On Block Ordering of Variables in Graphical Modelling

Abstract.  In graphical modelling, the existence of substantive background knowledge on block ordering of variables is used to perform structural learning within the family of chain graphs (CGs) in which every block corresponds to an undirected graph and edges joining vertices in different blocks are directed in accordance with the ordering. We show that this practice may lead to an inappropriate restriction of the search space and introduce the concept of labelled block ordering B corresponding to a family of B - consistent CGs in which every block may be either an undirected graph or a directed acyclic graph or, more generally, a CG. In this way we provide a flexible tool for specifying subsets of chain graphs, and we observe that the most relevant subsets of CGs considered in the literature are families of B -consistent CGs for the appropriate choice of B . Structural learning within a family of B -consistent CGs requires to deal with Markov equivalence. We provide a graphical characterization of equivalence classes of B -consistent CGs, namely the B - essential graphs , as well as a procedure to construct the B -essential graph for any given equivalence class of B -consistent chain graphs. Both largest CGs and essential graphs turn out to be special cases of B -essential graphs.     

On the Markov Equivalence of Chain Graphs, Undirected Graphs, and Acyclic Digraphs

Graphical Markov models use undirected graphs (UDGs), acyclic directed graphs (ADGs), or (mixed) chain graphs to represent possible dependencies among random variables in a multivariate distribution. Whereas a UDG is uniquely determined by its associated Markov model, this is not true for ADGs or for general chain graphs (which include both UDGs and ADGs as special cases). This paper addresses three questions regarding the equivalence of graphical Markov models: when is a given chain graph Markov equivalent (1) to some UDG? (2) to some (at least one) ADG? (3) to some decomposable UDG? The answers are obtained by means of an extension of Frydenberg’s (1990) elegant graph-theoretic characterization of the Markov equivalence of chain graphs.     

Joint response graphs and separation induced by triangular systems

Summary.  We consider joint probability distributions generated recursively in terms of univariate conditional distributions satisfying conditional independence restrictions. The independences are captured by missing edges in a directed graph. A matrix form of such a graph, called the generating edge matrix, is triangular so the distributions that are generated over such graphs are called triangular systems. We study consequences of triangular systems after grouping or reordering of the variables for analyses as chain graph models, i.e. for alternative recursive factorizations of the given density using joint conditional distributions. For this we introduce families of linear triangular equations which do not require assumptions of distributional form. The strength of the associations that are implied by such linear families for chain graph models is derived. The edge matrices of chain graphs that are implied by any triangular system are obtained by appropriately transforming the generating edge matrix. It is shown how induced independences and dependences can be studied by graphs, by edge matrix calculations and via the properties of densities. Some ways of using the results are illustrated.     

Graphical models for skew-normal variates

This paper explores the usefulness of the multivariate skew-normal distribution in the context of graphical models. A slight extension of the family recently discussed by Azzalini & Dalla Valle (1996 ) and Azzalini & Capitanio (1999 ) is described, the main motivation being the additional property of closure under conditioning. After considerations of the main probabilistic features, the focus of the paper is on the construction of conditional independence graphs for skew-normal variables. Necessary and sufficient conditions for conditional independence are stated, and the admissible structures of a graph under restriction on univariate marginal distribution are studied. Finally, parameter estimation is considered. It is shown how the factorization of the likelihood function according to a graph can be rearranged in order to obtain a parameter based factorization.     

Gaussian copula distributions for mixed data,with application in discrimination

The construction of a joint model for mixed discrete and continuous random variables that accounts for their associations is an important statistical problem in many practical applications. In this paper, we use copulas to construct a class of joint distributions of mixed discrete and continuous random variables. In particular, we employ the Gaussian copula to generate joint distributions for mixed variables. Examples include the robit-normal and probit-normal-exponential distributions, the first for modelling the distribution of mixed binary-continuous data and the second for a mixture of continuous, binary and trichotomous variables. The new class of joint distributions is general enough to include many mixed-data models currently available. We study properties of the distributions and outline likelihood estimation; a small simulation study is used to investigate the finite-sample properties of estimates obtained by full and pairwise likelihood methods. Finally, we present an application to discriminant analysis of multiple correlated binary and continuous data from a study involving advanced breast cancer patients.     

