We show that the large-eddy motions in turbulent fluid flow obey a modified hydrodynamic equation with a stochastic turbulent stress whose distribution is a causal functional of the large-scale velocity field itself. We do so by means of an exact procedure of "statistical filtering" of the Navier-Stokes equations, which formally solves the closure problem, and we discuss the relation of our analysis with the "decimation theory" of Kraichnan. We show that the statistical filtering procedure can be formulated using field-theoretic path-integral methods within the Martin-Siggia-Rose (MSR) formalism for classical statistical dynamics.