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A Robustification Approach to Stability and to Uniform Particle Approximation of Nonlinear Filters: The Example of PseudoMixing Signals
, 2002
"... We propose a new approach to study the stability of the optimal filter w.r.t. its initial condition, by introducing a "robust" filter, which is exponentially stable and which approximates the optimal filter uniformly in time. The "robust" filter is obtained here by truncation of ..."
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Cited by 37 (3 self)
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We propose a new approach to study the stability of the optimal filter w.r.t. its initial condition, by introducing a "robust" filter, which is exponentially stable and which approximates the optimal filter uniformly in time. The "robust" filter is obtained here by truncation of the likelihood function, and the robustification result is proved under the assumption that the Markov transition kernel satisfies a pseudomixing condition (weaker than the usual mixing condition), and that the observations are "sufficiently good". This robustification approach allows us to prove also the uniform convergence of several particle approximations to the optimal filter, in some cases of nonergodic signals.
R.Liptser, Stability of nonlinear filters in nonmixing
"... The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by, so called, mixing condition. The ..."
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Cited by 18 (7 self)
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The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by, so called, mixing condition. The latter is formulated in terms of the transition probability density of the signal. The most restrictive requirement of the mixing condition is a uniform positiveness of this density. We show that this requirement might be weakened regardless of an observation process structure.
Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear
, 2003
"... ABSTRACT. – In this work we study connections between various asymptotic properties of the nonlinear filter. It is assumed that the signal has a unique invariant probability measure. The key property of interest is expressed in terms of a relationship between the observation σ field and the tail σ f ..."
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Cited by 10 (0 self)
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ABSTRACT. – In this work we study connections between various asymptotic properties of the nonlinear filter. It is assumed that the signal has a unique invariant probability measure. The key property of interest is expressed in terms of a relationship between the observation σ field and the tail σ field of the signal, in the stationary filtering problem. This property can be viewed as the permissibility of the interchange of the order of the operations of maximum and countable intersection for certain σfields. Under suitable conditions, it is shown that the above property is equivalent to various desirable properties of the filter such as (a) uniqueness of invariant measure for the signal, (b) uniqueness of invariant measure for the pair (signal, filter), (c) a finite memory property of the filter, (d) a property of finite time dependence between the signal and observation σ fields and (e) asymptotic stability of the filter. Previous works on the asymptotic stability of the filter for a variety of filtering models then identify a rich class of filtering problems for which the above equivalent properties hold. 2003 Éditions scientifiques et médicales Elsevier SAS
On a role of predictor in the filtering stability
 Electron. Comm. Probab
"... Abstract. When is a nonlinear filter stable with respect to its initial condition? In spite of the recent progress, this question still lacks a complete answer in general. Currently available results indicate that stability of the filter depends on the signal ergodic properties and the observation p ..."
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Cited by 9 (1 self)
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Abstract. When is a nonlinear filter stable with respect to its initial condition? In spite of the recent progress, this question still lacks a complete answer in general. Currently available results indicate that stability of the filter depends on the signal ergodic properties and the observation process regularity and may fail if either of the ingredients is ignored. In this note we address the question of stability in a particular weak sense and show that the estimates of certain functions are always stable. This is verified without dealing directly with the filtering equation and turns to be inherited from certain onestep predictor estimates. 1.
R.: A complete solution to Blackwell’s unique ergodicity problem for hidden Markov chains
, 2010
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Stability of nonlinear filters in nonmixing case
, 2004
"... The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is an ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by the, so called, mixing conditi ..."
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Cited by 6 (1 self)
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The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is an ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by the, so called, mixing condition. The latter is formulated in terms of the transition probability density of the signal. The most restrictive requirement of the mixing condition is the uniform positiveness of this density. We show that it might be relaxed regardless of an observation process structure. 1. Introduction and the main result. This paper addresses the stability problem of the nonlinear filtering equation with respect to its initial condition. We consider a homogeneous ergodic Markov chain (Xn)n≥0 with values in S ⊆ Rd regarded as a signal to be filtered from observation of (Yn)n≥1,
Ergodicity and stability of the conditional distributions of nondegenerate markov chains
 Ann. Appl. Probab
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INTRINSIC METHODS IN FILTER STABILITY
"... Abstract. The purpose of this article is to survey some intrinsic methods for studying the stability of the nonlinear filter. By ‘intrinsic ’ we mean methods which directly exploit the fundamental representation of the filter as a conditional expectation through classical probabilistic techniques su ..."
