Bayesian Estimation of Graphical Gaussian Models with Edges and Vertices Symmetries

We consider the Bayesian analysis of undirected graphical Gaussian models with edges and vertices symmetries. The graphical Gaussian models with equality constraints on the precision matrix, that is the inverse covariance matrix, introduced by Hojsgaard and Lauritzen as RCON models. The models can be represented by colored graphs, where edges or vertices have the same coloring if the corresponding elements of the precision matrix are equal. In this thesis, we define a conjugate prior distribution for RCON models. We will, therefore, call this conjugate prior the colored G-Wishart.

We first develop a sampling scheme for the colored G-Wishart distribution. This sampling method is based on the Metropolis-Hastings algorithm and the Cholesky decomposition of matrices. In order to assess the accuracy of the Metropolis-Hastings sampling method, we compute the normalizing constants of the colored G-Wishart distribution for some special graphs: general trees, star graphs, a complete graph on 3 vertices and a simple decomposable model on 4 vertices with various symmetry constraints. By differentiating the analytic expression of normalizing constants, we can obtain the true mean of the colored G-Wishart distribution for these particular graphs. Moreover, we conduct a number of simulations to compare the true mean of the colored G-Wishart distribution with the sample mean obtained from a number of iterations of our Metropolis-Hastings algorithm.

Then, we give three methods for estimating the normalizing constant of the colored G-Wishart distribution. The three methods are the Monte Carlo method, the importance sampling, and the Laplace approximation. We furthermore apply these methods on the model search for a real dataset using Bayes factors.

At last, we propose the distributed Bayesian estimate of the precision matrix in colored graphical Gaussian models. We also study the asymptotic behaviour of our proposed estimate under the regular asymptotic regime when the number of variables p is fixed and under the double asymptotic regime when both p and the sample size n grow to infinity.