A Learning Algorithm Based on Experts' Opinions

The problem of efficient management of the information provided by many sources or equivalently of the aggregation of various models concerning a situation is considered. Such problems arise very often in decision making where the issue of modeling of the random factors introducing uncertainty arises. In this work we propose an aggregation scheme that allows a decision maker to combine various information sources or expert opinions concerning a random experiment into an "average" opinion which provides the best available single model for the situation at hand. The proposed method is based on variational arguments and proposes the best model as the minimizer of a properly selected weighted loss functional in the space of probability measures. Two proposals are investigated for the choice of this loss functional, (i) a weighted entropy functional including contributions from all information sources, and (ii) a weighted Wasserstein distance functional, leading to a barycentric model in the space of probability measures. Moreover, a machine learning type procedure for model selection, based in the proposed approach is developed, where the weights are chosen either by the proposed heuristic method where the weights are updated according to the information sources (experts) validity (credibility based on performance) or by standard optimization techniques. Finally, the proposed models are compared and evaluated through some simulated experiments