Frequentist model averaging for undirected Gaussian graphical models

Advances in information technologies have made network data increasingly frequent in a spectrum of big data applications, which is often explored by probabilistic graphical models. To precisely estimate the precision matrix, we propose an optimal model averaging estimator for Gaussian graphs. We prove that the proposed estimator is asymptotically optimal when candidate models are misspecified. The consistency and the asymptotic distribution of model averaging estimator, and the weight convergence are also studied when at least one correct model is included in the candidate set. Furthermore, numerical simulations and a real data analysis on yeast genetic data are conducted to illustrate that the proposed method is promising.     

Using graphical chain models to analyze differences in structural correlates of undernutrition in Benin and Bangladesh

Undernutrition among children is one of the most important health problems in developing countries. In order to understand the complex pathways affecting undernutrition which is crucial for policy interventions, one needs to explicitly model the dependence chain of immediate, intermediate, and underlying factors affecting undernutrition. Graphical chain models are used here to investigate the determinants of undernutrition in Benin and Bangladesh. While the dependence chain affecting undernutrition contains many common elements, the influence of demographic, cultural, and socioeconomic factors seems to have stronger direct and indirect influences in Benin than in Bangladesh, where many socioeconomic and gender related factors have a more direct influence on undernutrition.     

Bayesian inference of causal effects from observational data in Gaussian graphical models

We assume that multivariate observational data are generated from a distribution whose conditional independencies are encoded in a Directed Acyclic Graph (DAG). For any given DAG, the causal effect of a variable onto another one can be evaluated through intervention calculus. A DAG is typically not identifiable from observational data alone. However, its Markov equivalence class (a collection of DAGs) can be estimated from the data. As a consequence, for the same intervention a set of causal effects, one for each DAG in the equivalence class, can be evaluated. In this paper, we propose a fully Bayesian methodology to make inference on the causal effects of any intervention in the system. Main features of our method are: (a) both uncertainty on the equivalence class and the causal effects are jointly modeled; (b) priors on the parameters of the modified Cholesky decomposition of the precision matrices across all DAG models are constructively assigned starting from a unique prior on the complete (unrestricted) DAG; (c) an efficient algorithm to sample from the posterior distribution on graph space is adopted; (d) an objective Bayes approach, requiring virtually no user specification, is used throughout. We demonstrate the merits of our methodology in simulation studies, wherein comparisons with current state‐of‐the‐art procedures turn out to be highly satisfactory. Finally we examine a real data set of gene expressions for Arabidopsis thaliana.     

Fast Bayesian inference in large Gaussian graphical models

Despite major methodological developments, Bayesian inference in Gaussian graphical models remains challenging in high dimension due to the tremendous size of the model space. This article proposes a method to infer the marginal and conditional independence structures between variables by multiple testing, which bypasses the exploration of the model space. Specifically, we introduce closed‐form Bayes factors under the Gaussian conjugate model to evaluate the null hypotheses of marginal and conditional independence between variables. Their computation for all pairs of variables is shown to be extremely efficient, thereby allowing us to address large problems with thousands of nodes as required by modern applications. Moreover, we derive exact tail probabilities from the null distributions of the Bayes factors. These allow the use of any multiplicity correction procedure to control error rates for incorrect edge inclusion. We demonstrate the proposed approach on various simulated examples as well as on a large gene expression data set from The Cancer Genome Atlas.     

Application of Covariance Analysis to Ophtalmologic Research

Using covariance analysis as an adjusted regression analysis it is possible, with the assumption of homogeneity and normality of variance, to test several regression functions for homogeneity, if the given groups of data material are independent. This well-known analysis is applied to a number of examination methods of ophthalmology. In the cases of ophthalmo-dynamography (ODG) and ophthalmo-dynamometry (ODM), the paper answers the question to what extent the result of ophthalmologic measurement is influenced by posture (sitting — lying — standing). Contrary to several authors' practice of combining the regression lines for the systolic and diastolic blood-pressure because of the improvement of the correlation coefficient to be achieved in this way, it was possible to show that generally such an approach is not recommendable. Patients suffering from a certain type of arteriosclerosis distinguish themselves by deviations of their ODG data from the normal blood-pressure relation of the arteria carotis interna and the arteria brachialis. These differences can be tested for statistical significance by means of covariance analysis.     

Model selection and estimation in the Gaussian graphical model

We propose penalized likelihood methods for estimating the concentrationmatrix in the Gaussian graphical model. The methods lead toa sparse and shrinkage estimator of the concentration matrixthat is positive definite, and thus conduct model selectionand estimation simultaneously. The implementation of the methodsis nontrivial because of the positive definite constraint onthe concentration matrix, but we show that the computation canbe done effectively by taking advantage of the efficient maxdetalgorithm developed in convex optimization. We propose a BIC-typecriterion for the selection of the tuning parameter in the penalizedlikelihood methods. The connection between our methods and existingmethods is illustrated. Simulations and real examples demonstratethe competitive performance of the new methods.