This literature review discusses different methods under the general rubric of learning Bayesian networks from data, and includes some overlapping work on more general probabilistic networks. Connections are drawn between the statistical, neural network, and uncertainty communities, and between the different methodological communities, such as Bayesian, description length, and classical statistics. Basic concepts for learning and Bayesian networks are introduced and methods are then reviewed. Methods are discussed for learning parameters of a probabilistic network, for learning the structure, and for learning hidden variables. The presentation avoids formal definitions and theorems, as these are plentiful in the literature, and instead illustrates key concepts with simplified examples. Keywords--- Bayesian networks, graphical models, hidden variables, learning, learning structure, probabilistic networks, knowledge discovery. I. Introduction Probabilistic networks or probabilistic gra...

Undirected graphs and acyclic digraphs (ADGs), as well as their mutual extension to chain graphs, are widely used to describe dependencies 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. Whereas the undirected graph associated with a dependence model is uniquely determined, 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. Here it is shown that each Markov-equivalence class is uniquely determined by a single chain graph, the essential graph, that is itself simultaneously Markov equivalent to all ADGs in the equivalence class. Essential graphs are characterized, a polynomial-time algorithm for their construction is given, and their applications to model selection and other statistical

Graphical Markov models use graphs, either undirected, directed, or mixed, to represent possible dependences among statistical variables. Applications of undirected graphs (UDGs) include models for spatial dependence and image analysis, while acyclic directed graphs (ADGs), which are especially convenient for statistical analysis, arise in such fields as genetics and psychometrics and as models for expert systems and Bayesian belief networks. Lauritzen, Wermuth, and Frydenberg (LWF) introduced a Markov property for chain graphs, which are mixed graphs that can be used to represent simultaneously both causal and associative dependencies and which include both UDGs and ADGs as special cases. In this paper an alternative Markov property (AMP) for chain graphs is introduced, which in some ways is a more direct extension of the ADG Markov property than is the LWF property for chain graph. 1 INTRODUCTION Graphical Markov models use graphs, either undirected, directed, or mixed, to represent...

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 clas...

Two or more Bayesian network structures are Markov equivalent when the corresponding acyclic digraphs encode the same set of conditional independencies. Therefore, the search space of Bayesian network structures may be organized in equivalence classes, where each of them represents a different set of conditional independencies. The collection of sets of conditional independencies obeys a partial order, the so-called “inclusion order.” This paper discusses in depth the role that the inclusion order plays in learning the structure of Bayesian networks. In particular, this role involves the way a learning algorithm traverses the search space. We introduce a condition for traversal operators, the inclusion boundary condition, which, when it is satisfied, guarantees that the search strategy can avoid local maxima. This is proved under the assumptions that the data is sampled from a probability distribution which is faithful to an acyclic digraph, and the length of the sample is unbounded. The previous discussion leads to the design of a new traversal operator and two new learning algorithms in the context of heuristic search and the Markov Chain Monte Carlo method. We carry out a set of experiments with synthetic and real-world data that show empirically the benefit of striving for the inclusion order when learning Bayesian networks from data.

Chain graphs combine directed and undirected graphs and their underlying mathematics combines properties of the two. This paper gives a simplified definition of chain graphs based on a hierarchical combination of Bayesian (directed) and Markov (undirected) networks. Examples of a chain graph are multivariate feed-forward networks, clustering with conditional interaction between variables, and forms of Bayes classifiers. Chain graphs are then extended using the notation of plates so that samples and data analysis problems can be represented in a graphical model as well. Implications for learning are discussed in the conclusion. 1 Introduction Probabilistic networks are a notational device that allow one to abstract forms of probabilistic reasoning without getting lost in the mathematical detail of the underlying equations. They offer a framework whereby many forms of probabilistic reasoning can be combined and performed on probabilistic models without careful hand programming. Efforts ...

Abstract. Multivariate Gaussian graphical models are defined in terms of Markov properties, i.e., conditional independences associated with the underlying graph. Thus, model selection can be performed by testing these conditional independences, which are equivalent to specified zeroes among certain (partial) correlation coefficients. For concentration graphs, covariance graphs, acyclic directed graphs, and chain graphs (both LWF and AMP), we apply Fisher’s z-transformation, ˇ Sidák’s correlation inequality, and Holm’s step-down procedure, to simultaneously test the multiple hypotheses obtained from the Markov properties. This leads to a simple method for model selection that controls the overall error rate for incorrect edge inclusion. In practice, we advocate partitioning the simultaneous p-values into three disjoint sets, a significant set S, an indeterminate set I, and a non-significant set N. Then our SIN model selection method selects two graphs, a graph whose edges correspond to the union of S and I, and a more conservative graph whose edges correspond to S only. Prior information about the presence and/or absence of particular edges can be incorporated readily. 1.

This paper deals with chain graphs (CGs) which allow both directed and undirected edges. This class of graphs, introduced by Lauritzen and Wermuth [15], generalizes both UGs and DAGs. To establish the semantics of CGs one should associate an independency model to every CG. Some steps were already made. Lauritzen and Wermuth [16] intended to use CGs to describe independency models for strictly positive probability distributions and introduced the concept of the chain Markov property which is analogous to the concept of causal input list for DAGs. Lauritzen and Frydenberg [17, 9] generalized the concept of moral graph and introduced a moralization criterion for reading independency statements from a CG. Frydenberg [9] characterized CGs with the same Markov ON CHAIN GRAPH MODELS 3 property (that is producing the same CG-model) and Andersson, Madigan and Perlman [3] used special CGs to represent uniquely classes of Markov equivalent DAGs. Whittaker [31] in his book gave several examples of the use of CGs, and other recent works also deal with them [6, 20, 23, 30], the most comprehensive account is provided by the book [19]. Several results proved here were already presented (without proof) in our previous conference contribution [5]. An alternative approach to the generalization of UGs and DAGs was started by Cox and Wermuth [7] who introduced a wider class of joint-response chain graphs which allow also 'dashed' directed and undirected edges in addition to the classic 'solid' directed and undirected edges treated in this paper. Andersson, Madigan and Perlman [1] introduced an alternative Markov property to give an interpretation to those joint-response CGs which combine dashed directed edges with solid undirected edges (of course, another independency model is associated...

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 graphtheoretic 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 (exponentially large) equivalence class [D]. These results do not require the existence or positivity of joint densities. 1. Introduction. The use of directed graphs to represent possible dependencies among statistical variables dates back to Wright (1921) and has generated considerable research activity in the social and natural sciences. Since 1980, particular attention has been directed at

The application of certain Bayesian techniques, such as the Bayes factor and model averaging, requires the specification of prior distributions on the parameters of alternative models. We propose a new method for constructing compatible priors on the parameters of models nested in a given DAG (Directed Acyclic Graph) model, using a conditioning approach. We define a class of parameterisations consistent with the modular structure of the DAG and derive a procedure, invariant within this class, which we name reference conditioning.