Nonlinear Bayesian estimation using Gaussian sum approximations

Knowledge of the probability density function of the state conditioned on all available measurement data provides the most complete possible description of the state, and from this density any of the common types of estimates (e.g., minimum variance or maximum a posteriori) can be determined. Except in the linear Gaussian case, it is extremely difficult to determine this density function. In this paper an approximation that permits the explicit calculation of the a posteriori density from the Bayesian recursion relations is discussed and applied to the solution of the nonlinear filtering problem. In particular, it is noted that a weighted sum of Gaussian probability density functions can be used to approximate arbitrarily closely another density function. This representation provides the basis for procedure that is developed and discussed.