Grid Based Nonlinear Filtering Revisited: Recursive Estimation & Asymptotic Optimality

Abstract

We revisit the development of grid based recursive approximate filtering of general Markov processes in discrete time, partially observed in conditionally Gaussian noise. The grid based filters considered rely on two types of state quantization: The Markovian type and the marginal type. We propose a set of novel, relaxed sufficient conditions, ensuring strong and fully characterized pathwise convergence of these filters to the respective MMSE state estimator. In particular, for marginal state quantizations, we introduce the notion of conditional regularity of stochastic kernels, which, to the best of our knowledge, constitutes the most relaxed condition proposed, under which asymptotic optimality of the respective grid based filters is guaranteed. Further, we extend our convergence results, including filtering of bounded and continuous functionals of the state, as well as recursive approximate state prediction. For both Markovian and marginal quantizations, the whole development of the respective grid based filters relies more on linear-algebraic techniques and less on measure theoretic arguments, making the presentation considerably shorter and technically simpler.

A Bayesian Approach to Model Calibration and Parameter Estimation in Potential Drop Measuring

“…In terms of finite state space models, grid-based methods can be utilised to provide an optimal solution [22,23]. However, a variety of approximate filtering and smoothing methods for nonlinear and infinite state space models exist, which encompass inter alia Gaussian filtering and smoothing methods [19], Monte Carlo Sampling approaches [23] and approximate gridbased methods [24]. The former can be considered as local approaches, the two latter as global approaches [25].…”

On pathwise convergence of particle & grid based nonlinear filters: Feller vs conditional regularity

“…for all t ∈ N, where [A] (:, i) extracts the i-th column of the matrix A of arbitrary dimensions. In order to analytically study the asymptotic consistency of the approximate filter defined above, the simple notion of conditional regularity was recently introduced in [18]. The respective definition follows.…”

“…Definition 1. (Conditional Regularity of Stochastic Kernels [18]) Consider the stochastic (or Markov) kernel K :…”

“…On the other hand, for the case of grid based filters, Egoroff's Theorem is purely quantitatively justified. In fact, for fixed T , convergence to the MMSE optimal nonlinear filter occurs uniformly in a set of outcomes that approximates the certain event, at an exponential rate in N (for more details, see [22] and [18]). In this way, the approximation error is characterized for almost all possible paths of the observations process, providing a strong convergence guarantee, which enhances filter performance, at least in theory.…”

“…For grid based filters, a condition which provides satisfying convergence guarantees is conditional regularity of stochastic kernels [18]. Under the assumption of conditional regularity, grid based filters converge to the optimal MMSE state estimator in a relatively strong sense, i.e., compactly in time and uniformly in a set of outcomes of probability measure almost 1.…”