In many contemporary engineering problems, model uncertainty is inherent because accurate system identification is virtually impossible owing to system complexity or lack of data on account of availability, time, or cost. The situation can be treated by assuming that the true model belongs to an uncertainty class of models. In this context, an intrinsically Bayesian robust (IBR) filter is one that is optimal relative to the cost function (in the classical sense) and the prior distribution over the uncertainty class (in the Bayesian sense). IBR filters have previously been found for both Wiener and granulometric morphological filtering. In this paper, we derive the IBR Kalman filter that performs optimally relative to an uncertainty class of state-space models. Introducing the notion of Bayesian innovation process and the Bayesian orthogonality principle, we show how the problem of designing an IBR Kalman filter can be reduced to a recursive system similar to the classical Kalman recursive equations, except with "effective" counterparts, such as the effective Kalman gain matrix. After deriving the recursive IBR Kalman equations for discrete time, we use the limiting method to obtain the IBR Kalman-Bucy equations for continuous time. Finally, we demonstrate the utility of the proposed framework for two real world problems: sensor networks and gene regulatory network inference.