  This paper concerns numerical assessment of Monte Carlo error in particle
filters. We show that by keeping track of certain key features of the
genealogical structure arising from resampling operations, it is possible to
estimate variances of a number of standard Monte Carlo approximations which
particle filters deliver. All our estimators can be computed from a single run
of a particle filter with no further simulation. We establish that as the
number of particles grows, our estimators are weakly consistent for asymptotic
variances of the Monte Carlo approximations and some of them are also
non-asymptotically unbiased. The asymptotic variances can be decomposed into
terms corresponding to each time step of the algorithm, and we show how to
consistently estimate each of these terms. When the number of particles may
vary over time, this allows approximation of the asymptotically optimal
allocation of particle numbers.
