In this talk we shall consider a rigorous and systematic approach to uncertainty in problems of filtering in discrete time, using the structure of nonlinear expectations and risk measures. We shall show that, under general conditions relating the perception of uncertainty and the observation filtration, one has a forward recursion which describes the uncertainty over the current state of an unobserved process. In the setting of a discrete-time hidden Markov chain, or of the Kalman filter, we shall also obtain simple approximations which can be implemented in real time.