Pushing Kalman’s Idea to the Extremes

The paper focuses on the fundamental idea of Kalman's seminal paper: how to solve the filtering problem from the only knowledge of the first two moments of the noise terms. In this paper, by exploiting set of distributions based filtering, we solve this problem without introducing additional assumptions on the distributions of the noise terms (e.g., Gaussianity) or on the final form of the estimator (e.g., linear estimator). Given the moments (e.g., mean and variance) of random variable X, it is possible to define the set of all distributions that are compatible with the moments information. This set of distributions can be equivalently characterized by its extreme distributions which is a family of mixtures of Dirac's deltas. The lower and upper expectation of any function g of X are obtained in correspondence of these extremes and can be computed by solving a linear programming problem. The filtering problem can then be solved by running iteratively this linear programming problem.