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), img and q(xn|z1:n) in (15.41) can be expanded so that (15.41) becomes

(16.1) equation

The reason we have img in the denominator instead of img 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 img resulting in

(16.2)

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

(16.3)

so that the importance ...

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