Structure Discovery in Bayesian Networks: Algorithms and Applications
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Bayesian networks are a class of probabilistic graphical models that have been widely used in various tasks for probabilistic inference and causal modeling. A Bayesian network provides a compact, flexible, and interpretable representation of a joint probability distribution. When the network structure is unknown but there are observational data at hand, one can try to learn the network structure from the data. This is called structure discovery.
Structure discovery in Bayesian networks is a host of several interesting problem variants. In the optimal Bayesian network learning problem (we call this structure learning), one aims to find a Bayesian network that best explains the data and then utilizes this optimal Bayesian network for predictions or inferences. In others, we are interested in finding the local structural features that are highly probable (we call this structure discovery). Both structure learning and structure discovery are considered very hard because existing approaches to these problems require highly intensive computations.
In this dissertation, we develop algorithms to achieve more accurate, efficient and scalable structure discovery in Bayesian networks and demonstrate these algorithms in applications of systems biology and educational data mining. Specifically, this study is conducted in five directions.
First of all, we propose a novel heuristic algorithm for Bayesian network structure learning that takes advantage of the idea of curriculum learning and learns Bayesian network structures by stages. We prove theoretical advantages of our algorithm and also empirically show that it outperforms the state-of-the-art heuristic approach in learning Bayesian network structures.
Secondly, we develop an algorithm to efficiently enumerate the k-best equivalence classes of Bayesian networks where Bayesian networks in the same equivalence class are equally expressive in terms of representing probability distributions. We demonstrate our algorithm in the task of Bayesian model averaging. Our approach goes beyond the maximum-a-posteriori (MAP) model by listing the most likely network structures and their relative likelihood and therefore has important applications in causal structure discovery.
Thirdly, we study how parallelism can be used to tackle the exponential time and space complexity in the exact Bayesian structure discovery. We consider the problem of computing the exact posterior probabilities of directed edges in Bayesian networks. We present a parallel algorithm capable of computing the exact posterior probabilities of all possible directed edges with optimal parallel space efficiency and nearly optimal parallel time efficiency. We apply our algorithm to a biological data set for discovering the yeast pheromone response pathways.
Fourthly, we develop novel algorithms for computing the exact posterior probabilities of ancestor relations in Bayesian networks. Existing algorithm assumes an order-modular prior over Bayesian networks that does not respect Markov equivalence. Our algorithm allows uniform prior and respects the Markov equivalence. We apply our algorithm to a biological data set for discovering protein signaling pathways.
Finally, we introduce Combined student Modeling and prerequisite Discovery (COMMAND), a novel algorithm for jointly inferring a prerequisite graph and a student model from student performance data. COMMAND learns the skill prerequisite relations as a Bayesian network, which is capable of modeling the global prerequisite structure and capturing the conditional independence between skills. Our experiments on simulations and real student data suggest that COMMAND is better than prior methods in the literature. COMMAND is useful for designing intelligent tutoring systems that assess student knowledge or that offer remediation interventions to students.