# 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
##### References:

show all references

##### References:
 [1] Deng Lu, Maria De Iorio, Ajay Jasra, Gary L. Rosner. Bayesian inference for latent chain graphs. Foundations of Data Science, 2020, 2 (1) : 35-54. doi: 10.3934/fods.2020003 [2] Michael Herty, Lorenzo Pareschi, Giuseppe Visconti. Mean field models for large data–clustering problems. Networks & Heterogeneous Media, 2020, 15 (3) : 463-487. doi: 10.3934/nhm.2020027 [3] Roberto Natalini, Thierry Paul. On the mean field limit for Cucker-Smale models. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021164 [4] Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic & Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385 [5] Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475 [6] Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381 [7] Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. Kinetic & Related Models, 2019, 12 (6) : 1329-1358. doi: 10.3934/krm.2019052 [8] Fabio Bagagiolo, Rosario Maggistro, Raffaele Pesenti. Origin-to-destination network flow with path preferences and velocity controls: A mean field game-like approach. Journal of Dynamics & Games, 2021, 8 (4) : 359-380. doi: 10.3934/jdg.2021007 [9] Evangelos Evangelou. Approximate Bayesian inference for geostatistical generalised linear models. Foundations of Data Science, 2019, 1 (1) : 39-60. doi: 10.3934/fods.2019002 [10] Oliver J. Maclaren, Helen M. Byrne, Alexander G. Fletcher, Philip K. Maini. Models, measurement and inference in epithelial tissue dynamics. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1321-1340. doi: 10.3934/mbe.2015.12.1321 [11] Chjan C. Lim. Extremal free energy in a simple mean field theory for a coupled Barotropic fluid - rotating sphere system. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 361-386. doi: 10.3934/dcds.2007.19.361 [12] William Chad Young, Adrian E. Raftery, Ka Yee Yeung. A posterior probability approach for gene regulatory network inference in genetic perturbation data. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1241-1251. doi: 10.3934/mbe.2016041 [13] Samuel N. Cohen. Uncertainty and filtering of hidden Markov models in discrete time. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 4-. doi: 10.1186/s41546-020-00046-x [14] Franco Flandoli, Matti Leimbach. Mean field limit with proliferation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3029-3052. doi: 10.3934/dcdsb.2016086 [15] Yu Yang, Dongmei Xiao. Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 195-211. doi: 10.3934/dcdsb.2010.13.195 [16] Napoleon Bame, Samuel Bowong, Josepha Mbang, Gauthier Sallet, Jean-Jules Tewa. Global stability analysis for SEIS models with n latent classes. Mathematical Biosciences & Engineering, 2008, 5 (1) : 20-33. doi: 10.3934/mbe.2008.5.20 [17] Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic & Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429 [18] Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic & Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621 [19] T. S. Evans, A. D. K. Plato. Network rewiring models. Networks & Heterogeneous Media, 2008, 3 (2) : 221-238. doi: 10.3934/nhm.2008.3.221 [20] Dong-Mei Zhu, Wai-Ki Ching, Robert J. Elliott, Tak-Kuen Siu, Lianmin Zhang. Hidden Markov models with threshold effects and their applications to oil price forecasting. Journal of Industrial & Management Optimization, 2017, 13 (2) : 757-773. doi: 10.3934/jimo.2016045

2018 Impact Factor: 1.313