Zhu, Xiaoyuan (Queens College, City University of New York) | Yuan, Changhe (Queens College, City University of New York)

Most Relevant Explanation (MRE) is a new inference task in Bayesian networks that finds the most relevant partial instantiation of target variables as an explanation for given evidence by maximizing the Generalized Bayes Factor (GBF). No exact algorithm has been developed for solving MRE previously. This paper fills the void and introduces a breadth-first branch-and-bound MRE algorithm based on a novel upper bound on GBF. The bound is calculated by decomposing the computation of the score to a set of Markov blankets of subsets of evidence variables. Our empirical evaluations show that the proposed algorithm scales up exact MRE inference significantly.

Most Relevant Explanation (MRE) is an inference task in Bayesian networks that finds the most relevant partial instantiation of target variables as an explanation for given evidence by maximizing the Generalized Bayes Factor (GBF). No exact MRE algorithm has been developed previously except exhaustive search. This paper fills the void by introducing two Breadth-First Branch-and- Bound (BFBnB) algorithms for solving MRE based on novel upper bounds of GBF. One upper bound is created by decomposing the computation of GBF using a target blanket decomposition of evidence variables. The other upper bound improves the first bound in two ways. One is to split the target blankets that are too large by converting auxiliary nodes into pseudo-targets so as to scale to large problems. The other is to perform summations instead of maximizations on some of the target variables in each target blanket. Our empirical evaluations show that the proposed BFBnB algorithms make exact MRE inference tractable in Bayesian networks that could not be solved previously.

Fan, Xiannian (City University of New York) | Yuan, Changhe (City University of New York) | Malone, Brandon (University of Helsinki)

A recent breadth-first branch and bound algorithm (BFBnB)for learning Bayesian network structures (Maloneet al. 2011) uses two bounds to prune the searchspace for better efficiency; one is a lower bound calculatedfrom pattern database heuristics, and the otheris an upper bound obtained by a hill climbing search.Whenever the lower bound of a search path exceeds theupper bound, the path is guaranteed to lead to suboptimalsolutions and is discarded immediately. This paperintroduces methods for tightening the bounds. Thelower bound is tightened by using more informed variablegroupings when creating the pattern databases, andthe upper bound is tightened using an anytime learningalgorithm. Empirical results show that these boundsimprove the efficiency of Bayesian network learning bytwo to three orders of magnitude.

Yuan, Changhe, Malone, Brandon

Recently two search algorithms, A* and breadth-first branch and bound (BFBnB), were developed based on a simple admissible heuristic for learning Bayesian network structures that optimize a scoring function. The heuristic represents a relaxation of the learning problem such that each variable chooses optimal parents independently. As a result, the heuristic may contain many directed cycles and result in a loose bound. This paper introduces an improved admissible heuristic that tries to avoid directed cycles within small groups of variables. A sparse representation is also introduced to store only the unique optimal parent choices. Empirical results show that the new techniques significantly improved the efficiency and scalability of A* and BFBnB on most of datasets tested in this paper.

Malone, Brandon, Yuan, Changhe, Hansen, Eric A., Bridges, Susan

Previous work has shown that the problem of learning the optimal structure of a Bayesian network can be formulated as a shortest path finding problem in a graph and solved using A* search. In this paper, we improve the scalability of this approach by developing a memory-efficient heuristic search algorithm for learning the structure of a Bayesian network. Instead of using A*, we propose a frontier breadth-first branch and bound search that leverages the layered structure of the search graph of this problem so that no more than two layers of the graph, plus solution reconstruction information, need to be stored in memory at a time. To further improve scalability, the algorithm stores most of the graph in external memory, such as hard disk, when it does not fit in RAM. Experimental results show that the resulting algorithm solves significantly larger problems than the current state of the art.