Interactive Statistical Mechanics and Nonlinear Filtering

Newton, Nigel J.

doi:10.1007/s10955-008-9622-z

Cited by 9 publications

(12 citation statements)

References 30 publications

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“…They clarify informationtheoretic properties of estimators, and their metrics are appropriate "multiobjective" measures of approximation error. The results also have bearing on the theory of non-equilibrium statistical mechanics, in which rates of entropy production can be associated with rates of information supply [19], and hence with the quadratic variation of a process of "mesoscopic states" in a particular pseudo-Riemannian metric. The development of approximations is beyond the scope of this paper.…”

Section: Discussionmentioning

confidence: 95%

“…Notions of information supply and dissipation for nonlinear filters are defined in [18,19]. The supply at time t is the mutual information I(X; Y t 0 ), and the dissipation is the X t -conditional variant, EI(X; Y t 0 |X t ).…”

Section: Discussionmentioning

confidence: 99%

“…where G is as defined in (19), and we have used (20) and (18) in the third step, and (17) in the final step. The final integrand here is the Y s -conditional mean-square error for the filter's estimate of h s (X s ).…”

Section: Quadratic Variationmentioning

confidence: 99%

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Information geometric nonlinear filtering

Newton

2015

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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This paper develops information geometric representations for nonlinear filters in continuous time. The posterior distribution associated with an abstract nonlinear filtering problem is shown to satisfy a stochastic differential equation on a Hilbert information manifold. This supports the Fisher metric as a pseudo-Riemannian metric. Flows of Shannon information are shown to be connected with the quadratic variation of the process of posterior distributions in this metric. Apart from providing a suitable setting in which to study such information-theoretic properties, the Hilbert manifold has an appropriate topology from the point of view of multi-objective filter approximations. A general class of finite-dimensional exponential filters is shown to fit within this framework, and an intrinsic evolution equation, involving Amari's −1-covariant derivative, is developed for such filters. Three example systems, one of infinite dimension, are developed in detail.

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Section: Discussionmentioning

confidence: 95%

Section: Discussionmentioning

confidence: 99%

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Information geometric nonlinear filtering

Newton

2015

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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“…The scarce few contributions to this subject that we were able to trace over the past 50 years are the works of Bucy & Joseph [14], Kailath [42], Weidemann & Stear [71], Duncan [24], Zakai & Ziv [72], Tomita et al [70], Galdos & Gustafson [28] and Newton [64,65,63]. Following the prevalent mathematical trend in stochastic filtering theory, most of these works, except [72,28], did not consider model error in the filter and sought optimal information estimates on the filter performance without addressing the practically important issue of suboptimal but achievable estimates.…”

Section: Introductionmentioning

confidence: 99%

“…In [42,24,70] various bounds for the mutual information between the truth and its noisy observations were derived in terms of the optimal least-squares estimates. More recently, the statistical mechanical properties of Kalman filters were discussed in [63] and in [64,65] for linear and nonlinear filters; in those papers, the signal-filter pair is described by a dissipative system with rates of information supply and dissipation, and with information flow from the observation process into the filter state.…”

Section: Introductionmentioning

confidence: 99%

Quantifying Bayesian filter performance for turbulent dynamical systems through information theory

Branicki

Majda

2014

Communications in Mathematical Sciences

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Abstract. Incomplete knowledge of the true dynamics and its partial observability pose a notoriously difficult problem in many scientific applications which require predictions of high-dimensional dynamical systems with instabilities and energy fluxes across a wide range of scales. In such cases assimilation of real data into the modeled dynamics is necessary for mitigating model error and for improving the stability and predictive skill of imperfect models. However, the practically implementable data assimilation/filtering strategies are also imperfect and not optimal due to the formidably complex nature of the underlying dynamics. Here, the connections between information theory and the filtering problem are exploited in order to establish bounds on the filter error statistics, and to systematically study the statistical accuracy of various Kalman filters with model error for estimating the dynamics of spatially extended, partially observed turbulent systems. The effects of model error on filter stability and accuracy in this high-dimensional setting are analyzed through appropriate information measures which naturally extend the common path-wise estimates of filter performance, like the mean-square error or pattern correlation, to the statistical superensemble setting that involves all possible initial conditions and all realizations of noisy observations of the truth signal. Particular emphasis is on the notion of practically achievable filter skill which requires trade-offs between different facets of filter performance; a new information criterion is introduced in this context. This information-theoretic framework for assessment of filter performance has natural generalizations to Kalman filtering with non-Gaussian statistically exactly solvable forecast models. Here, this approach is utilized to study the performance of imperfect, reduced-order filters involving Gaussian forecast models which use various spatio-temporal discretizations to approximate the dynamics of the stochastically forced advection-diffusion equation; important examples in this configuration include effects of biases due to model error in the filter estimates for the mean dynamics which are quantified through appropriate information measures.

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Nonlinear Filtering and Information Geometry: A Hilbert Manifold Approach

Newton

2018

Information Geometry and Its Applications

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Interactive Statistical Mechanics and Nonlinear Filtering

Cited by 9 publications

References 30 publications

Information geometric nonlinear filtering

Information geometric nonlinear filtering

Quantifying Bayesian filter performance for turbulent dynamical systems through information theory

Nonlinear Filtering and Information Geometry: A Hilbert Manifold Approach

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