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Cholesky decomposition of a hyper inverse Wishart matrix. (English) Zbl 0974.62047
Summary: The canonical parameter of a covariance selection model is the inverse covariance matrix \(\Sigma^{-1}\) whose zero pattern gives the conditional independence structure characterising the model. We consider the upper triangular matrix \(\Phi\) obtained by the Cholesky decomposition \(\Sigma^{-1} =\Phi^T \Phi\). This provides an interesting alternative parametrisation of decomposable models since its upper triangle has the same zero structure as \(\Sigma^{-1}\) and its elements have an interpretation as parameters of certain conditional distributions.
For a distribution for \(\Sigma\), the strong hyper-Markov property is shown to be characterised by the mutual independence of the rows of \(\Phi\). This is further used to generalise to the hyper inverse Wishart distribution some well-known properties of the inverse Wishart distribution. In particular we show that a hyper inverse Wishart matrix can be decomposed into independent normal and chi-squared random variables, and we describe a family of transformations under which the family of hyper inverse Wishart distributions is closed.

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
05C90 Applications of graph theory
15B52 Random matrices (algebraic aspects)
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