  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.
