The Annals of Statistics
- Ann. Statist.
- Volume 46, Number 1 (2018), 90-118.
Exact formulas for the normalizing constants of Wishart distributions for graphical models
Gaussian graphical models have received considerable attention during the past four decades from the statistical and machine learning communities. In Bayesian treatments of this model, the $G$-Wishart distribution serves as the conjugate prior for inverse covariance matrices satisfying graphical constraints. While it is straightforward to posit the unnormalized densities, the normalizing constants of these distributions have been known only for graphs that are chordal, or decomposable. Up until now, it was unknown whether the normalizing constant for a general graph could be represented explicitly, and a considerable body of computational literature emerged that attempted to avoid this apparent intractability. We close this question by providing an explicit representation of the $G$-Wishart normalizing constant for general graphs.
Ann. Statist., Volume 46, Number 1 (2018), 90-118.
Received: June 2014
Revised: January 2017
First available in Project Euclid: 22 February 2018
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Bartlett decomposition bipartite graph Cholesky decomposition chordal graph directed acyclic graph $G$-Wishart distribution Gaussian graphical model generalized hypergeometric function of matrix argument moral graph normalizing constant Wishart distribution
Uhler, Caroline; Lenkoski, Alex; Richards, Donald. Exact formulas for the normalizing constants of Wishart distributions for graphical models. Ann. Statist. 46 (2018), no. 1, 90--118. doi:10.1214/17-AOS1543. https://projecteuclid.org/euclid.aos/1519268425
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