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

arXiv: https://arxiv.org/abs/1602.02279 · HTML: https://mlsanity.com/paper/1602.02279/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Feb 2016.

## Built on by (most cited first, 1 of 1)

- [Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyond](https://mlsanity.com/paper/2502.04575/) (2025)

## Abstract (arXiv)

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 Lotkatextendash Volterra system and a simple AR(1) model.