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

TL;DR

This paper introduces an efficient algorithm for approximating the second moment of particle filter estimates of the normalizing constant, with cost independent of the number of particles, ensuring unbiasedness and strong consistency.

Contribution

The paper presents a novel $O(M)$ algorithm for second moment approximation that outperforms simple averaging methods in efficiency and variance control.

Findings

Algorithm is unbiased and strongly consistent.

Cost is independent of particle number $N$.

Performance demonstrated on stochastic models.

Abstract

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.