Beskos et al. We 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 of space-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.
The space-time particle filters REF also rely on factorization of the conditional posterior, but Beskos et al. demonstrate that the filters provide consistent estimates as the number of particles increases.
A stable particle filter for a class of high-dimensional state-space models
Advances in Applied Probability
Mathematics