Unbiased filtering of a class of partially observed diffusions

Abstract

In this article we consider a Monte-Carlo-based method to filter partially observed diffusions observed at regular and discrete times. Given access only to Euler discretizations of the diffusion process, we present a new procedure which can return online estimates of the filtering distribution with no time-discretization bias and finite variance. Our approach is based upon a novel double application of the randomization methods of Rhee and Glynn (Operat. Res.63, 2015) along with the multilevel particle filter (MLPF) approach of Jasra et al. (SIAM J. Numer. Anal.55, 2017). A numerical comparison of our new approach with the MLPF, on a single processor, shows that similar errors are possible for a mild increase in computational cost. However, the new method scales strongly to arbitrarily many processors.

Unbiased Estimation of the Vanilla and Deterministic Ensemble Kalman-Bucy Filters

“…In the current work we apply an analogous setting cost-wise. In fact, provided that the probability distributions P L and P P satisfy condition (3.5), the error-to-cost rates are obtained in the same way as in [25]. If we set N p = N 0 2 p , ∆ l = 2 −l , P P (p) ∝ N −1 p (p + 1) log 2 (p + 2) 2 and P L (l) ∝ ∆ l (l + 1) log 2 (l + 2) 2 , and we choose M = O(ε −2 ) (with ε > 0) so that the variance is O(ε 2 ) then the cost to obtain M samples is O(ε −2 | log(ε)| 2+δ ) for any δ > 0 ([25] page 14).…”

“…Double randomization schemes have already been applied in [25] to the particle filter. In the current work we apply an analogous setting cost-wise.…”

“…This involves placing a probability distribution over the level of time-discretization, and then to sample from that particular level. For filtering this has been considered in the contexts of different works related to partially observed diffusions, discretized models and gradient estimates [5,21,22,25,34]. However as of yet these techniques have not been applied to Kalman-based filtering, which thus acts as the motivation for this article.…”

Unbiased estimation of the Hessian for partially observed diffusions

“…In particular, these methods allow us to unbiasedly estimate an expectation of a functional, by randomizing on the level of the time-discretization in a type of multilevel Monte Carlo (MLMC) approach [17], where there is a coupling between different levels. As a result, this methodology has been considered in the context of both filtering and Bayesian computation [18][19][20][21] and gradient estimation [22]. One advantage of this approach is that, with couplings, it is relatively simple to use & implement computationally, while exploiting such methodologies on a range of different model problems or set-ups.…”

“…It is important to emphasize that with inverse Hessian, which is required for Newton methodologies, we can debias both the C-CCPF and the normalΔPF. This can be achieved by using the same techniques which are presented in the work of Jasra et al [21].…”