Central Limit Theorems for Coupled Particle Filters

Abstract

In this article we prove new central limit theorems (CLT) for several coupled particle filters (CPFs). CPFs are used for the sequential estimation of the difference of expectations w.r.t. filters which are in some sense close. Examples include the estimation of the filtering distribution associated to different parameters (finite difference estimation) and filters associated to partially observed discretized diffusion processes (PODDP) and the implementation of the multilevel Monte Carlo (MLMC) identity. We develop new theory for CPFs and based upon several results, we propose a new CPF which approximates the maximal coupling (MCPF) of a pair of predictor distributions. In the context of ML estimation associated to PODDP with discretization ∆ l we show that the MCPF and the approach in [19] have, under assumptions, an asymptotic variance that is upper-bounded by an expression that is (almost) O(∆ l ), uniformly in time. The O(∆ l ) rate preserves the so-called forward rate of the diffusion in some scenarios which is not the case for the CPF in [21].

Multilevel Particle Filters for the Non-Linear Filtering Problem in Continuous Time

“…Remark 4.1. We note that the constant C in Theorem 4.1 depends upon t. As seen in [13], the task of bounding the asymptotic variance uniformly in t (for models as in [15]) is particularly difficult and one expects even more arduous calculations for the finite-sample variance. All of our below discussion does not consider t, although this is of course a very important issue.…”

“…) given in (9). However, this representation is by no means unique, nor, as discussed in [13] for the purposes of multilevel estimation optimal in any sense.…”

“…The approach corresponds to a so-called maximal coupling of the resampling indices, which ensures that one can consistently estimate differences such as [η l t − η l−1 t ](ϕ). When d x = 1, an alternative scheme which uses Wasserstein coupling is used in [14] (see also [13]). This latter procedure is considered in Section 5, but is not mathematically analyzed.…”

Unbiased Filtering of a Class of Partially Observed Diffusions

“…The method can sometimes provide errors that are uniform in time (e.g. [5]) and is most effective when the hidden state is in moderate dimension (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15). In the context of diffusions, the problem is even more challenging than usual, because the transition density of the diffusion process is seldom available up-to a non-negative and unbiased estimator, which precludes the use of exact simulation methods such as in [7].…”

“…This approach can be further enhanced by using a PF version of the popular multilevel Monte Carlo (MLMC) method of [8,11], called the multilevel particle filter (MLPF); see e.g. [1,13,14]. The basic notion of this methodology is to introduce a hierarchy of time-discretized filters and a collapsing sum representation of the most precise time-discretization, and then to approximate the representation using independent coupled particle filters (CPFs).…”