A PC algorithm variation for ordinal variables

Bayesian networks are graphical models that represent the joint distribution of a set of variables using directed acyclic graphs. The graph can be manually built by domain experts according to their knowledge. However, when the dependence structure is unknown (or partially known) the network has to be estimated from data by using suitable learning algorithms. In this paper, we deal with a constraint-based method to perform Bayesian networks structural learning in the presence of ordinal variables. We propose an alternative version of the PC algorithm, which is one of the most known procedures, with the aim to infer the network by accounting for additional information inherent to ordinal data. The proposal is based on a nonparametric test, appropriate for ordinal variables. A comparative study shows that, in some situations, the proposal discussed here is a slightly more efficient solution than the PC algorithm.     

Efficient Bayesian Inference for Multivariate Probit Models With Sparse Inverse Correlation Matrices

We propose a Bayesian approach for inference in the multivariate probit model, taking into account the association structure between binary observations. We model the association through the correlation matrix of the latent Gaussian variables. Conditional independence is imposed by setting some off-diagonal elements of the inverse correlation matrix to zero and this sparsity structure is modeled using a decomposable graphical model. We propose an efficient Markov chain Monte Carlo algorithm relying on a parameter expansion scheme to sample from the resulting posterior distribution. This algorithm updates the correlation matrix within a simple Gibbs sampling framework and allows us to infer the correlation structure from the data, generalizing methods used for inference in decomposable Gaussian graphical models to multivariate binary observations. We demonstrate the performance of this model and of the Markov chain Monte Carlo algorithm on simulated and real datasets. This article has online supplementary materials.     

The planning of marketing strategies in consumer credit: an approach based on graphical chain models for ordinal variables

This paper explores the potential of graphical chain models in defining marketing strategies in the consumer credit field. When ordinal variables are involved, the potential can be increased by exploiting the ordinal scale of variables. In the presented research, the effectiveness of ordinal methods is analysed in three important activities for building a score-card: (1) in coarse classifying of variables, (2) in model selection and (3) in measuring the prediction accuracy of the model.     

High-Dimensional Mixed Graphical Models

While graphical models for continuous data (Gaussian graphical models) and discrete data (Ising models) have been extensively studied, there is little work on graphical models for datasets with both continuous and discrete variables (mixed data), which are common in many scientific applications. We propose a novel graphical model for mixed data, which is simple enough to be suitable for high-dimensional data, yet flexible enough to represent all possible graph structures. We develop a computationally efficient regression-based algorithm for fitting the model by focusing on the conditional log-likelihood of each variable given the rest. The parameters have a natural group structure, and sparsity in the fitted graph is attained by incorporating a group lasso penalty, approximated by a weighted lasso penalty for computational efficiency. We demonstrate the effectiveness of our method through an extensive simulation study and apply it to a music annotation dataset (CAL500), obtaining a sparse and interpretable graphical model relating the continuous features of the audio signal to binary variables such as genre, emotions, and usage associated with particular songs. While we focus on binary discrete variables for the main presentation, we also show that the proposed methodology can be easily extended to general discrete variables.     

Model based clustering for mixed data: clustMD

A model based clustering procedure for data of mixed type, clustMD, is developed using a latent variable model. It is proposed that a latent variable, following a mixture of Gaussian distributions, generates the observed data of mixed type. The observed data may be any combination of continuous, binary, ordinal or nominal variables. clustMD employs a parsimonious covariance structure for the latent variables, leading to a suite of six clustering models that vary in complexity and provide an elegant and unified approach to clustering mixed data. An expectation maximisation (EM) algorithm is used to estimate clustMD; in the presence of nominal data a Monte Carlo EM algorithm is required. The clustMD model is illustrated by clustering simulated mixed type data and prostate cancer patients, on whom mixed data have been recorded.     

Precision Matrix Estimation by Inverse Principal Orthogonal Decomposition

We investigate the structure of a large precision matrix in Gaussian graphical models by decomposing it into a low rank component and a remainder part with sparse precision matrix.Based on the decomposition,we propose to estimate the large precision matrix by inverting a principal orthogonal decomposition(IPOD).The IPOD approach has appealing practical interpretations in conditional graphical models given the low rank component,and it connects to Gaussian graphical models with latent variables.Specifically,we show that the low rank component in the decomposition of the large precision matrix can be viewed as the contribution from the latent variables in a Gaussian graphical model.Compared with existing approaches for latent variable graphical models,the IPOD is conveniently feasible in practice where only inverting a low-dimensional matrix is required.To identify the number of latent variables,which is an objective of its own interest,we investigate and justify an approach by examining the ratios of adjacent eigenvalues of the sample covariance matrix?Theoretical properties,numerical examples,and a real data application demonstrate the merits of the IPOD approach in its convenience,performance,and interpretability.     

Positivity for Gaussian graphical models

Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient condition in terms of the graph for when a subdeterminant is zero for all covariance matrices that belong to the Gaussian graphical model. Here we extend this result to give explicit cancellation-free formulas for the expansions of non-zero subdeterminants.     

