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 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. We present an application to this problem of a novel dynamical mean field approximation, the Extended Plefka Expansion, which is based on a path integral description of the stochastic dynamics. 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 hidden-to-hidden variance -- conditioned on observations -- gives the expected error at each node when the hidden time courses are predicted based on the observations. We assess the accuracy of the Extended Plefka Expansion in predicting these single node variances as well as error correlations over time, focussing on the role of the system size and the number of observed nodes.