Bayesian Posterior Learning of Mixed Graphical Models
Abstract
Mixed Graphical Models (MGMs) provide a flexible framework for structure learning from heterogeneous data by treating sets of both continuous and discrete. Bayesian inference for MGMs remains challenging due to the combinatorial complexity of graph space exploration and posterior computation. In this paper, we model discrete components through latent Gaussian variables and consider two likelihood specifications: a copula-based ranked likelihood, yielding the copula-MGM, and a probit formulation based on cut-off points, yielding the probit-MGM. We then propose the Mixed Graph WWA, a class of MCMC methods for posterior simulation in Bayesian MGMs. Building upon the WWA algorithm, we develop two specialized algorithms: copula-WWA for copula-MGMs and probit-WWA for probit-MGMs. Both methods exploit the latent Gaussian representations to perform posterior inference through a Gibbs sampling scheme that alternates between latent-variable augmentation and graph-structure updates. Through extensive simulation studies we demonstrate that the proposed methods achieve graph recovery accuracy comparable to or better than existing approaches, including copula-BD MCMC and probit-BD MCMC based on the Birth-Death MCMC methodology, while maintaining efficient posterior exploration and favorable effective sample size per unit computational time. We further illustrate the practical utility of our approach through an application to the PAM$50$ breast cancer gene expression dataset, where the inferred MGMs reveal meaningful dependencies between gene expression profiles and cancer subtypes. These results highlight the effectiveness of the Mixed Graph WWA method as a scalable and principled tool for Bayesian structure learning in MGMs.
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BibTeX
@misc{guo2026bayesian,
title = {{Bayesian Posterior Learning of Mixed Graphical Models}},
author = {Erdong Guo and Alexandros Beskos and Maria De Iorio},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23857},
eprint = {2609.23857},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23857},
}
RIS
TY - UNPB
TI - Bayesian Posterior Learning of Mixed Graphical Models
AU - Erdong Guo
AU - Alexandros Beskos
AU - Maria De Iorio
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.23857
UR - https://arxiv.org/abs/2609.23857
AB - Mixed Graphical Models (MGMs) provide a flexible framework for structure learning from heterogeneous data by treating sets of both continuous and discrete. Bayesian inference for MGMs remains challenging due to the combinatorial complexity of graph space exploration and posterior computation. In this paper, we model discrete components through latent Gaussian variables and consider two likelihood specifications: a copula-based ranked likelihood, yielding the copula-MGM, and a probit formulation based on cut-off points, yielding the probit-MGM. We then propose the Mixed Graph WWA, a class of MCMC methods for posterior simulation in Bayesian MGMs. Building upon the WWA algorithm, we develop two specialized algorithms: copula-WWA for copula-MGMs and probit-WWA for probit-MGMs. Both methods exploit the latent Gaussian representations to perform posterior inference through a Gibbs sampling scheme that alternates between latent-variable augmentation and graph-structure updates. Through extensive simulation studies we demonstrate that the proposed methods achieve graph recovery accuracy comparable to or better than existing approaches, including copula-BD MCMC and probit-BD MCMC based on the Birth-Death MCMC methodology, while maintaining efficient posterior exploration and favorable effective sample size per unit computational time. We further illustrate the practical utility of our approach through an application to the PAM$50$ breast cancer gene expression dataset, where the inferred MGMs reveal meaningful dependencies between gene expression profiles and cancer subtypes. These results highlight the effectiveness of the Mixed Graph WWA method as a scalable and principled tool for Bayesian structure learning in MGMs.
ER -