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Abstract:
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In Gaussian graphical models, the zero entries in the precision matrix determine the dependence structure, so estimating that sparse precision matrix is an important and challenging problem. We propose a novel empirical version of the G-Wishart prior for sparse precision matrices, where the prior mode is informed by the data in a suitable way. Paired with a prior on the graph structure, a marginal posterior distribution for the same is obtained that takes the form of a ratio of two G-Wishart normalizing constants. We show that, thanks to the data-driven prior centering, this ratio can be easily and accurately computed using a Laplace approximation, which leads to fast and efficient posterior sampling even in high-dimensions. Numerical results demonstrate the proposed method's superior performance, in terms of speed and accuracy, across a variety of settings, and theoretical support is provided in the form of a posterior concentration rate theorem.
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