15.2 General Concepts Importance Sampling

As seen in Part II of this book, almost all of the Gaussian estimation (tracking) methods for nonlinear systems involve some kind of numerical evaluation of density-weighted moment integrals. Some of these numerical methods consist of weighted sums of a function at specific deterministic sigma points.

If the weighting density is multivariate and non-Gaussian or unknown, in general deterministic numerical methods cannot be used except for some cases with known analytical densities [13]. Recent (over the past 30 years) developments in Monte Carlo methods of integration for general density-weighted integrals have opened the floodgates for applications to estimation and tracking. The most influential of these methods revolve around the concept of independent importance sampling.

For a more complete discussion of the origins and fundamental developments of both general Monte Carlo integration methods and sequential importance sampling methods, the reader is referred to the books by Robert and Casella [14] and Doucet et al. [15] and the articles by Doucet [16] and Liu and Chen [17] and the references contained therein.

In Chapter 12, we solved the Gaussian density-weighted integrals by generating Monte Carlo samples from Gaussian densities thus reducing the integrals to weighted sums of functions evaluated at the sample points. For unknown or non-Gaussian densities, the process of generating Monte Carlo samples from that density can be accomplished ...

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