Bayesian Estimation and Tracking: A Practical Guide

16.1 General Concept of Sequential Importance Sampling

Remember that in all of the Gaussian Kalman filter estimators developed in Part II, the estimates at time t_n depended in an analytical way on the estimates at time t_n−1. We would like to develop a similar recursive formulation for the importance sampling approach for estimation when the noise densities are non-Gaussian.

Recursive estimation methods based on the sequential estimation of the weights are called sequential importance sampling (SIS) particle filters. Using the Chapman–Kolmogorov theorem (3.24), and q(x_n|z_1:n) in (15.41) can be expanded so that (15.41) becomes

(16.1)

The reason we have in the denominator instead of results from the fact that x_n−1 cannot be conditioned on a future observation. Since the state evolution equation is assumed to be a first-order Markov process independent of the observations, we can let resulting in

(16.2)

For a Markov process, we can also make the assumption that

(16.3)

so that the importance ...