Reducing the Bias in Blocked Particle Filtering for High-Dimensional   Systems
  Particle filtering is a powerful approximation method that applies to state estimation in nonlinear and non-Gaussian dynamical state-space models. Unfortunately, the approximation error depends exponentially on the system dimension. This means that an incredibly large number of particles may be needed to appropriately control the error in very large scale filtering problems. The computational burden required is often prohibitive in practice. Rebeschini and Van Handel (2013) analyse a new approach for particle filtering in large-scale dynamic random fields. Through a suitable localisation operation they reduce the dependence of the error to the size of local sets, each of which may be considerably smaller than the dimension of the original system. The drawback is that this localisation operation introduces a bias. In this work, we propose a modified version of Rebeschini and Van Handel's blocked particle filter. We introduce a new degree of freedom allowing us to reduce the bias. We do this by enlarging the space during the update phase and thus reducing the amount of dependent information thrown away due to localisation. By designing an appropriate tradeoff between the various tuning parameters it is possible to reduce the total error bound via allowing a temporary enlargement of the update operator without really increasing the overall computational burden. 
Francesco Bertoli and Adrian N. Bishop
math.OC