gauge freedom in information geometry

Naudts

Zhang

2018

Info. Geo.

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The standard model of information geometry, expressed as Fisher-Rao metric and Amari-Chensov tensor, reflects an embedding of probability density by log-transform. The present paper studies parametrized statistical models and the induced geometry using arbitrary embedding functions, comparing single-function approaches (Eguchi's U-embedding and Naudts' deformed-log or phi-embedding) and a twofunction embedding approach (Zhang's conjugate rho-tau embedding). In terms of geometry, the rho-tau embedding of a parametric statistical model defines both a Riemannian metric, called "rho-tau metric", and an alpha-family of rho-tau connections, with the former controlled by a single function and the latter by both embedding functions ρ and τ in general. We identify conditions under which the rho-tau metric becomes Hessian and hence the ±1 rho-tau connections are dually flat. For any choice of rho and tau there exist models belonging to the phi-deformed exponential family for which the rho-tau metric is Hessian. In other cases the rho-tau metric may be only conformally equivalent with a Hessian metric. Finally, we show a formulation of the maximum entropy framework which yields the phi-exponential family as the solution.

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Section: Goals Organization and Notationsmentioning

confidence: 99%

Rho–tau embedding and gauge freedom in information geometry

Naudts

Zhang

2018

Info. Geo.

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“…In a context quite similar to our own, a new type of chart has been introduced by N. Newton in [29,30,31], namely q → q − 1 + log q − E µ [log q]. This map is restricted to densities which are in L 2 (µ) and such that log p ∈ L 1 (µ).…”

Section: The Exponential Statistical Manifold and The L 2 Approachmentioning

confidence: 99%

“…Given the work of N. Newton [29,30,31] on finding an infinite dimensional manifold structure on the space of measures that combines the exponential manifold structure of G. Pistone and co-authors and the L 2 full-space structure used by D. Brigo and coauthors, further work is to be done to see how dimensionality reduction based on Newton's framework would look like and would relate to this paper.…”

Section: Conclusion and Further Workmentioning

confidence: 99%

Dimensionality Reduction for Measure Valued Evolution Equations in Statistical Manifolds

Brigo

Pistone

2016

Computational Information Geometry

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We propose a dimensionality reduction method for infinite-dimensional measurevalued 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. arXiv:1601.04189v3 [math.PR]

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“…For further references and a detailed literature review see the proceedings paper [18], where the maximum likelihood eigenfunctions result for the exponential families case is presented under the different statistical manifolds geometry of Pistone and Sempi [35], based on Orlicz spaces and charts, rather than on the minimal L 2 structure we use here. For general approaches that combine the L 2 geometry used here and the Orlicz-based geometry with applications to filtering see for example [31,32,33]. Notice however that our earlier proceedings paper [18] does not provide conditions for the existence of the solution of the original equation in the given function space, contrary to our existence result for the L 2 structure here, but uses the L 2 case itself to proceed, so that the present paper presents the only fully rigorous and consistent analysis on the eigenfunctions maximum-likelihood theorem, see the L 2 existence discussion in Section 3.4 in particular.…”

Section: Global Optimality -Metric Projection Localization Localizationmentioning

confidence: 99%

“…Expressing the partial derivatives of square roots via the chain rule and integrating by parts gives immediately the second equation in (4). Finally, while here we use the two maps p → √ p and p → p, one might use different maps in the spirit of the theory of deformed logarithms, see [30], see also [31,32,33].…”

Section: General Tangent Linear (Vector-field) Projection Of Fpkmentioning

confidence: 99%

Optimal approximations of the Fokker-Planck-Kolmogorov equation: projection, maximum likelihood eigenfunctions and Galerkin methods

Brigo¹,

Pistone²

2016

Preprint

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We study optimal finite dimensional approximations of the generally infinite-dimensional Fokker-Planck-Kolmogorov (FPK) equation, finding the curve in a given finite-dimensional family that best approximates the exact solution evolution. For a first local approximation we assign a manifold structure to the family and a metric. We then project the vector field of the partial differential equation (PDE) onto the tangent space of the chosen family, thus obtaining an ordinary differential equation for the family parameter. A second global approximation will be based on projecting directly the exact solution from its infinite dimensional space to the chosen family using the nonlinear metric projection. This will result in matching expectations with respect to the exact and approximating densities for particular functions associated with the chosen family, but this will require knowledge of the exact solution of FPK. A first way around this is a localized version of the metric projection based on the assumed density approximation. While the localization will remove global optimality, we will show that the somewhat arbitrary assumed density approximation is equivalent to the mathematically rigorous vector field projection. More interestingly we study the case where the approximating family is defined based on a number of eigenfunctions of the exact equation. In this case we show that the local vector field projection provides also the globally optimal approximation in metric projection, and for some families this coincides with a Galerkin method. We study exponential and mixture families, and the metrics for the vector field projection are respectively the Hellinger-Fisher-Rao and the direct L 2 distances. For the metric projection we use respectively relative entropy and L 2 direct distance. In the eigenfunctions case we derive the exact maximum likelihood density for FPK. Our results are based on the differential geometric approach to statistics and systems theory, and applications include filtering.

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Information geometric nonlinear filtering

Cited by 11 publications

References 22 publications

Rho–tau embedding and gauge freedom in information geometry

Rho–tau embedding and gauge freedom in information geometry

Dimensionality Reduction for Measure Valued Evolution Equations in Statistical Manifolds

Optimal approximations of the Fokker-Planck-Kolmogorov equation: projection, maximum likelihood eigenfunctions and Galerkin methods

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