Network Inference with Hidden Units

TL;DR

This paper develops exact and approximate learning rules for inferring connections in stochastic dynamical networks with hidden units, using mean field theory to handle computational complexity.

Contribution

It introduces exact learning rules for networks with hidden units and proposes a mean field approximation for large systems, advancing network inference methods.

Findings

Exact learning rules derived for both binary and continuous hidden units.

Mean field approximation effectively handles large system inference.

Numerical results illustrate the accuracy of the approximate methods.

Abstract

We derive learning rules for finding the connections between units in stochastic dynamical networks from the recorded history of a ``visible'' subset of the units. We consider two models. In both of them, the visible units are binary and stochastic. In one model the ``hidden'' units are continuous-valued, with sigmoidal activation functions, and in the other they are binary and stochastic like the visible ones. We derive exact learning rules for both cases. For the stochastic case, performing the exact calculation requires, in general, repeated summations over an number of configurations that grows exponentially with the size of the system and the data length, which is not feasible for large systems. We derive a mean field theory, based on a factorized ansatz for the distribution of hidden-unit states, which offers an attractive alternative for large systems. We present the results of some numerical calculations that illustrate key features of the two models and, for the stochastic case, the exact and approximate calculations.