We consider the numerical approximation of the filtering problem in high dimensions, that
is, when the hidden state lies in Rd with large d. 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 provides 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 independent and identically distributed structure and
a model of an L-Markovian structure (L ≥ 1, L independent of d) in the d-dimensional
space direction, when we show that the algorithm exhibits certain stability properties
as d increases at a cost O(nN d2), where n is the time parameter and N is the number
of Monte Carlo samples, which 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.