Gauge Freedom within the Class of Linear Feedback Particle Filters

Abstract

Feedback particle filters (FPFs) are Monte-Carlo approximations of the solution of the filtering problem in continuous time. The samples or particles evolve according to a feedback control law in order to track the posterior distribution. However, it is known that by itself, the requirement to track the posterior does not lead to a unique algorithm. Given a particle filter, another one can be constructed by applying a time-dependent transformation of the particles that keeps the posterior distribution invariant. Here, we characterize this gauge freedom within the class of FPFs for the linear-Gaussian filtering problem, and thereby extend previously known parametrized families of linear FPFs.

A Unification of Weighted and Unweighted Particle Filters

“…Observe how particle distribution interpolates between the prior distribution (when η = 1) and the posterior distribution (when η = 0, α = 0). It is easy to confirm that in the latter case, where β remains as the only free parameter, the PF resulting from Proposition 5 corresponds to the unweighted linear PF stated in [1], equation (17), with one BM, if it is indeed rewritten in terms of v. Notably, v = 0 recovers the deterministic linear FPF introduced in [24].…”

“…This paper also touches on the notion of non-uniqueness in designing the PF, which has been discussed in the literature but mainly restricted to the linear-Gaussian case and freedom of the particle movements. For example, [1] provides a systematic exploration of the non-uniqueness within the class of linear FPF in terms of gauge transforms and [25] studies the non-uniqueness of the feedback control law in particle dynamics for different types of ensemble Kalman filters. In this work, however, we examine the non-uniqueness in the nonlinear case and in a more general setting that includes weight dynamics.…”

“…where the linear operators {A t , B t , C t } have been defined as A t φ(x) := γ(x, t)φ(x) + u(x, t) + k(x, t)ε(x, t) + v(x, t)ζ(x, t) φ ′ (x) (5.21) + 1 2 k(x, t) 2 + v(x, t) 2 φ ′′ (x), B t φ(x) := ε(x, t)φ(x) + k(x, t)φ ′ (x), (5.22) C t φ(x) := ζ(x, t)φ(x) + v(x, t)φ ′ (x). (5.23) Step (b): Writing the SDE (5.20) in integral form and taking the conditional expectation of both sides gives the following:…”