  We consider the numerical approximation of the filtering problem in high
dimensions, that is, when the hidden state lies in $\mathbb{R}^d$ 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
\emph{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,
we expect that the state-space particle filter will scale much better with $d$
than the standard filter. We illustrate this analytically for a model of a
simple i.i.d. structure and one of a Markovian structure in the $d$-dimensional
space-direction, when we show that the algorithm exhibits certain stability
properties as $d$ increases at a cost $\mathcal{O}(nNd^2)$, where $n$ is the
time parameter and $N$ is the number of Monte Carlo samples, that are fixed and
independent of $d$. Similar results are expected to hold, under a more general
structure than the i.i.d.~one. independently of the dimension. 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.
