  We propose a new algorithm for approximating the non-asymptotic second moment
of the marginal likelihood estimate, or normalizing constant, provided by a
particle filter. The computational cost of the new method is $O(M)$ per time
step, independently of the number of particles $N$ in the particle filter,
where $M$ is a parameter controlling the quality of the approximation. This is
in contrast to $O(MN)$ for a simple averaging technique using $M$ i.i.d.
replicates of a particle filter with $N$ particles. We establish that the
approximation delivered by the new algorithm is unbiased, strongly consistent
and, under standard regularity conditions, increasing $M$ linearly with time is
sufficient to prevent growth of the relative variance of the approximation,
whereas for the simple averaging technique it can be necessary to increase $M$
exponentially with time in order to achieve the same effect. Numerical examples
illustrate performance in the context of a stochastic Lotka\textendash Volterra
system and a simple AR(1) model.
