# American Institute of Mathematical Sciences

2014, 11(1): 149-165. doi: 10.3934/mbe.2014.11.149

## Network inference with hidden units

 1 Department of Mathematics, Stockholm University, Kräftriket, S-106 91 Stockholm, Sweden 2 Nordita, Stockholm University and KTH, Roslagstullsbacken 23, S-106 91 Stockholm, Sweden

Received  December 2012 Revised  May 2013 Published  September 2013

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.
Citation: Joanna Tyrcha, John Hertz. Network inference with hidden units. Mathematical Biosciences & Engineering, 2014, 11 (1) : 149-165. doi: 10.3934/mbe.2014.11.149
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