Structural Dimension Reduction in Bayesian Networks
Pei Heng, Yi Sun, Jianhua Guo
TL;DR
This work tackles the computational cost of probabilistic inference in large Bayesian networks by introducing structural dimension reduction via the directed convex hull (d-convex hull). It proves an equivalence between minimum collapsible sets and d-convex hulls under faithfulness and develops the CMDSA algorithm to find the minimal d-convex hull containing a set of interest in polynomial time. The approach enables localized Bayesian networks that preserve inference with respect to the variables of interest, and experiments show strong dimension reduction and improved inference efficiency, especially in high-dimensional or sparse real networks. The contribution provides a practical, theory-grounded path to scalable BN inference and offers open-source code for further adoption and development.
Abstract
This work introduces a novel technique, named structural dimension reduction, to collapse a Bayesian network onto a minimum and localized one while ensuring that probabilistic inferences between the original and reduced networks remain consistent. To this end, we propose a new combinatorial structure in directed acyclic graphs called the directed convex hull, which has turned out to be equivalent to their minimum localized Bayesian networks. An efficient polynomial-time algorithm is devised to identify them by determining the unique directed convex hulls containing the variables of interest from the original networks. Experiments demonstrate that the proposed technique has high dimension reduction capability in real networks, and the efficiency of probabilistic inference based on directed convex hulls can be significantly improved compared with traditional methods such as variable elimination and belief propagation algorithms. The code of this study is open at \href{https://github.com/Balance-H/Algorithms}{https://github.com/Balance-H/Algorithms} and the proofs of the results in the main body are postponed to the appendix.