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.     

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.     

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.     

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.     

Formulas for counting acyclic digraph Markov equivalence classes

Multivariate Markov dependencies between different variables often can be represented graphically using acyclic digraphs (ADGs). In certain cases, though, different ADGs represent the same statistical model, thus leading to a set of equivalence classes of ADGs that constitute the true universe of available graphical models. Building upon the previously known formulas for counting the number of acyclic digraphs and the number of equivalence classes of size 1, formulas are developed to count ADG equivalence classes of arbitrary size, based on the chordal graph configurations that produce a class of that size. Theorems to validate the formulas as well as to aid in determining the appropriate chordal graphs to use for a given class size are included.     

A SINful approach to Gaussian graphical model selection

Multivariate Gaussian graphical models are defined in terms of Markov properties, i.e., conditional independences, corresponding to missing edges in the graph. Thus model selection can be accomplished by testing these independences, which are equivalent to zero values of corresponding partial correlation coefficients. For concentration graphs, acyclic directed graphs, and chain graphs (both LWF and AMP classes), we apply Fisher's z-transform, Šidák's correlation inequality, and Holm's step-down procedure to simultaneously test the multiple hypotheses specified by these zero values. This simple method for model selection controls the overall error rate for incorrect edge inclusion. Prior information about the presence and/or absence of particular edges can be readily incorporated.     

On the Validity of the Markov Interpretation of Path Diagrams of Gaussian Structural Equations Systems with Correlated Errors

Pearl's d -separation concept and the ensuing Markov property is applied to graphs which may have, between each two different vertices i and j , any subset of { i ← j , i → j , i ↔ j } as edges. The class of graphs so obtained is closed under marginalization. Furthermore, the approach permits a direct proof of this theorem: "The distribution of a multivariate normal random vector satisfying a system of linear simultaneous equations is Markov w.r.t. the path diagram of the linear system".     

Flowgraph Models for Generalized Phase Type Distributions Having Non-Exponential Waiting Times

ABSTRACT. Aalen (1995) introduced phase type distributions based on Markov processes for modelling disease progression in survival analysis. For tractability and to maintain the Markov property, these use exponential waiting times for transitions between states. This article extends the work of Aalen (1995) by generalizing these models to semi-Markov processes with non-exponential waiting times. The generalization allows more realistic modelling of the stages of a disease where the Markov property and exponential waiting times may not hold. Flowgraph models are introduced to provide a closed form for the distributions in situations involving non-exponential waiting times. Flowgraph models work where traditional methods of stochastic processes are intractable. Saddlepoint approximations are used in the analysis. Together, generalized phase type distributions, flowgraphs, and saddlepoint approximations create exciting and innovative prospects for the analysis of survival data.     

Markov Properties for Acyclic Directed Mixed Graphs

We consider acyclic directed mixed graphs, in which directed edges ( x → y ) and bi-directed edges ( x ↔ y ) may occur. A simple extension of Pearl's d -separation criterion, called m -separation, is applied to these graphs. We introduce a local Markov property which is equivalent to the global property resulting from the m -separation criterion for arbitrary distributions.     

Bayes Estimates of Markov Trends in Possibly Cointegrated Series

Stylized facts show that average growth rates of U.S. per capita consumption and income differ in recession and expansion periods. Because a linear combination of such series does not have to be a constant mean process, standard cointegration analysis between the variables to examine the permanent income hypothesis may not be valid. To model the changing growth rates in both series, we introduce a multivariate Markov trend model that accounts for different growth rates in consumption and income during expansions and recessions and across variables within both regimes. The deviations from the multivariate Markov trend are modeled by a vector autoregression (VAR) model. Bayes estimates of this model are obtained using Markov chain Monte Carlo methods. The empirical results suggest the existence of a cointegration relation between U.S. per capita disposable income and consumption, after correction for a multivariate Markov trend. This result is also obtained when per capita investment is added to the VAR.     

Nonparametric Bayesian analysis for multi-site hidden Markov model

The hidden Markov model (HMM) provides an attractive framework for modeling long-term persistence in a variety of applications including pattern recognition. Unlike typical mixture models, hidden Markov states can represent the heterogeneity in data and it can be extended to a multivariate case using a hierarchical Bayesian approach. This article provides a nonparametric Bayesian modeling approach to the multi-site HMM by considering stick-breaking priors for each row of an infinite state transition matrix. This extension has many advantages over a parametric HMM. For example, it can provide more flexible information for identifying the structure of the HMM than parametric HMM analysis, such as the number of states in HMM. We exploit a simulation example and a real dataset to evaluate the proposed approach.     

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.     