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The IPSP algorithm is an efficient algorithm for computing maximum likelihood estimation of Gaussian graphical models. It first divides clique marginals of graphical models into several groups, and then it adjusts clique marginals in each group locally. This paper uses the IIPS algorithm on junction tree to replace local adjustment on each group in the IPSP algorithm and propose a resulting algorithm called IPSP-JT to reduce the complexity of the IPSP algorithm. Moreover, we give a graph with minimum edges used by IIPS to adjust locally, and we prove its existence and uniqueness and construct a local junction tree. Numerical experiments show that the IPSP-JT algorithm runs faster than the IPSP algorithm for large Gaussian graphical models.     

Learning the Structure of Mixed Graphical Models

We consider the problem of learning the structure of a pairwise graphical model over continuous and discrete variables. We present a new pairwise model for graphical models with both continuous and discrete variables that is amenable to structure learning. In previous work, authors have considered structure learning of Gaussian graphical models and structure learning of discrete models. Our approach is a natural generalization of these two lines of work to the mixed case. The penalization scheme involves a novel symmetric use of the group-lasso norm and follows naturally from a particular parameterization of the model. Supplementary materials for this article are available online.     

Modelling and inference with Conditional Gaussian Probabilistic Decision Graphs

Probabilistic Decision Graphs (PDGs) are probabilistic graphical models that represent a factorisation of a discrete joint probability distribution using a “decision graph”-like structure over local marginal parameters. The structure of a PDG enables the model to capture some context specific independence relations that are not representable in the structure of more commonly used graphical models such as Bayesian networks and Markov networks. This sometimes makes operations in PDGs more efficient than in alternative models. PDGs have previously been defined only in the discrete case, assuming a multinomial joint distribution over the variables in the model. We extend PDGs to incorporate continuous variables, by assuming a Conditional Gaussian (CG) joint distribution. We also show how inference can be carried out in an efficient way.     

A Gaussian Mixed Model for Learning Discrete Bayesian Networks

In this paper, we address the problem of learning discrete Bayesian networks from noisy data. A graphical model based on a mixture of Gaussian distributions with categorical mixing structure coming from a discrete Bayesian network is considered. The network learning is formulated as a maximum likelihood estimation problem and performed by employing an EM algorithm. The proposed approach is relevant to a variety of statistical problems for which Bayesian network models are suitable—from simple regression analysis to learning gene/protein regulatory networks from microarray data.     

Copula selection for graphical models in continuous Estimation of Distribution Algorithms

This paper presents the use of graphical models and copula functions in Estimation of Distribution Algorithms (EDAs) for solving multivariate optimization problems. It is shown in this work how the incorporation of copula functions and graphical models for modeling the dependencies among variables provides some theoretical advantages over traditional EDAs. By means of copula functions and two well known graphical models, this paper presents a novel approach for defining new EDAs. Either dependence is modeled by a copula function chosen from a predefined set of six functions that aim to cover a wide range of inter-relations. It is also shown how the use of mutual information in the learning of graphical models implies a natural way of employing copula entropies. The experimental results on separable and non-separable functions show that the two new EDAs, which adopt copula functions to model dependencies, perform better than their original version with Gaussian variables.     

Visualization of Categorical Response Models: From Data Glyphs to Parameter Glyphs

The multinomial logit model is the most widely used model for nominal multi-category responses. One problem with the model is that many parameters are involved, and another that interpretation of parameters is much harder than for linear models because the model is nonlinear. Both problems can profit from graphical representations. We propose to visualize the effect strengths by star plots, where one star collects all the parameters connected to one term in the linear predictor. In simple models, one star refers to one explanatory variable. In contrast to conventional star plots, which are used to represent data, the plots represent parameters and are considered as parameter glyphs. The set of stars for a fitted model makes the main features of the effects of explanatory variables on the response variable easily accessible. The method is extended to ordinal models and illustrated by several datasets. Supplementary materials are available online.     

Uncertainty quantification for functional dependent random variables

This paper proposes a new methodology to model uncertainties associated with functional random variables. This methodology allows to deal simultaneously with several dependent functional variables and to address the specific case where these variables are linked to a vectorial variable, called covariate. In this case, the proposed uncertainty modelling methodology has two objectives: to retain both the most important features of the functional variables and their features which are the most correlated to the covariate. This methodology is composed of two steps. First, the functional variables are decomposed on a functional basis. To deal simultaneously with several dependent functional variables, a Simultaneous Partial Least Squares algorithm is proposed to estimate this basis. Second, the joint probability density function of the coefficients selected in the decomposition is modelled by a Gaussian mixture model. A new sparse method based on a Lasso penalization algorithm is proposed to estimate the Gaussian mixture model parameters and reduce their number. Several criteria are introduced to assess the methodology performance: its ability to approximate the functional variables probability distribution, their dependence structure and their features which explain the covariate. Finally, the whole methodology is applied on a simulated example and on a nuclear reliability test case.     

