An algorithm for approximating the second moment of the normalizing constant estimate from a particle filter

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.