Bayesian inference for high-dimensional decomposable graphs
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In this paper, we consider high-dimensional Gaussian graphical models where the true underlying graph is decomposable. A hierarchical G-Wishart prior is proposed to conduct a Bayesian inference for the precision matrix and its graph structure. Although the posterior asymptotics using the G-Wishart prior has received increasing attention in recent years, most of results assume moderate high-dimensional settings, where the number of variables p is smaller than the sample size n. However, this assumption might not hold in many real applications such as genomics, speech recognition and climatology. Motivated by this gap, we investigate asymptotic properties of posteriors under the high-dimensional setting where p can be much larger than n. The pairwise Bayes factor consistency, posterior ratio consistency and graph selection consistency are obtained in this high-dimensional setting. Furthermore, the posterior convergence rate for precision matrices under the matrix ℓ_1-norm is derived, which turns out to coincide with the minimax convergence rate for sparse precision matrices. A simulation study confirms that the proposed Bayesian procedure outperforms competitors.
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