Abstract
Graphical models are an important tool in exploring relationships between variables in complex, multivariate data. Methods for learning such graphical models are well-developed in the case where all variables are either continuous or discrete, including in high dimensions. However, in many applications, data span variables of different types (e.g., continuous, count, binary, ordinal, etc.), whose principled joint analysis is nontrivial. Latent Gaussian copula models, in which all variables are modeled as transformations of underlying jointly Gaussian variables, represent a useful approach. Recent advances have shown how the binary-continuous case can be tackled, but the general mixed variable type regime remains challenging. In this work, we make the simple but useful observation that classical ideas concerning polychoric and polyserial correlations can be leveraged in a latent Gaussian copula framework. Building on this observation, we propose a flexible and scalable methodology for data with variables of entirely general mixed type. We study the key properties of the approaches theoretically and empirically.
Funding Statement
This work was partly supported by the Helmholtz AI project “Scalable and Interpretable Models for Complex And Structured Data” (SIMCARD), the UK Medical Research Council (MC-UU-00002/17) and the National Institute for Health Research (Cambridge Biomedical Research Centre at the Cambridge University Hospitals NHS Foundation Trust).
This project also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme [grant agreement No 883818].
Acknowledgments
We thank the anonymous reviewers whose insightful comments and constructive feedback significantly enhanced the quality and clarity of this paper. We further want to thank Hongjian Shi for his helpful insights on the subject. Empirical results include work conducted using the UK Biobank Resource.
Citation
Konstantin Göbler. Mathias Drton. Sach Mukherjee. Anne Miloschewski. "High-dimensional undirected graphical models for arbitrary mixed data." Electron. J. Statist. 18 (1) 2339 - 2404, 2024. https://doi.org/10.1214/24-EJS2254
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