Hyper-inverse Wishart (HIW) distributions feature centrally in the analysis of Gaussian graphical models and related studies on structured variance matrices and covariance selection. The main contribution of this paper is to introduce and exemplify an efficient method for direct sampling from an arbitrary HIW distribution. This is of general interest as a simulation tool and, in particular, provides completion of the simulation toolbox for Bayesian exploration and analysis of Gaussian graphical models under HIW priors. The direct sampling method relies on properties of the junction tree representation of graphs, together with well-known matrix results for the inverse Wishart distribution. An example drawn from finance demonstrates application in a context where inferences on a structured covariance model are required. The simulation theory and method applies to both decomposable and non-decomposable graphical models, and can be easily embedded in MCMC and other simulation-based analyses that deal with uncertainty about the graphical model form as well as variance-covariance parameters.

• The original technical report is available here (pdf). The revised and substantially updated paper is published in Biometrika 2007.

• Erratum: There is a minor typo in the published paper in Section 3.2, page 652, first displayed equation at the top of the page. The degree of freedom parameter in the IW distribution there should be b+|S_i| rather than b+|R_i|. (Thanks to Hao Wang for spotting this).

• Software implementing the HIW simulation method is freely available at the HIW Sampler software link. page.

Research partially supported by grants from the National Science Foundation. Any opinions, findings and conclusions or recommendations expressed in this work are those of the authors and do not necessarily reflect the views of the NSF.