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16
Markov equivalence for ancestral graphs
, 2004
"... Ancestral graph models can encode conditional independence relations that arise in directed acyclic graph (DAG) models with latent and selection variables. However, for any 3JJ.cestral graph, there may be several other graphs to which it is Markov equivalent. We state and prove conditions under whic ..."
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Cited by 16 (5 self)
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Ancestral graph models can encode conditional independence relations that arise in directed acyclic graph (DAG) models with latent and selection variables. However, for any 3JJ.cestral graph, there may be several other graphs to which it is Markov equivalent. We state and prove conditions under which two maximal ancestral graphs are Markov equivalent to each other, thereby extending analogous results for DAGs given by other authors. 'University of W2k'lhi.ng1;on Technical No. 466. Contents
2005) Towards Characterizing Markov Equivalence Classes of Directed Acyclic Graphs with Latent Variables. UAI
 Proceedings of the 21th Conference on Uncertainty in Artificial Intelligence, AUAI
, 2005
"... It is well known that there may be many causal explanations that are consistent with a given set of data. Recent work has been done to represent the common aspects of these explanations into one representation. In this paper, we address what is less well known: how do the relationships common to eve ..."
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Cited by 9 (5 self)
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It is well known that there may be many causal explanations that are consistent with a given set of data. Recent work has been done to represent the common aspects of these explanations into one representation. In this paper, we address what is less well known: how do the relationships common to every causal explanation among the observed variables of some DAG process change in the presence of latent variables? Ancestral graphs provide a class of graphs that can encode conditional independence relations that arise in DAG models with latent and selection variables. In this paper we present a set of orientation rules that construct the Markov equivalence class representative for ancestral graphs, given a member of the equivalence class. These rules are sound and complete. We also show that when the equivalence class includes a DAG, the equivalence class representative is the essential graph for the said DAG.
Estimating highdimensional intervention effects from observation data. The Ann
 of Stat
"... We assume that we have observational data generated from an unknown underlying directed acyclic graph (DAG) model. A DAG is typically not identifiable from observational data, but it is possible to consistently estimate the equivalence class of a DAG. Moreover, for any given DAG, causal effects can ..."
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Cited by 8 (2 self)
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We assume that we have observational data generated from an unknown underlying directed acyclic graph (DAG) model. A DAG is typically not identifiable from observational data, but it is possible to consistently estimate the equivalence class of a DAG. Moreover, for any given DAG, causal effects can be estimated using intervention calculus. In this paper, we combine these two parts. For each DAG in the estimated equivalence class, we use intervention calculus to estimate the causal effects of the covariates on the response. This yields a collection of estimated causal effects for each covariate. We show that the distinct values in this set can be consistently estimated by an algorithm that uses only local information of the graph. This local approach is computationally fast and feasible in highdimensional problems. We propose to use summary measures of the set of possible causal effects to determine variable importance. In particular, we use the minimum absolute value of this set, since that is a lower bound on the size of the causal effect. We demonstrate the merits of our methods in a simulation study and on a data set about riboflavin production. 1. Introduction. Our
Graphical methods for efficient likelihood inference in gaussian covariance models
 Journal of Machine Learning
, 2008
"... Abstract. In graphical modelling, a bidirected graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bidirected graph into a maximal ancestral graph that (i) represents the same independence structure as the origi ..."
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Cited by 8 (3 self)
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Abstract. In graphical modelling, a bidirected graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bidirected graph into a maximal ancestral graph that (i) represents the same independence structure as the original bidirected graph, and (ii) minimizes the number of arrowheads among all ancestral graphs satisfying (i). Here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bidirected edges. In Gaussian models, this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts. 1.
Causal reasoning with ancestral graphs
, 2008
"... Causal reasoning is primarily concerned with what would happen to a system under external interventions. In particular, we are often interested in predicting the probability distribution of some random variables that would result if some other variables were forced to take certain values. One promin ..."
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Cited by 6 (0 self)
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Causal reasoning is primarily concerned with what would happen to a system under external interventions. In particular, we are often interested in predicting the probability distribution of some random variables that would result if some other variables were forced to take certain values. One prominent approach to tackling this problem is based on causal Bayesian networks, using directed acyclic graphs as causal diagrams to relate postintervention probabilities to preintervention probabilities that are estimable from observational data. However, such causal diagrams are seldom fully testable given observational data. In consequence, many causal discovery algorithms based on datamining can only output an equivalence class of causal diagrams (rather than a single one). This paper is concerned with causal reasoning given an equivalence class of causal diagrams, represented by a (partial) ancestral graph. We present two main results. The first result extends Pearl (1995)’s celebrated docalculus to the context of ancestral graphs. In the second result, we focus on a key component of Pearl’s calculus—the property of invariance under interventions, and give stronger graphical conditions for this property than those implied by the first result. The second result also improves the earlier, similar results due to Spirtes et al. (1993).