A Unified Approach to the Characterization of Equivalence Classes of DAGs, Chain Graphs with no Flags and Chain Graphs

Abstract.  A Markov property associates a set of conditional independencies to a graph. Two alternative Markov properties are available for chain graphs (CGs), the Lauritzen–Wermuth–Frydenberg (LWF) and the Andersson–Madigan– Perlman (AMP) Markov properties, which are different in general but coincide for the subclass of CGs with no flags . Markov equivalence induces a partition of the class of CGs into equivalence classes and every equivalence class contains a, possibly empty, subclass of CGs with no flags itself containing a, possibly empty, subclass of directed acyclic graphs (DAGs). LWF-Markov equivalence classes of CGs can be naturally characterized by means of the so-called largest CGs , whereas a graphical characterization of equivalence classes of DAGs is provided by the essential graphs . In this paper, we show the existence of largest CGs with no flags that provide a natural characterization of equivalence classes of CGs of this kind, with respect to both the LWF- and the AMP-Markov properties. We propose a procedure for the construction of the largest CGs, the largest CGs with no flags and the essential graphs, thereby providing a unified approach to the problem. As by-products we obtain a characterization of graphs that are largest CGs with no flags and an alternative characterization of graphs which are largest CGs. Furthermore, a known characterization of the essential graphs is shown to be a special case of our more general framework. The three graphical characterizations have a common structure: they use two versions of a locally verifiable graphical rule. Moreover, in case of DAGs, an immediate comparison of three characterizing graphs is possible.     

An R package for the simulation of correlated discrete variables

A package for the stochastic simulation of discrete variables with assigned marginal distributions and correlation matrix is presented and discussed. The simulating mechanism relies upon the Gaussian copula, linking the discrete distributions together, and an iterative scheme recovering the correlation matrix for the copula that ensures the desired correlations among the discrete variables. Examples of its use are provided as well as three possible applications (related to probability, sampling, and inference), which illustrate the utility of the package as an efficient and easy-to-use tool both in statistical research and for didactic purposes.     

The geometry of Gaussian double Markovian distributions

Gaussian double Markovian models consist of covariance matrices constrained by a pair of graphs specifying zeros simultaneously in the matrix and its inverse. We study the semi-algebraic geometry of these models, in particular their dimension, smoothness, and connectedness as well as algebraic and combinatorial properties.     

Alternative Markov Properties for Chain Graphs

Graphical Markov models use graphs to represent possible dependences among statistical variables. Lauritzen, Wermuth, and Frydenberg (LWF) introduced a Markov property for chain graphs (CG): graphs that can be used to represent both structural and associative dependences simultaneously and that include both undirected graphs (UG) and acyclic directed graphs (ADG) as special cases. Here an alternative Markov property (AMP) for CGs is introduced and shown to be the Markov property satisfied by a block-recursive linear system with multivariate normal errors. This model can be decomposed into a collection of conditional normal models, each of which combines the features of multivariate linear regression models and covariance selection models, facilitating the estimation of its parameters. In the general case, necessary and sufficient conditions are given for the equivalence of the LWF and AMP Markov properties of a CG, for the AMP Markov equivalence of two CGs, for the AMP Markov equivalence of a CG to some ADG or decomposable UG, and for other equivalences. For CGs, in some ways the AMP property is a more direct extension of the ADG Markov property than is the LWF property.     

Stratified Gaussian graphical models

Gaussian graphical models represent the backbone of the statistical toolbox for analyzing continuous multivariate systems. However, due to the intrinsic properties of the multivariate normal distribution, use of this model family may hide certain forms of context-specific independence that are natural to consider from an applied perspective. Such independencies have been earlier introduced to generalize discrete graphical models and Bayesian networks into more flexible model families. Here, we adapt the idea of context-specific independence to Gaussian graphical models by introducing a stratification of the Euclidean space such that a conditional independence may hold in certain segments but be absent elsewhere. It is shown that the stratified models define a curved exponential family, which retains considerable tractability for parameter estimation and model selection.     

