Machine Learning

, Volume 71, Issue 2–3, pp 185–217 | Cite as

Learning the structure of dynamic Bayesian networks from time series and steady state measurements

  • Harri Lähdesmäki
  • Ilya Shmulevich


Dynamic Bayesian networks (DBN) are a class of graphical models that has become a standard tool for modeling various stochastic time-varying phenomena. In many applications, the primary goal is to infer the network structure from measurement data. Several efficient learning methods have been introduced for the inference of DBNs from time series measurements. Sometimes, however, it is either impossible or impractical to collect time series data, in which case, a common practice is to model the non-time series observations using static Bayesian networks (BN). Such an approach is obviously sub-optimal if the goal is to gain insight into the underlying dynamical model. Here, we introduce Bayesian methods for the inference of DBNs from steady state measurements. We also consider learning the structure of DBNs from a combination of time series and steady state measurements. We introduce two different methods: one that is based on an approximation and another one that provides exact computation. Simulation results demonstrate that dynamic network structures can be learned to an extent from steady state measurements alone and that inference from a combination of steady state and time series data has the potential to improve learning performance relative to the inference from time series data alone.


Dynamic Bayesian networks Steady state analysis Bayesian inference Markov chain Monte Carlo Trans-dimensional Markov chain Monte Carlo 


