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Hyper Inverse Wishart Distribution for Non‐decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models

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  • ALBERTO ROVERATO

Abstract

While conjugate Bayesian inference in decomposable Gaussian graphical models is largely solved, the non‐decomposable case still poses difficulties concerned with the specification of suitable priors and the evaluation of normalizing constants. In this paper we derive the DY‐conjugate prior (Diaconis & Ylvisaker, 1979) for non‐decomposable models and show that it can be regarded as a generalization to an arbitrary graph G of the hyper inverse Wishart distribution (Dawid & Lauritzen, 1993). In particular, if G is an incomplete prime graph it constitutes a non‐trivial generalization of the inverse Wishart distribution. Inference based on marginal likelihood requires the evaluation of a normalizing constant and we propose an importance sampling algorithm for its computation. Examples of structural learning involving non‐decomposable models are given. In order to deal efficiently with the set of all positive definite matrices with non‐decomposable zero‐pattern we introduce the operation of triangular completion of an incomplete triangular matrix. Such a device turns out to be extremely useful both in the proof of theoretical results and in the implementation of the Monte Carlo procedure.

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  • Alberto Roverato, 2002. "Hyper Inverse Wishart Distribution for Non‐decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 29(3), pages 391-411, September.
    • Handle: RePEc:bla:scjsta:v:29:y:2002:i:3:p:391-411
    DOI: 10.1111/1467-9469.00297
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    17. Carter, Christopher K. & Wong, Frederick & Kohn, Robert, 2011. "Constructing priors based on model size for nondecomposable Gaussian graphical models: A simulation based approach," Journal of Multivariate Analysis, Elsevier, vol. 102(5), pages 871-883, May.
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    20. Bekker, Andriëtte & van Niekerk, Janet & Arashi, Mohammad, 2017. "Wishart distributions: Advances in theory with Bayesian application," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 272-283.
    21. Codazzi, Laura & Colombi, Alessandro & Gianella, Matteo & Argiento, Raffaele & Paci, Lucia & Pini, Alessia, 2022. "Gaussian graphical modeling for spectrometric data analysis," Computational Statistics & Data Analysis, Elsevier, vol. 174(C).
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    23. Christine Peterson & Francesco C. Stingo & Marina Vannucci, 2015. "Bayesian Inference of Multiple Gaussian Graphical Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 159-174, March.

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