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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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