  We consider situations where the applicability of sequential Monte Carlo
particle filters is compromised due to the expensive evaluation of the particle
weights. To alleviate this problem, we propose a new particle filter algorithm
based on the multilevel approach. We show that the resulting multilevel
bootstrap particle filter (MLBPF) retains the strong law of large numbers as
well as the central limit theorem of classical particle filters under mild
conditions. Our numerical experiments demonstrate up to 85\% reduction in
computation time compared to the classical bootstrap particle filter, in
certain settings. While it should be acknowledged that this reduction is highly
application dependent, and a similar gain should not be expected for all
applications across the board, we believe that this substantial improvement in
certain settings makes MLBPF an important addition to the family of sequential
Monte Carlo methods.
