Bayesian Network Learning with Parameter Constraints

The task of learning models for many real-world problems requires incorporating domain knowledge into learning algorithms, to enable accurate learning from a realistic volume of training data. This paper considers a variety of types of domain knowledge for constraining parameter estimates when learning Bayesian networks. In particular, we consider domain knowledge that constrains the values or relationships among subsets of parameters in a Bayesian network with known structure. We incorporate a wide variety of parameter constraints into learning procedures for Bayesian networks, by formulating this task as a constrained optimization problem. The assumptions made in module networks, dynamic Bayes nets and context specific independence models can be viewed as particular cases of such parameter constraints. We present closed form solutions or fast iterative algorithms for estimating parameters subject to several specific classes of parameter constraints, including equalities and inequalities among parameters, constraints on individual parameters, and constraints on sums and ratios of parameters, for discrete and continuous variables. Our methods cover learning from both frequentist and Bayesian points of view, from both complete and incomplete data. We present formal guarantees for our estimators, as well as methods for automatically learning useful parameter constraints from data. To validate our approach, we apply it to the domain of fMRI brain image analysis. Here we demonstrate the ability of our system to first learn useful relationships among parameters, and then to use them to constrain the training of the Bayesian network, resulting in improved cross-validated accuracy of the learned model. Experiments on synthetic data are also presented.