A stable particle filter for a class of high-dimensional state-space models

Abstract

Beskos et al. AbstractWe consider the numerical approximation of the filtering problem in high dimensions, that is when the hidden state lies in R d with d large. For low dimensional problems, one of the most popular numerical procedures for consistent inference is the class of approximations termed particle filters or sequential Monte Carlo methods. However, in high dimensions, standard particle filters (e.g. the bootstrap particle filter) can have a cost that is exponential in d for the algorithm to be stable in an appropriate sense. We develop a new particle filter, called the space-time particle filter, for a specific family of state-space models in discrete time. This new class of particle filters provide consistent Monte Carlo estimates for any fixed d, as do standard particle filters. Moreover, when there is a spatial mixing element in the dimension of the state-vector, the space-time particle filter will scale much better with d than the standard filter for a class of filtering problems. We illustrate this analytically for a model of a simple i.i.d. structure and one ofspace-direction, when we show that the algorithm exhibits certain stability properties as d increases at a cost O(nN d 2 ), where n is the time parameter and N is the number of Monte Carlo samples, that are fixed and independent of d. Our theoretical results are also supported by numerical simulations on practical models of complex structures. The results suggest that it is indeed possible to tackle some high dimensional filtering problems using the space-time particle filter that standard particle filters cannot handle.

Langevin and Hamiltonian Based Sequential MCMC for Efficient Bayesian Filtering in High-Dimensional Spaces

“…As a consequence, the classical Leapfrog integrator can be used for the SmHMC. technique [6], [7], [12]. More importantly, unlike all the other methods (SIR and SMCMC-based ones), this bias will never tend asymptotically (with the number of samples N ) to zero as long as the block size is less than the dimension of the state.…”

“…as the dimension d increases from d = 10, 100, 1000, ...) and analogous degeneracy effects typically due to the high variability of the incremental weights defined in Eq. (12). This can be exacerbated when non-linear and non-trivial dependence structures are present between the state vector sub-dimensions.…”

“…However, this strategy introduces in the final estimates a difficult to quantify bias which depends on the position along the split state vector elements. Another strategy, called space-time Particle filter (STPF), has been recently proposed in [12]. The key idea of the STPF is to exploit a specific factorization of the posterior distribution to design a particle filter moving along both the space and time index (as opposed to traditional particle filter that moves only along the time index).…”

“…The key idea of the STPF is to exploit a specific factorization of the posterior distribution to design a particle filter moving along both the space and time index (as opposed to traditional particle filter that moves only along the time index). However, since the local particle filters are running along space dimension, a patch degeneracy (on the space dimension) effect can be expected as the dimension of the system increases [7], [12].…”

Sequential Monte Carlo Methods for System Identification**This work was supported by the projects Learning of complex dynamical systems (Contract number: 637-2014-466) and Probabilistic modeling of dynamical systems (Contract number: 621-2013-5524), both funded by the Swedish Research Council.

“…A known deficiency of the standard (bootstrap) particle filter is its inability to handle high-dimensional variables x t [Bickel et al, 2008], which is usually the case in for example spatio-temporal models. However, some very recent work has shown promising directions to tackle high-dimensional models in a consistent way using SMC [Naesseth et al, 2014, Beskos et al, 2014, Naesseth et al, 2015.…”