Recursive bayesian estimation using gaussian sums

• Authors: H. W. Sorenson;D. L. Alspach
• Venue: Automatica (Journal of IFAC)
• Year: 1971

The Bayesian recursion relations which describe the behavior of the a posteriori probability density function of the state of a time-discrete stochastic system conditioned on available measurement data cannot generally be solved in closed-form when the system is either non-linear or nongaussian. In this paper a density approximation involving convex combinations of gaussian density functions is introduced and proposed as a meaningful way of circumventing the difficulties encountered in evaluating these relations and in using the resulting densities to determine specific estimation policies. It is seen that as the number of terms in the gaussian sum increases without bound, the approximation converges uniformly to any density function in a large class. Further, any finite sum is itself a valid density function unlike many other approximations that have been investigated. The problem of determining the a posteriori density and minimum variance estimates for linear systems with nongaussian noise is treated using the gaussian sum approximation. This problem is considered because it can be dealt with in a relatively straightforward manner using the approximation but still contains most of the difficulties that one encounters in considering non-linear systems since the a posteriori density is nongaussian. After discussing the general problem from the point-of-view of applying gaussian sums, a numerical example is presented in which the actual statistics of the a posteriori density are compared with the values predicted by the gaussian sum and by the Kalman filter approximations.