This thesis proposes a generalization for the model class of labeled directed acyclic graphs (LDAGs) introduced in Pensar et al. (2013), which themselves are a generalization of ordinary Bayesian networks. LDAGs allow encoding of a more refined dependency structure compared to Bayesian networks with a single DAG augmented with labels. The labels correspond to context-specific independencies (CSIs) which must be present in every parameterization of an LDAG. The generalization of LDAGs developed in this thesis allows placement of partial context-specific independencies (PCSIs) into labels of an LDAG model, further increasing the space of encodable dependency structures. PCSIs themselves allow a set of random variables to be independent of another when restricted to a subset of their outcome space. The generalized model class is named PCSI-labeled directed acyclic graph (PLDAG).
Several properties of PLDAGs are studied, including PCSI-equivalence of two distinct models, which corresponds to Markov-equivalence of ordinary DAGs. The efficient structure learning algorithm introduced for LDAGs is extended to learn PLDAG models. This algorithm uses a non-reversible Markov chain Monte Carlo (MCMC) method for ordinary DAG structure learning combined with a greedy hill climbing approach. The performance of PLDAG learning is compared against LDAG and traditional DAG learning using three different measures: Kullback-Leibler divergence, number of free parameters in the model and the correctness of the learned DAG structure. The results show that PLDAGs further decreased the number of free parameters needed in the learned model compared to LDAGs yet maintaining the same level of performance with respect to Kullback-Leibler divergence. Also PLDAG and LDAG structure learning algorithms were able to learn the correct DAG structure with less data in traditional DAG structure learning task compared to the base MCMC algorithm.
