On the bias of the SIR filter in parameter estimation of the dynamics process of state space models

As a popular nonlinear estimation tool, the sampling importance resampling (SIR) filter has been applied with the expectation-maximization (EM) principle, including the typical maximum a posteriori (MAP) estimation and maximum likelihood (ML) estimation, for estimating the parameters of the state space model (SSM). This paper concentrates on an inevitable bias existing in the EM-SIR filter for estimating the dynamics process of the SSM. It is analyzed that the root reason for the bias is the sample impoverishment caused by the resampling procedure employed in the filter. A process noise simulated for the particle propagation that is larger than the real noise involved with the true state will be helpful to counteract sample impoverishment, thereby providing better filtering result. Correspondingly, the EM-SIR filter tends to yield a biased (larger-than-the-truth) estimate of the process noise if it is unknown and needs to be estimated. The bias is elaborated via a straightforward roughening approach by means of both qualitative logical deduction and quantitative numerical simulation. However, it seems hard to fully remove this bias in practice.