Collapsibility for Directed Acyclic Graphs

Abstract.  Collapsibility means that the same statistical result of interest can be obtained before and after marginalization over some variables. In this paper, we discuss three kinds of collapsibility for directed acyclic graphs (DAGs): estimate collapsibility, conditional independence collapsibility and model collapsibility. Related to collapsibility, we discuss removability of variables from a DAG. We present conditions for these three different kinds of collapsibility and relationships among them. We give algorithms to find a minimum variable set containing a variable subset of interest onto which a statistical result is collapsible.     

Bayesian spatio‐temporal models based on discrete convolutions

Abstract: The authors consider a class of models for spatio‐temporal processes based on convolving independent processes with a discrete kernel that is represented by a lower triangular matrix. They study two families of models. In the first one, spatial Gaussian processes with isotropic correlations are convoluted with a kernel that provides temporal dependencies. In the second family, AR(p) processes are convoluted with a kernel providing spatial interactions. The covariance structures associated with these two families are quite rich. Their covariance functions that are stationary and separable in space and time as well as time dependent nonseparable and nonisotropic ones.     

Efficient calculation of the normalizing constant of the autologistic and related models on the cylinder and lattice

Summary. Motivated by the autologistic model for the analysis of spatial binary data on the two-dimensional lattice, we develop efficient computational methods for calculating the normalizing constant for models for discrete data defined on the cylinder and lattice. Because the normalizing constant is generally unknown analytically, statisticians have developed various ad hoc methods to overcome this difficulty. Our aim is to provide computationally and statistically efficient methods for calculating the normalizing constant so that efficient likelihood-based statistical methods are then available for inference. We extend the so-called transition method to find a feasible computational method of obtaining the normalizing constant for the cylinder boundary condition. To extend the result to the free-boundary condition on the lattice we use an efficient path sampling Markov chain Monte Carlo scheme. The methods are generally applicable to association patterns other than spatial, such as clustered binary data, and to variables taking three or more values described by, for example, Potts models.     

Graphs for Margins of Bayesian Networks

Directed acyclic graph (DAG) models—also called Bayesian networks—are widely used in probabilistic reasoning, machine learning and causal inference. If latent variables are present, then the set of possible marginal distributions over the remaining (observed) variables is generally not represented by any DAG. Larger classes of mixed graphical models have been introduced to overcome this; however, as we show, these classes are not sufficiently rich to capture all the marginal models that can arise. We introduce a new class of hyper‐graphs, called mDAGs, and a latent projection operation to obtain an mDAG from the margin of a DAG. We show that each distinct marginal of a DAG model is represented by at least one mDAG and provide graphical results towards characterizing equivalence of these models. Finally, we show that mDAGs correctly capture the marginal structure of causally interpreted DAGs under interventions on the observed variables.     

Identifiable finite mixtures of location models for clustering mixed-mode data

For clustering mixed categorical and continuous data, Lawrence and Krzanowski (1996) proposed a finite mixture model in which component densities conform to the location model. In the graphical models literature the location model is known as the homogeneous Conditional Gaussian model. In this paper it is shown that their model is not identifiable without imposing additional restrictions. Specifically, for g groups and m locations, (g!)m–1 distinct sets of parameter values (not including permutations of the group mixing parameters) produce the same likelihood function. Excessive shrinkage of parameter estimates in a simulation experiment reported by Lawrence and Krzanowski (1996) is shown to be an artifact of the model's non-identifiability. Identifiable finite mixture models can be obtained by imposing restrictions on the conditional means of the continuous variables. These new identified models are assessed in simulation experiments. The conditional mean structure of the continuous variables in the restricted location mixture models is similar to that in the underlying variable mixture models proposed by Everitt (1988), but the restricted location mixture models are more computationally tractable.     

