This paper is concerned with uncertainty based characterizations of fundamental performance limitations (Bode formula) and input-output properties (observability) in feedback systems. A hidden Markov model (HMM) formulation is introduced, based on which a belief process (conditional pdf) is defined. The belief process expresses uncertainty in the state conditioned on history of noisy observations. Both the Bode formula and observability are related to the asymptotic dynamics of the belief process. These results are described for the case of Gaussian linear systems and for open-loop nonlinear systems with negative Lyapunov exponents.