A generalized Mahalanobis distance for mixed data

A distance for mixed nominal, ordinal and continuous data is developed by applying the Kullback-Leibler divergence to the general mixed-data model, an extension of the general location model that allows for ordinal variables to be incorporated in the model. The distance obtained can be considered as a generalization of the Mahalanobis distance to data with a mixture of nominal, ordinal and continuous variables. Moreover, it includes as special cases previous Mahalanobis-type distances developed by Bedrick et al. (Biometrics 56 (2000) 394) and Bar-Hen and Daudin (J. Multivariate Anal. 53 (1995) 332). Asymptotic results regarding the maximum likelihood estimator of the distance are discussed. The results of a simulation study on the level and power of the tests are reported and a real-data example illustrates the method.     

Multinomial logit models with implicit variable selection

The multinomial logit model is the most widely used model for the unordered multi-category responses. However, applications are typically restricted to the use of few predictors because in the high-dimensional case maximum likelihood estimates frequently do not exist. In this paper we are developing a boosting technique called multinomBoost that performs variable selection and fits the multinomial logit model also when predictors are high-dimensional. Since in multi-category models the effect of one predictor variable is represented by several parameters one has to distinguish between variable selection and parameter selection. A special feature of the approach is that, in contrast to existing approaches, it selects variables not parameters. The method can also distinguish between mandatory predictors and optional predictors. Moreover, it adapts to metric, binary, nominal and ordinal predictors. Regularization within the algorithm allows to include nominal and ordinal variables which have many categories. In the case of ordinal predictors the order information is used. The performance of boosting technique with respect to mean squared error, prediction error and the identification of relevant variables is investigated in a simulation study. The method is applied to the national Indonesia contraceptive prevalence survey and the identification of glass. Results are also compared with the Lasso approach which selects parameters.     

Bayesian Ising Graphical Model for Variable Selection

In this article, we propose a new Bayesian variable selection (BVS) approach via the graphical model and the Ising model, which we refer to as the “Bayesian Ising graphical model” (BIGM). The BIGM is developed by showing that the BVS problem based on the linear regression model can be considered as a complete graph and described by an Ising model with random interactions. There are several advantages of our BIGM: it is easy to (i) employ the single-site updating and cluster updating algorithm, both of which are suitable for problems with small sample sizes and a larger number of variables, (ii) extend this approach to nonparametric regression models, and (iii) incorporate graphical prior information. In our BIGM, the interactions are determined by the linear model coefficients, so we systematically study the performance of different scale normal mixture priors for the model coefficients by adopting the global-local shrinkage strategy. Our results indicate that the best prior for the model coefficients in terms of variable selection should place substantial weight on small, nonzero shrinkage. The methods are illustrated with simulated and real data. Supplementary materials for this article are available online.     

Estimation of Symmetry-Constrained Gaussian Graphical Models: Application to Clustered Dense Networks

We propose a model selection algorithm for high-dimensional clustered data. Our algorithm combines a classical penalized likelihood method with a composite likelihood approach in the framework of colored graphical Gaussian models. Our method is designed to identify high-dimensional dense networks with a large number of edges but sparse edge classes. Its empirical performance is demonstrated through simulation studies and a network analysis of a gene expression dataset.     

Efficient Computation of ℓ1 Regularized Estimates in Gaussian Graphical Models

In this article, I propose an effcient algorithm to compute ?₁ regularized maximum likelihood estimates in the Gaussian graphical model. These estimators, recently proposed in an earlier article by Yuan and Lin, conduct parameter estimation and model selection simultaneously and have been shown to enjoy nice properties in both large and finite samples. To compute the estimates, however, can be very challenging in practice because of the high dimensionality and positive definiteness constraint on the covariance matrix. Taking advantage of the recent advance in semidefinite programming, Yuan and Lin suggested a sophisticated interior-point algorithm to solve the optimization problem. Although a polynomial time algorithm, the optimization technique is known not to be scalable for high-dimensional problems. Alternatively, this article shows that the estimates can be computed by iteratively solving a sequence of ?₁ regularized quadratic programs. By effectively exploiting the sparsity of the graphical structure, I propose a new algorithm that can be applied to problems of larger scale. When combined with a path-following strategy, the new algorithm can be used to efficiently approximate the entire solution path of the ?₁ regularized maximum likelihood estimates, which also facilitates the choice of tuning parameter. I demonstrate the efficacy and usefulness of the proposed algorithm on a few simulations and real datasets.     

Qualitative inequalities for squared partial correlations of a Gaussian random vector

We describe various sets of conditional independence relationships, sufficient for qualitatively comparing non-vanishing squared partial correlations of a Gaussian random vector. These sufficient conditions are satisfied by several graphical Markov models. Rules for comparing degree of association among the vertices of such Gaussian graphical models are also developed. We apply these rules to compare conditional dependencies on Gaussian trees. In particular for trees, we show that such dependence can be completely characterised by the length of the paths joining the dependent vertices to each other and to the vertices conditioned on. We also apply our results to postulate rules for model selection for polytree models. Our rules apply to mutual information of Gaussian random vectors as well.