Profile Graphical Models

arXiv stat.ML / 3/31/2026

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Key Points

The paper introduces “profile graphical models,” a new class of graphical models that use a single graph to represent how an external risk factor changes the dependence/conditional structure among multivariate variables.
It generalizes existing graphical-model families by including multiple graphs and chain graphs as special cases, and it formalizes Markov properties for how conditional independencies vary across risk profiles.
The authors show structural/probabilistic connections by proving that their profile undirected graphical models are independence-compatible with two-block LWF chain graph models.
They propose a Bayesian learning framework for Gaussian profile undirected models using continuous spike-and-slab priors to capture shared sparsity across risk levels, along with a fast EM algorithm for inference.
Experiments and an application to protein network data in acute myeloid leukemia subtypes demonstrate improved parsimonious network learning and greater ability to capture patient heterogeneity versus competing approaches.

Abstract

We introduce a novel class of graphical models, termed profile graphical models, that represent, within a single graph, how an external factor influences the dependence structure of a multivariate set of variables. This class is quite general and includes multiple graphs and chain graphs as special cases. Profile graphical models capture the conditional distributions of a multivariate random vector given different levels of a risk factor, and learn how the conditional independence structure among variables may vary across these risk profiles; we formally define this family of models and establish their corresponding Markov properties. We derive key structural and probabilistic properties that underpin a more powerful inferential framework than existing approaches, underscoring that our contribution extends beyond a novel graphical representation.Furthermore, we show that the resulting profile undirected graphical models are independence-compatible with two-block LWF chain graph models.We then develop a Bayesian approach for Gaussian undirected profile graphical models based on continuous spike-and-slab priors to learn shared sparsity structures across different levels of the risk factor. We also design a fast EM algorithm for efficient inference. Inferential properties are explored through simulation studies, including the comparison with competing methods. The practical utility of this class of models is demonstrated through the analysis of protein network data from various subtypes of acute myeloid leukemia. Our results show a more parsimonious network and greater patient heterogeneity than its competitors, highlighting its enhanced ability to capture subject-specific differences.