  When considering a graphical Gaussian model ${\mathcal{N}}_G$ 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 ${\mathcal{N}}_G$ 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.
