Optimal Mixture Approximation of the Product of Mixtures

Gaussian mixture densities are a very common tool for describing arbitrarily structured uncertainties in various applications. Many of these applications have to deal with the fusion of uncertainties, an operation that is usually performed by multiplication of these densities. The product of Gaussian mixtures can be calculated exactly, but the number of mixture components in the resulting mixture increases exponentially. Hence, it is essential to approximate the resulting mixture with less components, to keep it tractable for further processing steps. This paper introduces an approach for approximating the exact product with a mixture that uses less components. The maximum approximation error can be chosen by the user. This choice allows to trade accuracy of the approximation for the number of mixture components used. This is possible due to the usage of a progressive processing scheme that calculates the product operation by means of a system of ordinary differential equations. The solution of this system yields the parameters of the desired Gaussian mixture.