Steen A. Andersson

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.

Bayesian model averaging and model selection for markov equivalence classes of acyclic digraphs

Acyclic digraphs (ADGs) are widely used to describe dependences among variables in multivariate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. There may, however, be many ADGs that determine the same dependence (= Markov) model. Thus, the family of all ADGs with a given set of vertices is naturally partitioned into Markov-equivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection or model averaging, that fail to take into account these equivalence classes, may incur substantial computational or other inefficiencies. Recent results have shown that each Markov-equivalence class is uniquely determined by a single chain graph, the essential graph, that is itself Markov-equivalent simultaneously to all ADGs in the equivalence class. Here we propose t...

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 the relation between conditional independence models determined by finite distributive lattices and by directed acyclic graphs

The relations among the classes of multivariate conditional independence models determined by directed acyclic graphs (DAG), undirected graphs (UDG), decomposable graphs (DEC), and finite distributive lattices (LCI) are investigated. First, LCI models that admit positive joint densities are characterized in terms of an appropriate factorization of the density. This factorization is then recognized as a particular form of the recursive factorization that characterizes DAG models, thereby establishing that the LCI models comprise a subclass of the class of DAG models. Precisely, the class of LCI models coincides with the subclass of transitive DAG models. Furthermore, the class of LCI models has nontrivial intersection with the class of DEC models. A series of examples illustrating these relations are presented.

A graphical characterization of lattice conditional independence models

Lattice conditional independence (LCI) models for multivariate normal data recently have been introduced for the analysis of non-monotone missing data patterns and of nonnested dependent linear regression models (≡ seemingly unrelated regressions). It is shown here that the class of LCI models coincides with a subclass of the class of graphical Markov models determined by acyclic digraphs (ADGs), namely, the subclass of transitive ADG models. An explicit graph-theoretic characterization of those ADGs that are Markov equivalent to some transitive ADG is obtained. This characterization allows one to determine whether a specific ADG D is Markov equivalent to some transitive ADG, hence to some LCI model, in polynomial time, without an exhaustive search of the (possibly superexponentially large) equivalence class [D]. These results do not require the existence or positivity of joint densities.

Lattice-ordered conditional independence models for missing data

Statistical inference for the parameters of a multivariate normal distribution Np([mu], [Sigma]) based on a sample with missing observations is straightforward when the missing data pattern is monotone (= nested), reducing to the analysis of several normal linear regression models by step-wise conditioning. When the missing data pattern is non-monotone, however, such analysis is impossible. It is shown here that every missing data pattern naturally determines a set of lattice-ordered conditional independence restrictions which, when imposed upon the unknown covariance matrix [Sigma], yields a factorization of the joint likelihood function as a product of (conditional) likelihood functions of normal linear regression models just as in the monotone case. From this factorization the maximum likelihood estimators of [mu] and [Sigma] (under the conditional independence restrictions) can be explicitly derived.

Characterizing Markov equivalence classes for AMP chain graph models

Chain graphs (CG) (= adicyclic graphs) use undirected and directed edges to represent both structural and associative dependences. Like acyclic directed graphs (ADGs), the CG associated with a statistical Markov model may not be unique, so CGs fall into Markov equivalence classes, which may be superexponentially large, leading to unidentifiability and computational inefficiency in model search and selection. It is shown here that, under the Andersson-Madigan-Perlman (AMP) interpretation of a CG, each Markov-equivalence class can be uniquely represented by a single distinguished CG, the AMP essential graph, that is itself simultaneously Markov equivalent to all CGs in the AMP Markov equivalence class. A complete characterization of AMP essential graphs is obtained. Like the essential graph previously introduced for ADGs, the AMP essential graph will play a fundamental role for inference and model search and selection for AMP CG models.

A Model for Ranking the Punitiveness of the States

There has recently been much interest in the measurement of imprisonment rates. Since this variable has such widespread importance in criminological research and policy, new methods are called for in expanding the procedures for evaluating levels of punitiveness as indicated by imprisonment rates. This paper presents a new model using logarithmic transformations to develop a system for ranking the punitiveness of the states. Comparisons are made between different approaches to specifying imprisonment rates including controls for crime rates and arrest rates. Results of the analyses indicate that the use of this model generates somewhat different rankings of punitiveness compared with those based on sample imprisonment rates or prisoner/arrest ratios.

On Riesz and Wishart distributions associated with decomposable undirected graphs

Classical Wishart distributions on the open convex cone of positive definite matrices and their fundamental features are extended to generalized Riesz and Wishart distributions associated with decomposable undirected graphs using the basic theory of exponential families. The families of these distributions are parameterized by their expectations/natural parameter and multivariate shape parameter and have a non-trivial overlap with the generalized Wishart distributions defined in Andersson and Wojnar (2004) [4,8]. This work also extends the Wishart distributions of type I in Letac and Massam (2007) [7] and, more importantly, presents an alternative point of view on the latter paper.

A test of social stress theory as applied to the study of imprisonment rates

The research presented in this article attempts to contribute to the ongoing study of the determinants of state imprisonment rates. Crime rates and unemployment rates have been studied extensively as they relate to imprisonment rates, and a number of demographic and social/structural factors have been analyzed as potential predictors of rates of imprisonment rates. In this study, the analysis of imprisonment rates was expanded through the consideration of other social stress variables and ameliorative factors that have been theorized to be associated with crime rates and imprisonment rates. The most interesting finding was that the variable prison costs, defined as the average cost per inmate per year, was found to have a significant effect on imprisonment rates. Percentage of Black population in the state and percentage of the population without health insurance were also found to have significant relationships with imprisonment rates, providing initial support for social stress theory of imprisonment.