  We present average performance results for dynamical inference problems in
large networks, where a set of nodes is hidden while the time trajectories of
the others are observed. Examples of this scenario can occur in signal
transduction and gene regulation networks. We focus on the linear stochastic
dynamics of continuous variables interacting via random Gaussian couplings of
generic symmetry. We analyze the inference error, given by the variance of the
posterior distribution over hidden paths, in the thermodynamic limit and as a
function of the system parameters and the ratio {\alpha} between the number of
hidden and observed nodes. By applying Kalman filter recursions we find that
the posterior dynamics is governed by an "effective" drift that incorporates
the effect of the observations. We present two approaches for characterizing
the posterior variance that allow us to tackle, respectively, equilibrium and
nonequilibrium dynamics. The first appeals to Random Matrix Theory and reveals
average spectral properties of the inference error and typical posterior
relaxation times, the second is based on dynamical functionals and yields the
inference error as the solution of an algebraic equation.
