Relative Entropy and Error Bounds for Filtering of Markov Processes

Clark, John M.; Ocone, Daniel; Coumarbatch, C.

doi:10.1007/pl00009856

Cited by 25 publications

(24 citation statements)

References 6 publications

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“…In a much more general setting, some results on the asymptotic properties of nonlinear filtering errors can be found in the work of Clark et al [8]. Their point of view is close to the one used in this paper (in particular, ergodicity of the signal process or observations of the white noise type are not assumed), but their results do not establish stability of the filter.…”

Section: Introductionmentioning

confidence: 84%

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Observability and nonlinear filtering

Handel

2008

Probab. Theory Relat. Fields

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This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process. We demonstrate that observability implies stability of the filter, i.e., the filtered estimates become insensitive to the initial measure at large times. For the special case where the signal is a finite-state Markov process and the observations are of the white noise type, a complete (necessary and sufficient) characterization of filter stability is obtained in terms of a slightly weaker detectability condition. In addition to observability, the role of controllability in filter stability is explored. Finally, the results are partially extended to non-compact signal state spaces

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Section: Introductionmentioning

confidence: 84%

“…The significance of the compactness assumption and some extensions of the results to the non-compact case are discussed in Sect. 8.…”

Section: Introductionmentioning

confidence: 95%

Observability and nonlinear filtering

Handel

2008

Probab. Theory Relat. Fields

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“…It has been established in [10] that the relative entropy of the filter is a decreasing sequence, but the analysis is in continuous time and the authors feel it is worth recreating the results here in discrete time. To this end, will extensively use the chain rule for relative entropy [17, Theorem 5.3.1]: Lemma 8.1.…”

Section: Relative Entropy Mergingmentioning

confidence: 96%

“…Relative entropy as a measure of discrepancy between the true filter and the incorrectly initialized filter is studied by Clark, Ocone, and Coumarbatch in [10]. Here they consider the filtering problem in continuous time with the associated nondegeneracy assumptions.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Our structural results in Section 7.1 are very general and prove a useful tool to analyse the root mechanisms of filter stability. iii) The relative entropy of the filter was established as a non-increasing sequence in [10], but its convergence to zero was not established, except for the specific case of the Beneš filter in [25] with a non-degenerate measurement channel. Theorem 4.5 establishes the equivalence between relative entropy merging and total variation, a result which is hinted at in the literature, see [8,Remark 4.2] or [30, Remark 5.9], but never proven fully.…”

Section: 1mentioning

confidence: 99%

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Stability of Non-Linear Filters, Observability and Relative Entropy

McDonald

Yüksel

2018

2018 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Filter stability is a classical problem for partially observed Markov processes (POMP). For a POMP, an incorrectly initialized non-linear filter is said to be stable if the filter eventually corrects itself with the arrival of new measurement information. In the literature, studies on the stability of non-linear filters either focus on the ergodic properties on the hidden Markov process, or the informativeness/observability properties of the measurement channel. While notions of observability exist in the literature, they are often difficult to verify and specific examples of observable systems are mostly restricted to additive noise models with additional strict regularity properties. In this paper, we introduce a general definition of observability for stochastic non-linear dynamical systems and compare it with related findings in the literature. Our observability notion involves a functional characterization which is easily computed for a variety of systems as we demonstrate. Under this observability definition we establish filter stability results for a variety of criteria including weak merging and total variation merging, both in expectation and in an almost sure sense, as well as relative entropy. We consider the implications between these notions, which unify various results in the literature in a concise manner. Our conditions, and the examples we study, complement and generalize the existing results on filter stability.

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Interactive Statistical Mechanics and Nonlinear Filtering

Newton

2008

J Stat Phys

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Relative Entropy and Error Bounds for Filtering of Markov Processes

Cited by 25 publications

References 6 publications

Observability and nonlinear filtering

Observability and nonlinear filtering

Stability of Non-Linear Filters, Observability and Relative Entropy

Interactive Statistical Mechanics and Nonlinear Filtering

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