Stability of Non-Linear Filters, Observability and Relative Entropy

Abstract

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.

Planning in Observable POMDPs in Quasipolynomial Time

“…The reason why uniform observability suffices for asymptotic stability is a classical result by Blackwell and Dubins [BD62] that (with no assumptions other than absolute continuity of the filter's initialization with respect to the truth) the filter's prediction of Y t+1:∞ at time t must merge with the true distribution as t → ∞. This connection has been observed several times [CL06,VH09a,MY18]; see [VH10] for an exposition.…”

“…Thus, to get any reasonable quantitative bounds it is necessary to assume that each state is reachable from every other state in a small num-ber of steps. To give another example, [MY18] assumes a bound on the Dobrushin coefficient of the transitions, which is violated if there are two states and one action such that the induced transition distributions are disjoint. In fact, [MY18] also assumes a similar bound on the observations, which is intuitively the opposite of observability: the bound gets worse as the observations become more informative.…”

Suboptimal Filtering Over Sensor Networks With Random Communication

“…The papers [19], [20], [21] consider a discrete-time linear dynamical system and associate randomness with the transmission times of the output measurements. A different toolset, based on relative entropy, is adopted in [22] to study the stability and convergence of filters under relaxed assumptions on observation channels. For continuous-time dynamical system driven by white noise, centralized continuous-discrete estimators are studied in [23].…”

Error Bounds for Locally Optimal Distributed Filters with Random Communication Graphs

“…The papers [21], [22], [23] consider a discrete-time linear dynamical system and associate randomness with the transmission times of the output measurements. A different toolset, based on relative entropy, is adopted in [24] to study the stability and convergence of filters under relaxed assumptions on observation channels. For continuous-time dynamical system driven by white noise, centralized continuous-discrete observer is proposed in [25].…”