Predictive coding appears to be one of the fundamental working principles of brain processing. Amongst other aspects, brains often predict the sensory consequences of their own actions. Predictive coding resembles Kalman filtering, where incoming sensory information is filtered to produce prediction errors for subsequent adaptation and learning. However, to generate prediction errors given motor commands, a suitable temporal forward model is required to generate predictions. While in engineering applications, it is usually assumed that this forward model is known, the brain has to learn it. When filtering sensory input and learning from the residual signal in parallel, a fundamental problem arises: the system can enter a delusional loop when filtering the sensory information using an overly trusted forward model. In this case, learning stalls before accurate convergence because uncertainty about the forward model is not properly accommodated. We present a Bayes-optimal solution to this generic and pernicious problem for the case of linear forward models, which we call Predictive Inference and Adaptive Filtering (PIAF). PIAF filters incoming sensory information and learns the forward model simultaneously. We show that PIAF is formally related to Kalman filtering and to the Recursive Least Squares linear approximation method, but combines these procedures in a Bayes optimal fashion. Numerical evaluations confirm that the delusional loop is precluded and that the learning of the forward model is more than 10-times faster when compared to a naive combination of Kalman filtering and Recursive Least Squares.