Bias of Particle Approximations to Optimal Filter Derivative

In many applications, a state-space model depends on a parameter which needs to be inferred from a data set. Quite often, it is necessary to perform the parameter inference online. In the maximum likelihood approach, this can be done using stochastic gradient search and the optimal filter derivative. However, the optimal filter and its derivative are not analytically tractable for a non-linear state-space model and need to be approximated numerically. In [Poyiadjis, Doucet and Singh, Biometrika 2011], a particle approximation to the optimal filter derivative has been proposed, while the corresponding $L_{p}$ error bonds and the central limit theorem have been provided in [Del Moral, Doucet and Singh, SIAM Journal on Control and Optimization 2015]. Here, the bias of this particle approximation is analyzed. We derive (relatively) tight bonds on the bias in terms of the number of particles. Under (strong) mixing conditions, the bounds are uniform in time and inversely proportional to the number of particles. The obtained results apply to a (relatively) broad class of state-space models met in practice.