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 tn depended in an analytical way on the estimates at time tn−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(xn|z1:n) in (15.41) can be expanded so that (15.41) becomes
The reason we have in the denominator instead of results from the fact that xn−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
For a Markov process, we can also make the assumption that
so that the importance ...