  The typical particle filtering approximation error is exponentially dependent
on the dimension of the model. Therefore, to control this error, an enormous
number of particles are required, which means a heavy computational burden that
is often so great it is simply prohibitive. Rebeschini and van Handel (2013)
consider particle filtering in a large-scale dynamic random field. Through a
suitable localisation operation, they prove that a modified particle filtering
algorithm can achieve an approximation error that is mostly independent of the
problem dimension. To achieve this feat, they inadvertently introduce a
systematic bias that is spatially dependent (in that the bias at one site is
dependent on the location of that site). This bias consequently varies
throughout field. In this work, a simple extension to the algorithm of
Rebeschini and van Handel is introduced which acts to average this bias term
over each site in the field through a kind of spatial smoothing. It is shown
that for a certain class of random field it is possible to achieve a completely
spatially uniform bound on the bias and that in any general random field the
spatial inhomogeneity is significantly reduced when compared to the case in
which spatial smoothing is not considered. While the focus is on spatial
averaging in this work, the proposed algorithm seemingly exhibits other
advantageous properties such as improved robustness and accuracy in those cases
in which the underlying dynamic field is time varying.
