  We introduce a weighted particle representation for the solution of the
filtering problem based on a suitably chosen variation of the classical de
Finetti theorem. This representation has important theoretical and numerical
applications. In this paper, we explore some of its theoretical consequences.
The first is to deduce the equations satisfied by the solution of the filtering
problem in three different frameworks: the signal independent Brownian
measurement noise model, the spatial observations with additive white noise
model and the cluster detection model in spatial point processes. Secondly we
use the representation to show that a suitably chosen filtering discretisation
converges to the filtering solution. Thirdly we study the leading error
coefficient for the discretisation. We show that it satisfies a stochastic
partial differential equation by exploiting the weighted particle
representation for both the approximation and the limiting filtering solution.