  1. Andrieu, C., Djurić, P. M., & Doucet, A. (2001). Model selection by MCMC computation. Signal Processing, 81(1), 19–37. CrossRefzbMATHGoogle Scholar
  2. Bernard, A., & Hartemink, A. (2005). Informative structure priors: joint learning of dynamic regulatory networks from multiple types of data. In Proceedings of pacific symposium on biocomputing (PSB 05) (Vol. 10, pp. 459–470). Singapore: World Scientific. CrossRefGoogle Scholar
  3. Brooks, S. P., Giudici, P., & Philippe, A. (2003). Nonparametric convergence assessment for MCMC model selection. Journal of Computational and Graphical Statistics, 12(1), 1–22. CrossRefMathSciNetGoogle Scholar
  4. Ching, W.-K., Zhang, S.-Q., Ng, M. K., & Akutsu, T. (2007). An approximation method for solving the steady-state probability distribution of probabilistic Boolean networks. Bioinformatics, 23(12), 1511–1518. CrossRefGoogle Scholar
  5. Çinlar, E. (1997). Introduction to stochastic processes (1st ed.). Englewood Cliffs: Prentice Hall. Google Scholar
  6. Cooper, G. F., & Herskovits, E. (1992). A Bayesian method for the induction of probabilistic networks from data. Machine Learning, 9(4), 309–347. zbMATHGoogle Scholar
  7. Cowell, R. G., Dawid, A. P., Lauritzen, S. L., & Spiegelhalter, D. J. (1999). Probabilistic networks and expert systems. New York: Springer. zbMATHGoogle Scholar
  8. Dean, T., & Kanazawa, K. (1989). A model for reasoning about persistence and causation. Computational Intelligence, 5(3), 142–150. CrossRefGoogle Scholar
  9. Dellaportas, P., & Forster, J. J. (1999). Markov chain Monte Carlo model determination for hierarchical and graphical log-linear models. Biometrika, 86(3), 615–633. CrossRefMathSciNetzbMATHGoogle Scholar
  10. Dobra, A., Hans, C., Jones, B., Nevins, J. R., Yao, G., & West, M. (2004). Sparse graphical models for exploring gene expression data. Journal of Multivariate Analysis, 90(1), 196–212. CrossRefMathSciNetzbMATHGoogle Scholar
  11. Fawcett, T. (2006). An introduction to ROC analysis. Pattern Recognition Letters, 27(8), 861–874. CrossRefMathSciNetGoogle Scholar
  12. Friedman, N. (2004). Inferring cellular networks using probabilistic graphical models. Science, 303, 799–805. CrossRefGoogle Scholar
  13. Friedman, N., & Koller, D. (2003). Being Bayesian about network structure. A Bayesian approach to structure discovery in Bayesian networks. Machine Learning, 50, 95–125. CrossRefzbMATHGoogle Scholar
  14. Friedman, N., Murphy, K., & Russell, S. (1998). Learning the structure of dynamic probabilistic networks. In Proceedings of fourteenth conference on uncertainty in artificial intelligence (UAI) (pp. 139–147). San Mateo: Morgan Kaufmann. Google Scholar
  15. Geiger, D., & Heckerman, D. (1997). A characterization of the Dirichlet distribution through global and local parameter independence. The Annals of Statistics, 25(3), 1344–1369. CrossRefMathSciNetzbMATHGoogle Scholar
  16. Gelman, A., Roberts, G. O., & Gilks, W. R. (1996). Efficient Metropolis jumping rules. In J. M. Bernardo, J. O. Berger, A. P. Dawid, & A. F. M. Smith (Eds.), Bayesian statistics (Vol. 5, pp. 599–607). Oxford: Oxford University Press. Google Scholar
  17. Giudici, P., Green, P. J., & Tarantola, C. (2000). Efficient model determination for discrete graphical models (Discussion paper No. 99-63). Available on-line at
  18. Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82(4), 711–732. CrossRefMathSciNetzbMATHGoogle Scholar
  19. Hartemink, A., Gifford, D., Jaakkola, T., & Young, R. (2001). Using graphical models and genomic expression data to statistically validate models of genetic regulatory networks. In Proceedings of pacific symposium on biocomputing (PSB 01) (Vol. 6, pp. 422–433). Singapore: World Scientific. Google Scholar
  20. Hartemink, A., Gifford, D., Jaakkola, T., & Young, R. (2002). Combining location and expression data for principled discovery of genetic regulatory network models. In Proceedings of pacific symposium on biocomputing (PSB 02) (Vol. 7, pp. 437–449). Singapore: World Scientific. Google Scholar
  21. Heckerman, D. (1998). A tutorial on learning with Bayesian networks. In M. I. Jordan (Ed.), Learning in graphical models (pp. 301–354). Cambridge: MIT Press. Google Scholar
  22. Heckerman, D., Geiger, D., & Chickering, D. (1995). Learning Bayesian networks: the combination of knowledge and statistical data. Machine Learning, 20(3), 197–243. zbMATHGoogle Scholar
  23. Husmeier, D. (2003). Sensitivity and specificity of inferring genetic regulatory interactions from microarray experiments with dynamic Bayesian networks. Bioinformatics, 19(17), 2271–2282. CrossRefGoogle Scholar
