Set-Valued Filtering and Smoothing

A theory of discrete-time optimal filtering and smoothing based on convex sets of probability distributions is presented. Rather than propagating a single conditional distribution as does conventional Bayesian estimation, a convex set of conditional distributions is evolved. For linear Gaussian systems, the convex set may be generated by a set of Gaussian distributions with equal covariance with means in a convex region of state space. The conventional point-valued Kalman filter is generalized to a set-valued Kalman filter, consisting of equations of evolution of a convex set of conditional means and a conditional covariance. The resulting estimator is an exact solution to the problem of running an infinity of Kalman filters and fixed-interval smoothers, each with different initial conditions. An application is presented to illustrate and interpret the estimator results.