Bias of the SIR filter in estimation of the state transition noise

TL;DR

This paper analyzes the bias in the SIR filter's estimation of state transition noise, showing it tends to overestimate the noise due to sample impoverishment and proposing that larger transition noise can improve particle diversity.

Contribution

It reveals the bias in the SIR filter's noise estimation and explains how a heavier-tailed proposal distribution can mitigate sample impoverishment effects.

Findings

SIR filter tends to overestimate transition noise when it is unknown.

Using a proposal with heavier tails reduces sample impoverishment.

Numerical simulations confirm the bias and the benefits of larger transition noise.

Abstract

This Note investigates the bias of the sampling importance resampling (SIR) filter in estimation of the state transition noise in the state space model. The SIR filter may suffer from sample impoverishment that is caused by the resampling and therefore will benefit from a sampling proposal that has a heavier tail, e.g. the state transition noise simulated for particle preparation is bigger than the true noise involved with the state dynamics. This is because a comparably big transition noise used for particle propagation can spread overlapped particles to counteract impoverishment, giving better approximation of the posterior. As such, the SIR filter tends to yield a biased (bigger-than-the-truth) estimate of the transition noise if it is unknown and needs to be estimated, at least, in the forward-only filtering estimation. The bias is elaborated via the direct roughening approach by means of both qualitative logical deduction and quantitative numerical simulation.