Decision based uncertainty propagation using adaptive Gaussian mixtures

Given a decision process based on the approximate probability density function returned by a data assimilation algorithm, an interaction level between the decision making level and the data assimilation level is designed to incorporate the information held by the decision maker into the data assimilation process. Here the information held by the decision maker is a loss function at a decision time which maps the state space onto real numbers which represent the threat associated with different possible outcomes or states. The new probability density function obtained will address the region of interest, the area in the state space with the highest threat, and will provide overall a better approximation to the true conditional probability density function within it. The approximation used for the probability density function is a Gaussian mixture and a numerical example is presented to illustrate the concept.