Gaussian approximation of Gaussian scale mixture

For a given positive random variable $V>0$ and a given $Z\sim N(0,1)$ independent of $V$, we compute the scalar $t_0$ such that the distance between $Z\sqrt{V}$ and $Z\sqrt{t_0}$ in the $L^2(\R)$ sense, is minimal. We also consider the same problem in several dimensions when $V$ is a random positive definite matrix.Comment: 13 page

Wishart distributions for decomposable graphs

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.Comment: Published at http://dx.doi.org/10.1214/009053606000001235 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayes factors and the geometry of discrete hierarchical loglinear models

A standard tool for model selection in a Bayesian framework is the Bayes factor which compares the marginal likelihood of the data under two given different models. In this paper, we consider the class of hierarchical loglinear models for discrete data given under the form of a contingency table with multinomial sampling. We assume that the Diaconis-Ylvisaker conjugate prior is the prior distribution on the loglinear parameters and the uniform is the prior distribution on the space of models. Under these conditions, the Bayes factor between two models is a function of their prior and posterior normalizing constants. These constants are functions of the hyperparameters $(m,\alpha)$ which can be interpreted respectively as marginal counts and the total count of a fictive contingency table. We study the behaviour of the Bayes factor when $\alpha$ tends to zero. In this study two mathematical objects play a most important role. They are, first, the interior $C$ of the convex hull $\bar{C}$ of the support of the multinomial distribution for a given hierarchical loglinear model together with its faces and second, the characteristic function $\mathbb{J}_C$ of this convex set $C$. We show that, when $\alpha$ tends to 0, if the data lies on a face $F_i$ of $\bar{C_i},i=1,2$ of dimension $k_i$, the Bayes factor behaves like $\alpha^{k_1-k_2}$. This implies in particular that when the data is in $C_1$ and in $C_2$, i.e. when $k_i$ equals the dimension of model $J_i$, the sparser model is favored, thus confirming the idea of Bayesian regularization.Comment: 37 page

A conjugate prior for discrete hierarchical log-linear models

In Bayesian analysis of multi-way contingency tables, the selection of a prior distribution for either the log-linear parameters or the cell probabilities parameters is a major challenge. In this paper, we define a flexible family of conjugate priors for the wide class of discrete hierarchical log-linear models, which includes the class of graphical models. These priors are defined as the Diaconis--Ylvisaker conjugate priors on the log-linear parameters subject to "baseline constraints" under multinomial sampling. We also derive the induced prior on the cell probabilities and show that the induced prior is a generalization of the hyper Dirichlet prior. We show that this prior has several desirable properties and illustrate its usefulness by identifying the most probable decomposable, graphical and hierarchical log-linear models for a six-way contingency table.Comment: Published in at http://dx.doi.org/10.1214/08-AOS669 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Moments of minors of Wishart matrices

For a random matrix following a Wishart distribution, we derive formulas for the expectation and the covariance matrix of compound matrices. The compound matrix of order $m$ is populated by all $m\times m$-minors of the Wishart matrix. Our results yield first and second moments of the minors of the sample covariance matrix for multivariate normal observations. This work is motivated by the fact that such minors arise in the expression of constraints on the covariance matrix in many classical multivariate problems.Comment: Published in at http://dx.doi.org/10.1214/07-AOS522 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org