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28
Logical and algorithmic properties of conditional independence and graphical models
 THE ANNALS OF STATISTICS
, 1993
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Axioms of Causal Relevance
 Artificial Intelligence
, 1996
"... This paper develops axioms and formal semantics for statements of the form "X is causally irrelevant to Y in context Z," which we interpret to mean "Changing X will not affect Y if we hold Z constant." The axiomization of causal irrelevance is contrasted with the axiomization of informational irr ..."
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Cited by 52 (13 self)
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This paper develops axioms and formal semantics for statements of the form "X is causally irrelevant to Y in context Z," which we interpret to mean "Changing X will not affect Y if we hold Z constant." The axiomization of causal irrelevance is contrasted with the axiomization of informational irrelevance, as in "Learning X will not alter our belief in Y , once we know Z." Two versions of causal irrelevance are analyzed, probabilistic and deterministic. We show that, unless stability is assumed, the probabilistic definition yields a very loose structure, that is governed by just two trivial axioms. Under the stability assumption, probabilistic causal irrelevance is isomorphic to path interception in cyclic graphs. Under the deterministic definition, causal irrelevance complies with all of the axioms of path interception in cyclic graphs, with the exception of transitivity. We compare our formalism to that of [Lewis, 1973], and offer a graphical method of proving theorems abou...
The Multiinformation Function As A Tool For Measuring Stochastic Dependence
 Learning in Graphical Models
, 1998
"... . Given a collection of random variables [¸ i ] i2N where N is a finite nonempty set, the corresponding multiinformation function ascribes the relative entropy of the joint distribution of [¸ i ] i2A with respect to the product of distributions of individual random variables ¸ i for i 2 A to every s ..."
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Cited by 32 (0 self)
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. Given a collection of random variables [¸ i ] i2N where N is a finite nonempty set, the corresponding multiinformation function ascribes the relative entropy of the joint distribution of [¸ i ] i2A with respect to the product of distributions of individual random variables ¸ i for i 2 A to every subset A ae N . We argue it is a useful tool for problems concerning stochastic (conditional) dependence and independence (at least in discrete case). First, it makes possible to express the conditional mutual information between [¸ i ] i2A and [¸ i ] i2B given [¸ i ] i2C (for every disjoint A; B; C ae N) which can be considered as a good measure of conditional stochastic dependence. Second, one can introduce reasonable measures of dependence of level r among variables [¸ i ] i2A (where A ae N , 1 r ! card A) which are expressible by means of the multiinformation function. Third, it enables one to derive theoretical results on (nonexistence of an) axiomatic characterization of stochastic c...
Formal Properties of Conditional Independence in Different Calculi of AI
"... In this paper formal properties of CI in different frameworks are studied. The first part is devoted to the comparison of three different frameworks for study CI: probability theory, theory of relational databases and Spohn's theory of ordinal conditional functions. Although CImodels arising in the ..."
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Cited by 22 (5 self)
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In this paper formal properties of CI in different frameworks are studied. The first part is devoted to the comparison of three different frameworks for study CI: probability theory, theory of relational databases and Spohn's theory of ordinal conditional functions. Although CImodels arising in these frameworks are very similar (they satisfy semigraphoid axioms) we give examples showing that their formal properties still differ (each other). On the other hand, we find that (within each of these frameworks) there exists no finite complete axiomatic characterization of CImodels by finding an infinite set of sound inference rules (the same in all three frameworks). In the second part further frameworks for CI are discussed: DempsterShafer theory, possibility theory and (general) Shenoy's theory of valuationbased systems.
Independency relationships and learning algorithms for singly connected networks
, 1998
"... Graphical structures such as Bayesian networks or M arkov networks are very useful tools for representing irrelevance or independency relationships, and they may be used to efficiently perform reasoning tasks. Singly connected networks are important specific cases where there is no more than one un ..."
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Cited by 18 (10 self)
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Graphical structures such as Bayesian networks or M arkov networks are very useful tools for representing irrelevance or independency relationships, and they may be used to efficiently perform reasoning tasks. Singly connected networks are important specific cases where there is no more than one undirected path connecting each pair of variables. The aim of this paper is to investigate the kind of properties that a dependency model must verify in order to be equivalent to a singly connected graph structure, as a way of driving automated discovery and construction of singly connected networks in data. The main results are the characterizations of those dependency models which are isomorphic to singly connected graphs (either via the dseparation criterion for directed acyclic graphs or via the separation criterion for undirected graphs), as well as the development of efficient algorithms for learning singly connected graph representations of dependency models.
