Density-ratio robustness in dynamic state estimation

The filtering problem is addressed by taking into account imprecision in the knowledge about the probabilistic relationships involved. Imprecision is modelled in this paper by a particular closed convex set of probabilities that is known with the name of density ratio class or constant odds-ratio (COR) model. The contributions of this paper are the following. First, we shall define an optimality criterion based on the squared-loss function for the estimates derived from a general closed convex set of distributions. Second, after revising the properties of the density ratio class in the context of parametric estimation, we shall extend these properties to state estimation accounting for system dynamics. Furthermore, for the case in which the nominal density of the COR model is a multivariate Gaussian, we shall derive closed-form solutions for the set of optimal estimates and for the credible region. Third, we discuss how to perform Monte Carlo integrations to compute lower and upper expectations from a COR set of densities. Then we shall derive a procedure that, employing Monte Carlo sampling techniques, allows us to propagate in time both the lower and upper state expectation functionals and, thus, to derive an efficient solution of the filtering problem. Finally, we empirically compare the proposed estimator with the Kalman filter. This shows that our solution is more robust to the presence of modelling errors in the system and that, hence, appears to be a more realistic approach than the Kalman filter in such a case.

► We solve the filtering problem when uncertainty is modelled by sets of probabilities.
► We consider in particular the set of probabilities known as density ratio class.
► Initial state, dynamics and measurement equations are modelled with this class.
► A closed form solution for the filtering problem is derived and widely discussed.
► Empirical evaluations and comparison w.r.t. Kalman Filter are discussed.