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

TLDR

Conjugate Bayesian inference is well understood for decomposable Gaussian graphical models, but non‑decomposable cases still face challenges in specifying priors and computing normalizing constants. The paper derives a DY‑conjugate prior for non‑decomposable models, extending the hyper inverse Wishart distribution to arbitrary graphs. To enable inference, the authors introduce an importance‑sampling algorithm for the normalizing constant and a triangular‑completion operation to efficiently handle positive‑definite matrices with non‑decomposable zero patterns. They demonstrate that for incomplete prime graphs the prior generalizes the inverse Wishart distribution, provide structural‑learning examples with non‑decomposable models, and show that the triangular‑completion device is essential for theoretical proofs and Monte Carlo implementation.

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.