Cholesky decomposition of a hyper inverse Wishart matrix

Abstract

The canonical parameter of a covariance selection model is the inverse covariance matrix Σ -1 whose zero pattern gives the conditional independence structure characterising the model. In this paper we consider the upper triangular matrix Φ obtained by the Cholesky decomposition Σ - 1 = Φ T Φ. This provides an interesting alternative parameterisation of decomposable models since its upper triangle has the same zero structure as Σ -1 and its elements have an interpretation as parameters of certain conditional distributions. For a distribution for Σ, the strong hyper-Markov property is shown to be characterised by the mutual independence of the rows of Φ. 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.