A theoretical study of Y structures for causal discovery
 Proceedings of the Conference on Uncertainty in Artificial Intelligence
, 2006
"... Causal discovery from observational data in the presence of unobserved variables is challenging. Identification of socalled Y substructures is a sufficient condition for ascertaining some causal relations in the large sample limit, without the assumption of no hidden common causes. An example of a ..."
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Cited by 5 (3 self)
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Causal discovery from observational data in the presence of unobserved variables is challenging. Identification of socalled Y substructures is a sufficient condition for ascertaining some causal relations in the large sample limit, without the assumption of no hidden common causes. An example of a Y substructure is A → C, B → C, C → D. This paper describes the first asymptotically reliable and computationally feasible scorebased search for discrete Y structures that does not assume that there are no unobserved common causes. For any parameterization of a directed acyclic graph (DAG) that has scores with the property that any DAG that can represent the distribution beats any DAG that can’t, and for two DAGs that represent the distribution, if one has fewer parameters than the other, the one with the fewest parameter wins. In this framework there is no need to assign scores to causal structures with unobserved common causes. The paper also describes how the existence of a Y structure shows the presence of an unconfounded causal relation, without assuming that there are no hidden common causes. 1
P.: A transformational characterization of markov equivalence for directed acyclic graphs with latent variables
 In: Proc. of the 21st Conference on Uncertainty in Artificial Intelligence (UAI
, 2005
"... The conditional independence relations present in a data set usually admit multiple causal explanations — typically represented by directed graphs — which are Markov equivalent in that they entail the same conditional independence relations among the observed variables. Markov equivalence between di ..."
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Cited by 3 (1 self)
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The conditional independence relations present in a data set usually admit multiple causal explanations — typically represented by directed graphs — which are Markov equivalent in that they entail the same conditional independence relations among the observed variables. Markov equivalence between directed acyclic graphs (DAGs) has been characterized in various ways, each of which has been found useful for certain purposes. In particular, Chickering’s transformational characterization is useful in deriving properties shared by Markov equivalent DAGs, and, with certain generalization, is needed to justify a search procedure over Markov equivalence classes, known as the GES algorithm. Markov equivalence between DAGs with latent variables has also been characterized, in the spirit of Verma and Pearl (1990), via maximal ancestral graphs (MAGs). The latter can represent the observable conditional independence relations as well as some causal features of DAG models with latent variables. However, no characterization of Markov equivalent MAGs is yet available that is analogous to the transformational characterization for Markov equivalent DAGs. The main contribution of the current paper is to establish such a characterization for directed MAGs, which we expect will have similar uses as Chickering’s characterization does for DAGs. 1
Orientation rules for constructing markov equivalence classes for maximal ancestral graphs
, 2005
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Maximum likelihood fitting of acyclic directed mixed graphs to binary data
 Proceedings of the 26th International Conference on Uncertainty in Artificial Intelligence
, 2010
"... mixed graphs to binary data ..."
Learning semimarkovian causal models using experiments
 IN: PROCEEDINGS OF THE THIRD EUROPEAN WORKSHOP ON PROBABILISTIC GRAPHICAL MODELS , PGM
, 2006
"... SemiMarkovian causal models (SMCMs) are an extension of causal Bayesian networks for modeling problems with latent variables. However, there is a big gap between the SMCMs used in theoretical studies and the models that can be learned from observational data alone. The result of standard algorithms ..."
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Cited by 1 (1 self)
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SemiMarkovian causal models (SMCMs) are an extension of causal Bayesian networks for modeling problems with latent variables. However, there is a big gap between the SMCMs used in theoretical studies and the models that can be learned from observational data alone. The result of standard algorithms for learning from observations, is a complete partially ancestral graph (CPAG), representing the Markov equivalence class of maximal ancestral graphs (MAGs). In MAGs not all edges can be interpreted as immediate causal relationships. In order to apply stateoftheart causal inference techniques we need to completely orient the learned CPAG and to transform the result into a SMCM by removing noncausal edges. In this paper we combine recent work on MAG structure learning from observational data with causal learning from experiments in order to achieve that goal. More specifically, we provide a set of rules that indicate which experiments are needed in order to transform a CPAG to a completely oriented SMCM and how the results of these experiments have to be processed. We will propose an alternative representation for SMCMs that can easily be parametrised and where the parameters can be learned with classical methods. Finally, we show how this parametrisation can be used to develop methods to efficiently perform both probabilistic and causal inference.