We present a new statistical approach to the problem of combining models with observations for prediction of physical phenomena. The approach is devised as follows. First, we augment the model with a Gaussian linear functional to represent various sources of
uncertainty in the model. Next, we develop a functional regression procedure to determine the posterior distribution of the Gaussian functional by utilizing observations and adjoint states. It allows us to compute the posterior distribution of both the output estimate and the state estimate. Furthermore, we describe sequential experimental design algorithms for choosing the observations to further reduce prediction uncertainty. We then consider the use of the reduced basis method to reduce the computational cost of both the analysis and the experimental design. We present several numerical examples to demonstrate our approach in comparison with Gaussian process regression.