Mixture inverse Gaussian for unobserved heterogeneity in the autoregressive conditional duration model

In this paper, we assume that the duration of a process has two different intrinsic components or phases which are independent. The first is the time it takes for a trade to be initiated in the market (for example, the time during which agents obtain knowledge about the market in which they are operating and accumulate information, which is coherent with Brownian motion) and the second is the subsequent time required for the trade to develop into a complete duration. Of course, if the first time is zero then the trade is initiated immediately and no initial knowledge is required. If we assume a specific compound Bernoulli distribution for the first time and an inverse Gaussian distribution for the second, the resulting convolution model has a mixture of an inverse Gaussian distribution with its reciprocal, which allows us to specify and test the unobserved heterogeneity in the autoregressive conditional duration (ACD) model.

Our proposals make it possible not only to capture various density shapes of the durations but also easily to accommodate the behaviour of the tail of the distribution and the non monotonic hazard function. The proposed model is easy to fit and characterizes the behaviour of the conditional durations reasonably well in terms of statistical criteria based on point and density forecasts.     

Accuracy of variational estimates for random graph mixture models

Variational and variational Bayes techniques are popular approaches for statistical inference of complex models but their theoretical properties are still not well known. Because of both unobserved variables and intricate dependency structures, mixture models for random graphs constitute a good case study. We first present four different variational estimates for the parameters of these models. We then compare their accuracy through simulation studies and show that the variational Bayes estimates seem the most accurate for moderate graph size. We finally re-analyse the regulatory network of Escherichia coli with this approach.     

Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models

Abstract. We propose an objective Bayesian method for the comparison of all Gaussian directed acyclic graphical models defined on a given set of variables. The method, which is based on the notion of fractional Bayes factor (BF), requires a single default (typically improper) prior on the space of unconstrained covariance matrices, together with a prior sample size hyper‐parameter, which can be set to its minimal value. We show that our approach produces genuine BFs. The implied prior on the concentration matrix of any complete graph is a data‐dependent Wishart distribution, and this in turn guarantees that Markov equivalent graphs are scored with the same marginal likelihood. We specialize our results to the smaller class of Gaussian decomposable undirected graphical models and show that in this case they coincide with those recently obtained using limiting versions of hyper‐inverse Wishart distributions as priors on the graph‐constrained covariance matrices.     

Most Likely Transformations

We propose and study properties of maximum likelihood estimators in the class of conditional transformation models. Based on a suitable explicit parameterization of the unconditional or conditional transformation function, we establish a cascade of increasingly complex transformation models that can be estimated, compared and analysed in the maximum likelihood framework. Models for the unconditional or conditional distribution function of any univariate response variable can be set up and estimated in the same theoretical and computational framework simply by choosing an appropriate transformation function and parameterization thereof. The ability to evaluate the distribution function directly allows us to estimate models based on the exact likelihood, especially in the presence of random censoring or truncation. For discrete and continuous responses, we establish the asymptotic normality of the proposed estimators. A reference software implementation of maximum likelihood‐based estimation for conditional transformation models that allows the same flexibility as the theory developed here was employed to illustrate the wide range of possible applications.     

Maximum Likelihood Estimation in Gaussian Chain Graph Models under the Alternative Markov Property

Abstract.  The Andersson–Madigan–Perlman (AMP) Markov property is a recently proposed alternative Markov property (AMP) for chain graphs. In the case of continuous variables with a joint multivariate Gaussian distribution, it is the AMP rather than the earlier introduced Lauritzen–Wermuth–Frydenberg Markov property that is coherent with data-generation by natural block-recursive regressions. In this paper, we show that maximum likelihood estimates in Gaussian AMP chain graph models can be obtained by combining generalized least squares and iterative proportional fitting to an iterative algorithm. In an appendix, we give useful convergence results for iterative partial maximization algorithms that apply in particular to the described algorithm.