Inference for dynamics of continuous variables: the extended Plefka expansion with hidden nodes

Abstract

We consider the problem of a subnetwork of observed nodes embedded into a\nlarger bulk of unknown (i.e. hidden) nodes, where the aim is to infer these\nhidden states given information about the subnetwork dynamics. The biochemical\nnetworks underlying many cellular and metabolic processes are important\nrealizations of such a scenario as typically one is interested in\nreconstructing the time evolution of unobserved chemical concentrations\nstarting from the experimentally more accessible ones. We present an\napplication to this problem of a novel dynamical mean field approximation, the\nExtended Plefka Expansion, which is based on a path integral description of the\nstochastic dynamics. As a paradigmatic model we study the stochastic linear\ndynamics of continuous degrees of freedom interacting via random Gaussian\ncouplings. The resulting joint distribution is known to be Gaussian and this\nallows us to fully characterize the posterior statistics of the hidden nodes.\nIn particular the equal-time hidden-to-hidden variance -- conditioned on\nobservations -- gives the expected error at each node when the hidden time\ncourses are predicted based on the observations. We assess the accuracy of the\nExtended Plefka Expansion in predicting these single node variances as well as\nerror correlations over time, focussing on the role of the system size and the\nnumber of observed nodes.\n