We consider the problem of a subnetwork of observed nodes embedded into a larger bulk of unknown (i.e. hidden) nodes, where the aim is to infer these hidden states given information about the subnetwork dynamics. The biochemical networks underlying many cellular and metabolic processes are important realizations of such a scenario as typically one is interested in reconstructing the time evolution of unobserved chemical concentrations starting from the experimentally more accessible ones. As a paradigmatic model we study the stochastic linear dynamics of continuous degrees of freedom interacting via random Gaussian couplings. The resulting joint distribution is known to be Gaussian and this allows us to fully characterize the posterior statistics of the hidden nodes. In particular the equal-time hiddento-hidden correlation – conditioned on observations - gives the expected error when the hidden time courses are predicted based on the observations. We study it in the stationary regime and in the infinite network size limit by resorting to a novel dynamical mean field approximation, the Extended Plefka Expansion, that is based on a path integral description of the stochastic dynamics. We analyze the phase diagram in the space of the relevant parameters, namely the ratio between the numbers of observed and hidden nodes, the degree of symmetry of the interactions and the relative amplitudes of the hidden-to-hidden and hidden-toobserved couplings with respect to the decay constant of the internal hidden dynamics. We assess in particular how the interplay of such structural properties of the system may affect the accuracy of the prediction by identifying critical regions for the inference error, i.e. parameters values for which the error would diverge.