  We present a theoretical framework and numerical methods for predicting the
large-scale properties of solutions of partial differential equations that are
too complex to be properly resolved. We assume that prior statistical
information about the distribution of the solutions is available, as is often
the case in practice. The quantities we can compute condition the prior
information and allow us to calculate mean properties of solutions in the
future. We derive approximate ways for computing the evolution of the
probabilities conditioned by what we can compute, and obtain ordinary
differential equations for the expected values of a set of large-scale
variables. Our methods are demonstrated on two simple but instructive examples,
where the prior information consists of invariant canonical distributions
