## Abstract

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

α 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 non-equilibrium 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.

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

α 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 non-equilibrium 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.

Original language | English |
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Article number | 10.1103/PhysRevE.95.012122 |

Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | PHYSICAL REVIEW E |

Volume | 95 |

Issue number | 1 |

DOIs | |

Publication status | Published - 13 Jan 2017 |

## Keywords

- Inference, Linear Dynamics, Kalman Filter, Random Matrix Theory, Dynamical Functional, Gaussian posterior distributions