PF-EIS & PF-MT: New Particle Filtering Algorithms for Multimodal Observation Likelihoods and Large Dimensional State Spaces

Consider tracking a state space model with multimodal observation likelihoods using a particle filter (PF). Under certain assumptions that imply narrowness of the state transition prior, many efficient importance sampling techniques have been proposed in literature. For large dimensional state spaces (LDSS), these assumptions may not always hold. But, it is usually true that at a given time, state change in all except a few dimensions is small. We use this fact to design a simple modification (PF-EIS) of an existing importance sampling technique. Also, importance sampling on an LDSS is expensive (requires large number of particles, N) even with the best technique. But if the "residual space" variance is small enough, we can replace importance sampling in residual space by mode tracking (PF-MT). This drastically reduces the importance sampling dimension for LDSS, hence greatly reducing the required N.