Robust Kalman and Bayesian Set-Valued Filtering and Model Validation for Linear Stochastic Systems*

Consider a linear stochastic filtering problem in which the probability measure specifying all randomness is only partially known. The deviation between the real and assumed probability models is constrained by a divergence bound between the respective probability measures under which the models are defined. This bound defines a so-called uncertainty set. A recursive set-valued filtering characterization is derived and is guaranteed (with probability one) to contain the true conditional posterior of the unknown, real world, filtering problem when the real world measure is within this uncertainty set. Some filtering approximations and related results are given. The set-valued characterization is related to the problem of robust model validation and model goodness-of-fit statistical hypothesis testing. It is shown how relevant terms involving the innovation sequence (re)appear in multiple settings from set-valued filtering to statistical model evaluation.