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Biometrika 2000 87(1):99-112; doi:10.1093/biomet/87.1.99
© 2000 by Biometrika Trust
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Cholesky decomposition of a hyper inverse Wishart matrix

A Roverato

Dipartimento di Economia Politica, University of Modena and Reggio Emilia, Viale J. Berengario n. 51, 41100 Modena, Italy E-mail: roverato@unimo.it

The canonical parameter of a covariance selection model is the inverse covariance matrix [sum ]-1 whose zero pattern gives the conditional independence structure characterising the model. In this paper we consider the upper triangular matrix {Phi} obtained by the Cholesky decomposition [sum ]-1 = {Phi}T{Phi}. This provides an interesting alternative parameterisation of decomposable models since its upper triangle has the same zero structure as [sum ]-1 and its elements have an interpretation as parameters of certain conditional distributions. For a distribution for [sum ], 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.

Key Words: Cholesky decomposition; Conjugate distribution; Covariance selection model; Decomposable graph; Hyper-Markov distribution; Strong hyper-Markov property


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