Characterizing and Identifying Separable Graphical Models

2026-07-01

AI Summary

This work addresses the challenge of unifying the representation of conditional independence structures induced by feedback, latent variables, and selection mechanisms. It proposes a class of separable graphical models based on mixed graphs containing directed, undirected, and bidirected edges, where the absence of an edge corresponds to a separating set between its endpoints. By introducing separable graphs and their essential forms, the framework subsumes several existing graphical models. The study establishes an equivalence among graph structure, separation properties, and canonical parametrization, and leverages this correspondence to design an algorithm for identifying equivalence classes. Under mild assumptions, the algorithm consistently recovers the separation-equivalence class of a separable graph, thereby providing both theoretical foundations and computational tools for modeling and learning complex dependency structures.

Abstract

We study a broad class of graphical models whose independencies correspond to vertex separation in mixed graphs with directed, undirected, and bidirected edges, that are capable of encoding independence structures arising from feedback, latent and selection mechanisms. In particular, we introduce separable graphs, in which each missing edge implies the existence of a separating set for its endpoints, and essentially separable graphs, those graphs separation equivalent to a separable graph. We show that these models include many existing graph families used to define graphical models an provide several characterizations of separable graphs and essentially separable graphs. We also provide multiple characterizations of separation equivalence for separable graphs. One is a graphical characterization in terms of ordinary graph properties, extending earlier results for specific subfamilies Another is a separational characterization depending only on graph separation properties. Finally, we provide a canonical representation for the equivalence classes of essentially separable graphs and develop an algorithm that, under suitable assumptions, identifies the equivalence class of any essentially separable graph.

Problem

• graphical models
• separable graphs
• vertex separation
• mixed graphs
• independence structures

Innovation

• separable graphs
• graphical models
• separation equivalence
• canonical representation
• mixed graphs