Block Kalman Filter: An Asymptotic Block Particle Filter In The Linear Gaussian Case

The curse of dimensionality in particle filtering can be mitigated by approximating the posterior distribution by a product of marginals on disjoint low dimensional subspaces of the state space. One such approach is known as the block particle filter in which the correction and resampling steps in particle filtering are run separately for estimating each marginal. In the linear and Gaussian case, the particle filter converges to the optimal Bayesian solution, i.e. the Kalman filter, as the number of particle increases. In this paper, we introduce the block based approach in the Kalman filter and show that the block particle filter asymptotically acts as the resulting block Kalman filter. It provides a relevant framework to disambiguate the bias incurred by the blocking step from the Monte Carlo estimation error.