Wishart exponential families on cones related to An graphs

Overview & Synopsis

Let G = A n be the graph corresponding to the graphical model of nearest neighbour interaction in a Gaussian character. We study Natural Exponential Families(NEF) of Wishart distributions on convex cones Q G and P G , where P G is the cone of positive definite real symmetric matrices with obligatory zeros prescribed by G, and Q G is the dual cone of P G. The Wishart NEF that we construct include Wishart distributions considered earlier by Lauritzen (1996) and Letac and Massam (2007) for models based on decomposable graphs. Our approach is however different and allows us to study the basic objects of Wishart NEF on the cones Q G and P G. We determine Riesz measures generating Wishart exponential families on Q G and P G , and we give the quadratic construction of these Riesz measures and exponential families. The mean, inverse-mean, covariance and variance functions, as well as moments of higher order are studied and their explicit formulas are given.