Wishart exponential families on cones related to tridiagonal matrices

Abstract

Let G be the graph corresponding to the graphical model of nearest neighbor 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 tridiagonal positive definite real symmetric matrices, and $Q_G$ is the dual cone of $P_G$. The Wishart NEF that we construct include Wishart distributions considered earlier for models based on decomposable(chordal) 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.

Appendix

We list here some properties of triangular matrices, used in proofs.

Lemma 7

• 1. Let $A= K^0$, where $K=A_{\{1:k\}}$ and let L be lower triangular and U upper triangular $n\times n$ matrices. Then $UAL=\left( U_{\{1:k\}}KL_{\{1:k\}}\right) ^0.$

• 2. Let M, L, U be matrices $n\times n$, with L lower triangular and U upper triangular. Then, for all $i=1,\ldots , n$, $(LMU)_{\{1:i\}}= L_{\{1:i\}}M_{\{1:i\}} U_{\{1:i\}}$ and $(UML)_{\{i:n\}}=U_{\{i:n\}}M_{\{i:n\}}L_{\{i:n\}}$.

• 3. If T is an invertible triangular matrix then $(T_{\{1:k\}})^{-1}=(T^{-1})_{\{1:k\}}$ for all $k=1,\ldots , n$.