  24. Husmeier, D. (2005). Introduction to learning Bayesian networks from data. In D. Husmeier, R. Dybowski, & S. Roberts (Eds.), Probabilistic modeling in bioinformatics and medical informatics (pp. 17–57). Berlin: Springer. CrossRefGoogle Scholar
  25. Imoto, S., Kim, S., Goto, T., Miyano, S., Aburatani, S., & Tashiro, K. (2003). Bayesian network and nonparametric heteroscedastic regression for nonlinear modeling of genetic network. Journal of Bioinformatics and Computational Biology, 1(2), 231–252. CrossRefGoogle Scholar
  26. Lähdesmäki, H., Hautaniemi, S., Shmulevich, I., & Yli-Harja, O. (2006). Relationships between probabilistic Boolean networks and dynamic Bayesian networks as models of gene regulatory networks. Signal Processing, 86(4), 814–834. CrossRefzbMATHGoogle Scholar
  27. Langville, A. N., & Meyer, C. D. (2006). Updating Markov chains with an eye on Google’s PageRank. SIAM Journal on Matrix Analysis and Applications, 27(4), 968–987. CrossRefMathSciNetzbMATHGoogle Scholar
  28. Lehoucq, R. B., Sorensen, D. C., & Yang, C. (1998). ARPACK users’ guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. Philadelphia: SIAM. Google Scholar
  29. Madigan, D., & York, J. (1995). Bayesian graphical models for discrete data. International Statistical Review, 63(2), 215–232. zbMATHCrossRefGoogle Scholar
  30. Markowetz, F. (2007). A bibliography on learning causal networks of gene interactions. Available on-line at
  31. Murphy, K. P. (2001). The Bayes net toolbox for Matlab. Computing Science and Statistics, 33, 1–20. Software is available on-line at MathSciNetGoogle Scholar
  32. Murphy, K. P. (2002). Dynamic Bayesian networks: representation, inference and learning. PhD thesis, University of California, Berkeley. Google Scholar
  33. Nikovski, D. (1998). Learning stationary temporal probabilistic networks. In Conference on automated learning and discovery. Google Scholar
  34. Pearl, J. (2000). Causality: models, reasoning, and inference. Cambridge: Cambridge University Press. zbMATHGoogle Scholar
  35. Pe’er, D., Regev, A., Elidan, G., & Friedman, N. (2001). Inferring subnetworks from perturbed expression profiles. Bioinformatics, 17(Suppl. 1), 215S–224S. Google Scholar
  36. Pournara, I. (2004). Reconstructing gene regulatory networks by passive and active Bayesian learning. PhD thesis, Birkbeck College, University of London. Google Scholar
  37. Pournara, I., & Wernisch, L. (2004). Reconstruction of gene networks using Bayesian learning and manipulation experiments. Bioinformatics, 20(17), 2934–2942. CrossRefGoogle Scholar
  38. Richardson, S., & Green, P. (1997). On Bayesian analysis of mixtures with an unknown number of components, with discussion. Journal of the Royal Statistical Society: Series B, 59(4), 731–792. CrossRefMathSciNetzbMATHGoogle Scholar
  39. Rissanen, J. J. (1978). Modeling by shortest data description. Automatica, 14, 465–471. CrossRefzbMATHGoogle Scholar
  40. Robert, C. P., & Casella, G. (2005). Monte Carlo statistical methods (2nd ed.), Berlin: Springer. Google Scholar
  41. Robert, C. P., Rydén, T., & Titterington, D. M. (2000). Bayesian inference in hidden Markov models through the reversible jump Markov chain Monte Carlo. Journal of the Royal Statistical Society: Series B, 62(1), 57–75. CrossRefMathSciNetzbMATHGoogle Scholar
  42. Sachs, K., Perez, O., Pe’er, D., Lauffenburger, D. A., & Nolan, G. P. (2005). Causal protein-signaling networks derived from multiparameter single-cell data. Science, 308(5721), 523–529. CrossRefGoogle Scholar
  43. Schäfer, J., & Strimmer, K. (2005). An empirical Bayes approach to inferring large-scale gene association networks. Bioinformatics, 21(5), 754–764. Google Scholar
  44. Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6(2), 461–464. CrossRefMathSciNetzbMATHGoogle Scholar
  45. Stewart, W. J. (1994). Introduction to the numerical solution of Markov chains (1st ed.). Princeton: Princeton University Press. zbMATHGoogle Scholar
  46. Werhli, A. V., Grzegorczyk, M., & Husmeier, D. (2006). Comparative evaluation of reverse engineering gene regulatory networks with relevance networks, graphical Gaussian models and Bayesian networks. Bioinformatics, 22(20), 2523–2531. CrossRefGoogle Scholar
  47. Wille, A., Zimmermann, P., Vranová, E., Fürholz, A., Laule, O., & Bleuler, S. (2004). Sparse graphical Gaussian modeling of the isoprenoid gene network in Arabidopsis thaliana. Genome Biology, 5(11), R92. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institute for Systems BiologySeattleUSA
  2. 2.Department of Signal ProcessingTampere University of TechnologyTampereFinland

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