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Title: Random covariance associated to a weighted graph Authors:  Gerard Letac - Universite Paul Sabatier (France) [presenting]
Abstract: Given an undirected graph with n vertices and with given weights $w(e)$ on the edge $e$, we consider a positive definite matrix $M$ of order $n$ such that the off diagonal entries are 0 or $-w(e)$. We provide the diagonal of $M$ with a family of distributions already considered by a group of physicists around 2010 called `Multivariate reciprocal inverse Gaussian laws' (MRIG). This family has many interesting properties: stability by marginalization and conditioning, simplicity of the Laplace transform and moments. The one dimensional margins are the familiar reciprocal inverse Gaussian (RIG). If the graph is connected the inverse of $M$ has positive coefficients and this makes MRIG an interesting choice for a priori probabilities on concentration matrices for Gaussian graphical models.