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Abstract. The purpose of this article is to survey some intrinsic methods for studying the stability of the nonlinear filter. By ‘intrinsic ’ we mean methods which directly exploit the fundamental representation of the filter as a conditional expectation through classical probabilistic techniques such as change of measure, martingale convergence, coupling, etc. Beside their conceptual appeal and the additional insight gained into the filter stability problem, these methods allow one to establish stability of the filter under weaker conditions compared to other methods, e.g., to go beyond strong mixing signals, to reveal connections between filter stability and classical notions of observability, and to discover links to martingale convergence and information theory. 1. Inroduction Consider a pair of random sequences (X, Y) = (Xn, Yn)n∈Z+, where the signal component Xn takes values in a Polish space 1 S and the observation component Yn takes values in R p for some p ≥ 1. The classical filtering problem is to compute the conditional distribution πn(·) = P(Xn ∈ · F Y 0,n), (1.1) where F Y k,n stands for the σalgebra of events generated by Ym, k ≤ m ≤ n (similarly, we will use below the σalgebra F X k,n generated by Xm, k ≤ m ≤ n). Once πn is found, the optimal mean square estimate of f(Xn) can be calculated as E(f(Xn)F Y ∫ 0,n) = f(x) πn(dx) for any function f with Ef(Xn)  2 < ∞. If both X and (X, Y) are Markov processes, πn satisfies a recursive filtering equation. Specifically, let Λ and ν denote the transition probability and the initial distribution of X, i.e., for A ∈ B(S) ν(A) = P(X0 ∈ A), Λ(Xn−1, A) = P(Xn ∈ AF X 0,n−1)
Parameter estimation and asymptotic stability in stochastic filtering
 Stochastic Process. Appl
"... In this paper, we study the problem of estimating a Markov chain X(signal) from its noisy partial information Y, when the transition probability kernel depends on some unknown parameters. Our goal is to compute the conditional distribution process P{XnYn,..., Y1}, referred to hereafter as the optim ..."
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Cited by 6 (1 self)
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In this paper, we study the problem of estimating a Markov chain X(signal) from its noisy partial information Y, when the transition probability kernel depends on some unknown parameters. Our goal is to compute the conditional distribution process P{XnYn,..., Y1}, referred to hereafter as the optimal filter. Following a standard Bayesian technique, we treat the parameters as a nondynamic component of the Markov chain. As a result, the new Markov chain is not going to be mixing, even if the original one is. We show that, under certain conditions, the optimal filters are still going to be asymptotically stable with respect to the initial conditions. Thus, by computing the optimal filter of the new system, we can estimate the signal adaptively. Key words: nonlinear filtering, asymptotic stability, ergodic decomposition, Bayesian estimators.
Model robustness of finite state nonlinear filtering over the infinite time horizon
 Ann. Appl. Probab
"... Abstract. We investigate the robustness of nonlinear filtering for continuous time finite state Markov chains, observed in white noise, with respect to misspecification of the model parameters. It is shown that the distance between the optimal filter and that with incorrect model parameters converge ..."
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Cited by 5 (5 self)
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Abstract. We investigate the robustness of nonlinear filtering for continuous time finite state Markov chains, observed in white noise, with respect to misspecification of the model parameters. It is shown that the distance between the optimal filter and that with incorrect model parameters converges to zero uniformly over the infinite time interval as the misspecified model converges to the true model, provided the signal obeys a mixing condition. The filtering error is controlled through the exponential decay of the derivative of the nonlinear filter with respect to its initial condition. We allow simultaneously for misspecification of the initial condition, of the transition rates of the signal, and of the observation function. The first two cases are treated by relatively elementary means, while the latter case requires the use of Skorokhod integrals and tools of anticipative stochastic calculus. 1.