Collapsibility of Graphical CG-Regression Models

Abstract.  CG-regressions are multivariate regression models for mixed continuous and discrete responses that result from conditioning in the class of conditional Gaussian (CG) models. Their conditional independence structure can be read off a marked graph. The property of collapsibility, in this context, means that the multivariate CG-regression can be decomposed into lower dimensional regressions that are still CG and are consistent with the corresponding subgraphs. We derive conditions for this property that can easily be checked on the graph, and indicate computational advantages of this kind of collapsibility. Further, a simple graphical condition is given for checking whether a decomposition into univariate regressions is possible.     

Parameterizations and Fitting of Bi-directed Graph Models to Categorical Data

Abstract.  We discuss two parameterizations of models for marginal independencies for discrete distributions which are representable by bi-directed graph models, under the global Markov property. Such models are useful data analytic tools especially if used in combination with other graphical models. The first parameterization, in the saturated case, is also known as thenation multivariate logistic transformation, the second is a variant that allows, in some (but not all) cases, variation-independent parameters. An algorithm for maximum likelihood fitting is proposed, based on an extension of the Aitchison and Silvey method.     

Hierarchical Bayesian Approach to a Multi-Site Hidden Markov Model

Multivariate data with a sequential or temporal structure occur in various fields of study. The hidden Markov model (HMM) provides an attractive framework for modeling long-term persistence in areas of pattern recognition through the extension of independent and identically distributed mixture models. Unlike in typical mixture models, the heterogeneity of data is represented by hidden Markov states. This article extends the HMM to a multi-site or multivariate case by taking a hierarchical Bayesian approach. This extension has many advantages over a single-site HMM. For example, it can provide more information for identifying the structure of the HMM than a single-site analysis. We evaluate the proposed approach by exploiting a spatial correlation that depends on the distance between sites.     

Fitting Gaussian Markov Random Fields to Gaussian Fields

This paper discusses the following task often encountered in building Bayesian spatial models: construct a homogeneous Gaussian Markov random field (GMRF) on a lattice with correlation properties either as present in some observed data, or consistent with prior knowledge. The Markov property is essential in designing computationally efficient Markov chain Monte Carlo algorithms to analyse such models. We argue that we can restate both tasks as that of fitting a GMRF to a prescribed stationary Gaussian field on a lattice when both local and global properties are important. We demonstrate that using the KullbackLeibler discrepancy often fails for this task, giving severely undesirable behaviour of the correlation function for lags outside the neighbourhood. We propose a new criterion that resolves this difficulty, and demonstrate that GMRFs with small neighbourhoods can approximate Gaussian fields surprisingly well even with long correlation lengths. Finally, we discuss implications of our findings for likelihood based inference for general Markov random fields when global properties are also important.     

The definition of tree-indexed Markov chains in random environment and their existence

This paper gives the definition of tree-indexed Markov chains in random environment with discrete state space, and then studies some equivalent theorems of tree-indexed Markov chains in random environment. Finally, we give the equivalence on tree-indexed Markov chains in Markov environment and double Markov chains indexed by a tree.     

Markov Chain Markov Field dynamics: Models and statistics

This study deals with time dynamics of Markov fields defined on a finite set of sites with state space <$>E<$>, focussing on Markov Chain Markov Field (MCMF) evolution. Such a model is characterized by two families of potentials: the instantaneous interaction potentials, and the time delay potentials. Four models are specified: auto-exponential dynamics (<$>E = {\of R}^+<$>), auto-normal dynamics (<$>E = {\of R}<$>), auto-Poissonian dynamics (<$>E = {\of N}<$>) and auto-logistic dynamics ( E qualitative and finite). Sufficient conditions ensuring ergodicity and strong law of large numbers are given by using a Lyapunov criterion of stability, and the conditional pseudo-likelihood statistics are summarized. We discuss the identification procedure of the two Markovian graphs and look for validation tests using martingale central limit theorems. An application to meteorological data illustrates such a modelling.     

Classical and Bayesian Analysis of Univariate and Multivariate Stochastic Volatility Models

In this paper, efficient importance sampling (EIS) is used to perform a classical and Bayesian analysis of univariate and multivariate stochastic volatility (SV) models for financial return series. EIS provides a highly generic and very accurate procedure for the Monte Carlo (MC) evaluation of high-dimensional interdependent integrals. It can be used to carry out ML-estimation of SV models as well as simulation smoothing where the latent volatilities are sampled at once. Based on this EIS simulation smoother, a Bayesian Markov chain Monte Carlo (MCMC) posterior analysis of the parameters of SV models can be performed.     

A New Model for Multivariate Markov Chains

We propose a new model for multivariate Markov chains of order one or higher on the basis of the mixture transition distribution (MTD) model. We call it the MTD‐Probit. The proposed model presents two attractive features: it is completely free of constraints, thereby facilitating the estimation procedure, and it is more precise at estimating the transition probabilities of a multivariate or higher‐order Markov chain than the standard MTD model.