Sparse high dimensional graphical model selection is a topic of much interest in modern day statistics. A popular approach is to apply 1 -penalties to either (1) parametric likelihoods, or, (2) regularized regression/pseudo-likelihoods, with the latter having the distinct advantage that they do not explicitly assume Gaussianity. As none of the popular methods proposed for solving pseudo-likelihood based objective functions have provable convergence guarantees, it is not clear if corresponding estimators exist or are even computable, or if they actually yield correct partial correlation graphs. This paper proposes a new pseudo-likelihood based graphical model selection method that aims to overcome some of the shortcomings of current methods, but at the same time retain all their respective strengths. In particular, we introduce a novel framework that leads to a convex formulation of the partial covariance regression graph problem, resulting in an objective function comprised of quadratic forms. The objective is then optimized via a coordinatewise approach. The specific functional form of the objective function facilitates rigorous convergence analysis leading to convergence guarantees; an important property that cannot be established using standard results, when the dimension is larger than the sample size, as is often the case in high dimensional applications. These convergence guarantees ensure that estimators are well-defined under very general conditions, and are always computable. In addition, the approach yields estimators that have good large sample properties and also respect symmetry.Furthermore, application to simulated/real data, timing comparisons and numerical convergence is demonstrated. We also present a novel unifying framework that places all graphical pseudo-likelihood methods as special cases of a more general formulation, leading to important insights.

We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions.Comment: This paper commented in: [arXiv:0808.3855], [arXiv:0808.3856], [arXiv:0808.3859], [arXiv:0808.3861]. Rejoinder in [arXiv:0808.3864]. Published in at http://dx.doi.org/10.1214/07-STS252 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Covariance estimation and selection for high-dimensional multivariate datasets is a fundamental problem in modern statistics. Gaussian directed acyclic graph (DAG) models are a popular class of models used for this purpose. Gaussian DAG models introduce sparsity in the Cholesky factor of the inverse covariance matrix, and the sparsity pattern in turn corresponds to specific conditional independence assumptions on the underlying variables. A variety of priors have been developed in recent years for Bayesian inference in DAG models, yet crucial convergence and sparsity selection properties for these models have not been thoroughly investigated. Most of these priors are adaptations/generalizations of the Wishart distribution in the DAG context. In this paper, we consider a flexible and general class of these 'DAG-Wishart' priors with multiple shape parameters. Under mild regularity assumptions, we establish strong graph selection consistency and establish posterior convergence rates for estimation when the number of variables p is allowed to grow at an appropriate sub-exponential rate with the sample size n.MSC 2010 subject classifications: Primary 62F15; secondary 62G20 P (|S ij − (Σ 0 ) ij | ≥ t) ≤ m 1 exp{−m 2 n(tǫ 0,n ) 2 }, |t| ≤ δ.

The striatum of the brain coordinates motor function. Dopamine-related drugs may be therapeutic to patients with striatal neurodegeneration, such as Huntington's disease (HD) and Parkinson's disease (PD), but these drugs have unwanted side effects. In addition to stimulating the release of norepinephrine, amphetamines, which are used for narcolepsy and hyperactivity disorder (ADHD), trigger dopamine release in the striatum. The GTPase Ras homolog-enriched in the striatum (Rhes) inhibits dopaminergic signaling in the striatum, is implicated in HD, and has a role in striatal motor control. We found that the guanine nucleotide exchange factor (GEF) RasGRP1 inhibited Rhes-mediated control of striatal motor activity in mice. RasGRP1 stabilized Rhes, increasing its synaptic accumulation in cultured striatal neurons.. Whereas partially Rhes-deficient (Rhes +/− ) mice had an enhanced locomotor response to amphetamine, this phenotype was attenuated by coincident depletion of RasGRP1. By proteomic analysis of striatal lysates from Rhes-heterozygous mice with wild-type or partial or complete knockout of Rasgrp1, we identified a diverse set of Rhes-interacting proteins, the "Rhesactome," and determined that RasGRP1 ** This manuscript has been accepted for publication in Science Signaling. This version has not undergone final editing. Please refer to the complete version of record at http://www.sciencesignaling.org/. The manuscript may not be reproduced or used in any manner that does not fall within the fair use provisions of the Copyright Act without the prior, written permission of AAAS.

Gaussian covariance graph models encode marginal independence among the components of a multivariate random vector by means of a graph $G$. These models are distinctly different from the traditional concentration graph models (often also referred to as Gaussian graphical models or covariance selection models) since the zeros in the parameter are now reflected in the covariance matrix $\Sigma$, as compared to the concentration matrix $\Omega =\Sigma^{-1}$. The parameter space of interest for covariance graph models is the cone $P_G$ of positive definite matrices with fixed zeros corresponding to the missing edges of $G$. As in Letac and Massam [Ann. Statist. 35 (2007) 1278--1323], we consider the case where $G$ is decomposable. In this paper, we construct on the cone $P_G$ a family of Wishart distributions which serve a similar purpose in the covariance graph setting as those constructed by Letac and Massam [Ann. Statist. 35 (2007) 1278--1323] and Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272--1317] do in the concentration graph setting. We proceed to undertake a rigorous study of these "covariance" Wishart distributions and derive several deep and useful properties of this class.Comment: Published in at http://dx.doi.org/10.1214/10-AOS841 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org