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Background  Inferring gene regulatory network (GRN) has been an important topic in Bioinformatics. Many computational methods infer the GRN from high-throughput expression data. Due to the presence of time delays in the regulatory relationships, High-Order Dynamic Bayesian Network (HO-DBN) is a good model of GRN. However, previous GRN inference methods assume  causal sufficiency , i.e. no unobserved common cause. This assumption is convenient but unrealistic, because it is possible that relevant factors have not even been conceived of and therefore un-measured. Therefore an inference method that also handles hidden common cause(s) is highly desirable. Also, previous methods for discovering hidden common causes either do not handle multi-step time delays or restrict that the parents of hidden common causes are not observed genes.    Results  We have developed a discrete HO-DBN learning algorithm that can infer also hidden common cause(s) from discrete time series expression data, with some assumptions on the conditional distribution, but is less restrictive than previous methods. We assume that each hidden variable has only observed variables as children and parents, with at least two children and possibly no parents. We also make the simplifying assumption that children of hidden variable(s) are not linked to each other. Moreover, our proposed algorithm can also utilize multiple short time series (not necessarily of the same length), as long time series are difficult to obtain.    Conclusions  We have performed extensive experiments using synthetic data on GRNs of size up to 100, with up to 10 hidden nodes. Experiment results show that our proposed algorithm can recover the causal GRNs adequately given the incomplete data. Using the limited real expression data and small subnetworks of the YEASTRACT network, we have also demonstrated the potential of our algorithm on real data, though more time series expression data is needed.

High-order dynamic Bayesian Network learning with hidden common causes for causal gene regulatory network