Efficient markov network structure discovery using independence tests
 In Proc SIAM Data Mining
, 2006
"... We present two algorithms for learning the structure of a Markov network from discrete data: GSMN and GSIMN. Both algorithms use statistical conditional independence tests on data to infer the structure by successively constraining the set of structures consistent with the results of these tests. GS ..."
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Cited by 16 (1 self)
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We present two algorithms for learning the structure of a Markov network from discrete data: GSMN and GSIMN. Both algorithms use statistical conditional independence tests on data to infer the structure by successively constraining the set of structures consistent with the results of these tests. GSMN is a natural adaptation of the GrowShrink algorithm of Margaritis and Thrun for learning the structure of Bayesian networks. GSIMN extends GSMN by additionally exploiting Pearl’s wellknown properties of conditional independence relations to infer novel independencies from known independencies, thus avoiding the need to perform these tests. Experiments on artificial and real data sets show GSIMN can yield savings of up to 70 % with respect to GSMN, while generating a Markov network with comparable or in several cases considerably improved quality. In addition
An Algorithm for Finding Minimum dSeparating Sets in Belief Networks
 Proceedings of the twelfth Conference of Uncertainty in Artificial Intelligence
, 1996
"... The criterion commonly used in directed acyclic graphs (dags) for testing graphical independence is the wellknown dseparation criterion. It allows us to build graphical representations of dependency models (usually probabilistic dependency models) in the form of belief networks, which make possibl ..."
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Cited by 14 (4 self)
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The criterion commonly used in directed acyclic graphs (dags) for testing graphical independence is the wellknown dseparation criterion. It allows us to build graphical representations of dependency models (usually probabilistic dependency models) in the form of belief networks, which make possible an easy interpretation and management of independence relationships, without reference to numerical parameters (conditional probabilities). In this paper we study the following combinatorial problem: to find the minimum dseparating set for two nodes in a dag. This set would represent the minimum information necessary to prevent these two nodes to influence each other. The solution of this basic problem and of some of its extensions can be useful in several ways, as we will see later. Our solution is based on a twosteps process: first, we reduce the original problem to the simpler one of finding a minimum separating set in an undirected graph, and second, we develop an algorithm for solvi...
Conditional Independence
, 1997
"... This article has been prepared as an entry for the Wiley Encyclopedia of Statistical Sciences (Update). It gives a brief overview of fundamental properties and applications of conditional independence. ESS Update A. P. Dawid Conditional Independence Ancillarity; axioms; graphical models; markov pro ..."
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Cited by 14 (0 self)
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This article has been prepared as an entry for the Wiley Encyclopedia of Statistical Sciences (Update). It gives a brief overview of fundamental properties and applications of conditional independence. ESS Update A. P. Dawid Conditional Independence Ancillarity; axioms; graphical models; markov properties; sufficiency. The concepts of independence and conditional independence (CI) between random variables originate in Probability Theory, where they are introduced as properties of an underlying probability measure P on the sample space (see CONDITIONAL PROBABILITY AND EXPECTATION). Much of traditional Probability Theory and Statistics involves analysis of distributions having such properties: for example, limit theorems for independent and identically distributed variables, or the theory of MARKOV PROCESSES. More recently, it has become apparent that it is fruitful to treat conditional independence (and its special case independence) as a primitive concept, with an intuitive meaning, ...
Bayesian Nets Are All There Is To Causal Dependence
 STOCHASTIC DEPENDENCE AND CAUSALITY, CSLI
, 2001
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Semigraphoids and Structures of Probabilistic Conditional Independence
, 1997
"... this paper, the semigraphoid closure of every couple of CIstatements is proved to be a CImodel. The substantial step to it is to show that every probabilistically sound inference rule for axiomatic characterization of CI properties (= axiom), having at most two antecedents, is a consequence of the ..."
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Cited by 12 (0 self)
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this paper, the semigraphoid closure of every couple of CIstatements is proved to be a CImodel. The substantial step to it is to show that every probabilistically sound inference rule for axiomatic characterization of CI properties (= axiom), having at most two antecedents, is a consequence of the semigraphoid inference rules. Moreover, all potential dominant triplets of the mentioned semigraphoid closure are found.