Discrete time nonlinear filters with informative observations are stable

Abstract

The nonlinear filter associated with the discrete time signal-observation model (X k , Y k ) is known to forget its initial condition as k → ∞ regardless of the observation structure when the signal possesses sufficiently strong ergodic properties. Conversely, it stands to reason that if the observations are sufficiently informative, then the nonlinear filter should forget its initial condition regardless of any properties of the signal. We show that for observations of additive type Y k = h(X k ) + ξ k with invertible observation function h (under mild regularity assumptions on h and on the distribution of the noise ξ k ), the filter is indeed stable in a weak sense without any assumptions at all on the signal process. If the signal satisfies a uniform continuity assumption, weak stability can be strengthened to stability in total variation.2000 Mathematics Subject Classification. Primary 93E11; secondary 60J05, 62M20, 93E15.

Stability of Non-Linear Filters, Observability and Relative Entropy

“…Note that this result is an improvement over [28,Theorem 2.2]. The theorem concluded bounded Lipschitz merging of the filter in expectation, which is weaker than weak convergence in expectation.…”

“…The result of Blackwell and Dubins [3] pairs with uniform observability, in that (3.2) directly follows from Blackwell and Dubins. Then uniform observability would imply filter stability in bounded Lipschitz distance [28]. van Handel proves this in [28], however the author only studied the measurement channel where h(x, z) = f (x) + z where f −1 is uniformly continuous and Z must have an everywhere non-zero characteristic function (e.g.…”

“…However, we would like to consider more stringent notions of stability for the filter, as well as stability in relative entropy for both the predictor and filter. Under different assumptions specific results can be developed , for example [28,Lemma 4.2] which establishes the total variation merging of the filter in expectation from that of the predictor using non-degeneracy. However, by examining the form of the Radon Nikodym derivative of P µ and P ν restricted and conditioned on different sigma fields, we can gain significant insight into how these different notions of stability relate to one another.…”

Robustness to Incorrect Priors in Infinite Horizon Stochastic Control

“…The problems we consider are also related to, in the control-free context, the theory and applications of non-linear filtering with incorrect initial specifications. Here, the problem is to identify conditions on when an incorrectly initialized non-linear filter asymptotically gets corrected with the accumulation of additional measurements; these often require strong ergodicity properties of the Markov process [8,9,11] or regularity properties (such as absolute continuity) of incorrect prior with respect to the true one and conditions on the measurement processes [24].…”

Phase transitions in nonlinear filtering

“…This is a mild condition in classical filtering models that serves mainly to rule out the singular case of noiseless observations: for example, the addition of any observation noise to the above counterexample would render the filter ergodic. On the other hand, even in the noiseless case, ergodicity is inherited in the absence of certain symmetries that are closely related to systems-theoretic notions of observability [48,49,51,9]. One can therefore conclude that while there exist elementary examples where the ergodicity of the model fails to be inherited by the filter, such examples must be very fragile as they require both a singular observation structure and the presence of unusual symmetries, either of which is readily broken by a small perturbation of the model.…”