An Algorithm for Finding Minimum d-Separating Sets in Belief Networks
This addresses a combinatorial problem in belief networks that could aid in interpreting and managing independence relationships, though it appears incremental as it builds on existing d-separation concepts.
The paper tackles the problem of finding the minimum d-separating set for two nodes in a directed acyclic graph, which represents the minimal number of variables needed to block influence between them, and presents a solution based on reducing it to a minimum separating set problem in an undirected graph and developing an algorithm.
The criterion commonly used in directed acyclic graphs (dags) for testing graphical independence is the well-known d-separation criterion. It allows us to build graphical representations of dependency models (usually probabilistic dependency models) in the form of belief networks, which make easy interpretation and management of independence relationships possible, without reference to numerical parameters (conditional probabilities). In this paper, we study the following combinatorial problem: finding the minimum d-separating set for two nodes in a dag. This set would represent the minimum information (in the sense of minimum number of variables) necessary to prevent these two nodes from influencing each other. The solution to this basic problem and some of its extensions can be useful in several ways, as we shall see later. Our solution is based on a two-step process: first, we reduce the original problem to the simpler one of finding a minimum separating set in an undirected graph, and second, we develop an algorithm for solving it.