  It is critical and challenging to design robust predictors for stochastic
dynamical systems (SDSs) with uncertainty quantification (UQ) in the
prediction. Specifically, robustness guarantees the worst-case performance when
the predictor's information set of the system is inadequate, and UQ
characterizes how confident the predictor is about the predictions. However, it
is difficult for traditional robust predictors to provide robust UQ because
they were designed to robustify the performance of point predictions. In this
paper, we investigate how to robustify the probabilistic prediction for SDS,
which can inherently provide robust distributional UQ. To characterize the
performance of probabilistic predictors, we generalize the concept of
likelihood function to likelihood functional, and prove that this metric is a
proper scoring rule. Based on this metric, we propose a framework to quantify
when the predictor is robust and analyze how the information set affects the
robustness. Our framework makes it possible to design robust probabilistic
predictors by solving functional optimization problems concerning different
information sets. In particular, we design a class of moment-based optimal
robust probabilistic predictors and provide a practical Kalman-filter-based
algorithm for implementation. Extensive numerical simulations are provided to
elaborate on our results.
