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We consider the problem of a subnetwork of observed nodes embedded into alarger bulk of unknown (i.e. hidden) nodes, where the aim is to infer thesehidden states given information about the subnetwork dynamics. The biochemicalnetworks underlying many cellular and metabolic processes are importantrealizations of such a scenario as typically one is interested inreconstructing the time evolution of unobserved chemical concentrationsstarting from the experimentally more accessible ones. We present anapplication to this problem of a novel dynamical mean field approximation, theExtended Plefka Expansion, which is based on a path integral description of thestochastic dynamics. As a paradigmatic model we study the stochastic lineardynamics of continuous degrees of freedom interacting via random Gaussiancouplings. The resulting joint distribution is known to be Gaussian and thisallows us to fully characterize the posterior statistics of the hidden nodes.In particular the equal-time hidden-to-hidden variance -- conditioned onobservations -- gives the expected error at each node when the hidden timecourses are predicted based on the observations. We assess the accuracy of theExtended Plefka Expansion in predicting these single node variances as well aserror correlations over time, focussing on the role of the system size and thenumber of observed nodes.

Inference for dynamics of continuous variables: the Extended Plefka Expansion with hidden nodes