Gérard Letac

Wishart distributions for decomposable graphs

When considering a graphical Gaussian model N G Markov with respect to a decomposable graph G, the parameter space of interest for the precision parameter is the cone P G of positive definite matrices with fixed zeros corresponding to the missing edges of G. The parameter space for the scale parameter of N G is the cone Q G , dual to P G , of incomplete matrices with submatrices corresponding to the cliques of G being positive definite. In this paper we construct on the cones Q G and P G two families of Wishart distributions, namely the Type I and Type II Wisharts. They can be viewed as generalizations of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272-1317]. We show that the Type I and II Wisharts have properties similar to those of the hyper and hyper inverse Wishart. Indeed, the inverse of the Type II Wishart forms a conjugate family of priors for the covariance parameter of the graphical Gaussian model and is strong directed hyper Markov for every direction given to the graph by a perfect order of its cliques, while the Type I Wishart is weak hyper Markov. Moreover, the inverse Type II Wishart as a conjugate family presents the advantage of having a multidimensional shape parameter, thus offering flexibility for the choice of a prior. Both Type I and II Wishart distributions depend on multivariate shape parameters. A shape parameter is acceptable if and only if it satisfies a certain eigenvalue property. We show that the sets of acceptable shape parameters for a noncomplete G have dimension equal to at least one plus the number of cliques in G. These families, as conjugate families, are richer than the traditional Diaconis-Ylvisaker conjugate families which all have a shape parameter set of dimension one. A decomposable graph which does not contain a three-link chain as an induced subgraph is said to be homogeneous. In this case, our Wisharts are particular cases of the Wisharts on homogeneous cones as defined by Andersson and Wojnar [J. Theoret. Probab. 17 (2004) 781-818] and the dimension of the shape parameter set is even larger than in the nonhomogeneous case: it is indeed equal to the number of cliques plus the number of distinct minimal separators. Using the model where G is a three-link chain, we show by computing a 7-tuple integral that in general we cannot expect the shape parameter sets to have dimension larger than the number of cliques plus one.

Explicit stationary distributions for compositions of random functions and products of random matrices

If (Yn)n=1/∞ is a sequence of i.i.d. random variables onE=(0,+∞) and iff is positive onE, this paper studies explicit examples of stationary distributions for the Markov chain (Wn)n=0∞ defined byWn=Ynf(Wn-1). The case wheref is a Moebius function(ax+b)/(cx+d) leads to products of certain random (2,2) matrices and to interesting random continued fractions. These explicit examples are built with a naive idea by considering genral exponential families onE, especially the families of beta distributions of the first and second kind.

The diagnonal multivariate natural exponential families and their classification

A natural exponential family (NEF)F in ℝn,n>1, is said to be diagonal if there existn functions,a1,...,an, on some intervals of ℝ, such that the covariance matrixVF(m) ofF has diagonal (a1(m1),...,an(mn)), for allm=(m1,...,mn) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ℝk and ℝn-k, for somek=1,...,n−1. This paper shows that there are only six types of irreducible diagonal NEFs in ℝn, that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ℝn, under what conditions is its projectionp(F) in ℝk, underp(x1,...,xn)∶=(x1,...,xk),k=1,...,n−1, still an NEF in ℝk? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofVF(m1,...,mn) does not depend on (mk+1,...,mn).

Exponential and bayesian conjugate families: Review and extensions

SummaryThe notion of a conjugate family of distributions plays a very important role in the Bayesian approach to parametric inference. One of the main features of such a family is that it is closed under sampling, but a conjugate family often provides prior distributions which are tractable in various other respects. This paper is concerned with the properties of conjugate families for exponential family models. Special attention is given to the class of natural exponential families having a quadratic variance function, for which the theory is particularly fruitful. Several classes of conjugate families have been considered in the literature and here we describe some of their most interesting features. Relationships between such classes are also discussed. Our aim is to provide a unified approach to the theory of conjugate families for exponential family likelihoods. An important aspect of the theory concerns reparameterisations of the exponential family under consideration. We briefly review the concept of a conjugate parameterisation, which provides further insight into many of the properties discussed throughout the paper. Finally, further implications of these results for Bayesian conjugate analysis of exponential families are investigated.

A characterization of the Wishart exponential families by an invariance property

AbstractE is the space of real symmetric (d, d) matrices, andS and

Laplace transforms which are negative powers of quadratic polynomials

\bar S

Additive Properties of the Dufresne Laws and Their Multivariate Extension

are the subsets ofE of positive definite and semipositive-definite matrices. Let there be ap in