Pearl, J. and Paz, A. (1987) Graphoids: a graph based logic for reasoning about relevancy relations. In Advance in Artificial Intelligence-II, 357-363. North-Holland.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... crucial inadequacy in common: They are relatively sensitive to imperfections within the examined area because most of these techniques require completely randomized samples as a precondition for their For a detailed discussion of the conditional independence property see the discussion in [17] and [18]. Parametric Models are characterized by certain assumptions (e.g. normally distributed sample, etc. Common approaches are the classical forms of Regression Analysis (linear or non linear; bi or multivariate) and the Analysis of Variance (ANOVA) application. In comparatively complex ....

J. Pearl and A. Paz, "Graphoids: A graph-based logic for reasoning about relevance relations", in B. du Bolay et al. (eds.), "Advances in artificial intelligence", vol. II, pp. 357-363, North Holland, Amsterdam, 1987.

....form independence relations embedded in joint probability distributions to allow for comparison and classification. J. Pearl and his co researchers were among the first to formalise properties of independence relations in an axiomatic system and to develop a logic for informational independence [Pearl Paz, 1985, Pearl Verma, 1987, Geiger Pearl, 1988] In this section, we review Pearl s axiomatic system. 2.1 Pearl s Axiomatic System for Informational Independence In the context of probability theory, the concept of independence is generally introduced in terms of numerical quantities: the ....

....independence. Any model in the set of models [A] of the system A is termed a semi graphoid independence relation; any model in [A ] is termed a graphoid independence relation. The terms graphoid and semi graphoid refer to the representation of independence relations in graphical structures [Pearl Paz, 1985, Pearl, 1988] Pearl s restricted axiomatic system for informational independence A serves to capture (at least) all independence relations that are embedded in a joint probability distribution; this property is stated more formally in the following lemma. Lemma 2.4 Let P be the set of all ....

J. Pearl and A. Paz (1985). GRAPHOIDS: a graph-based logic for reasoning about relevance relations, in: B. Du Boulay, D. Hogg, and L. Steels (eds). Advances in Artificial Intelligence 2, North-Holland.

....[10] instead. Still, those methods seem to have one crucial inadequacy in common: They are relatively sensitive to imperfections within the examined area because most of these techniques require For a detailed discussion of the conditional independence roperty see the discussion in [17] and [18]. Parametric Models are characterized by certain assumptions (e.g. normally distributed sample, etc. Common approaches are the classical forms of Regression Analysis (linear or nonlinear; bi or multivariate) and the Analysis of Variance (ANOVA) completely randomized samples as a precondifion ....

J. Pearl and A. Paz, "Graphoids: A graph-based logic for reasoning about relevance relations", in B. du Bolay et al. (eds.), "Advances in artificial intelligence", vol. II, pp. 357-363, North Holland, Amsterdam, 1987.

....the design and implementation of probabilistic reasoning or database systems, a crucial issue to consider is the implication problem. The implication problem has been extensively studied in both relational databases, including [2] 3] 24] 26] 27] and in Bayesian networks [13] 15] 30] [33], 36] 37] 41] 46] The implication problem is to test whether a given input set of independencies logically implies another independency . Traditionally, axiomatization was studied in an attempt to solve the implication problem for data and probabilistic conditional independencies. In this ....

J. Pearl and A. Paz, "Graphoids: Graph-based logic for reasoning about relevance relations," Univ. California, Tech. Rep. R-53-L, 1985.

....xed, would changing X alter Y Formal interest in this notion appears in the works of Suppes [8] and Salmon [7] who attempted to give it probabilistic interpretation. This paper pursues a logical approach, and starts with the work by Galles and Pearl [2] which, similar to the work on graphoids [5, 3], is based on a set of axioms and rules of inference that de nes a formal deductive system. In this system, causal relevance is expressed by logical formulas and new relevance sentences can be derived from old ones through rules of inference. Such deductive system requires an interpretation; that ....

J. Pearl and A. Paz (1987). Graphoids: A graphbased logic for reasoning about relevance relations. Advances in Articial Intelligence, vol. II. NorthHolland.

....4 The particular normalization we adopt is discussed in section 4. Theorem 1 The set of p and u independencies generated by any pair (p, u) has a perfect map. Proof. We follow the methodology of Bacchus and Grove (1995) that is, we appeal to a necessary and su#cient condition in Pearl and Paz (1989) and check that suitable generalizations of p independence and u independence both possess the following five properties: symmetry, decomposition, intersection, strong union and transitivity. We prove it in the case of utility; the proof for probability is analogous. Let A, B, C, D,R,R # , R ## ....

J. Pearl and A. Paz (1989), Graphoids: A GraphBased Logic for Reasoning About Relevance Relations. In B. Du Boulay (Ed.), Advances in Artificial Intelligence - II, North-Holland.

....measure satisfies the pairwise Markov property with respect to an undirected graph G if and only if it factorizes according to G. Proof: See Lauritzen (1996) 2 In fact, if (C5 ) holds, the global, local, and pairwise Markov properties coincide. This fact is stated in the theorem below, due to Pearl and Paz (1987); see also Pearl (1988) Theorem 1.4 (Pearl and Paz) If a probability distribution on X is such that (C5 ) holds for disjoint subsets A; B; C; D then (G) L) P) Proof: See Lauritzen (1996) 2 The global Markov property (G) is important because it gives a general criterion for deciding ....

Pearl, J. and Paz, A.: 1987, Graphoids: A graph based logic for reasoning about relevancy relations, in B. D. Boulay, D. Hogg and L. Steel (eds), Advances in Artificial Intelligence---II, North-Holland, Amsterdam, pp. 357--363.

....in which all dependencies (and, by default, all independencies) are clearly indicated by the arcs connecting the nodes. This property is so important that the name of independence networks has been suggested as the most adequate denomination for this scheme [14] There is a solid graph theory [19, 20] that constitutes the axiomatic framework of BNs. The purpose of this paper is to present BNs from the viewpoint of distributed computation. Our objective is not to make an exhaustive review of the field, but rather to show how knowledge representation and reasoning (evidence propagation) can be ....

J. Pearl and A. Paz. GRAPHOIDS: A graph-based logic for reasoning about relevance relations. Technical Report (R--53--L), Cognitive Systems Laboratory, University of California, Los Angeles, 1985.

....is the possibility distribution defined on corresponding to the graph. A belief network, then, represents the conditional independence relations that exist in a given domain. Now, conditional independence is a relationship between variables or groups of variables that has the following properties [36]: 1. Trivial independence: I(XjZj; 2. Symmetry: I(XjZjY ) I(Y jZjX) 3. Decomposition: I(XjZjY [ W ) I(XjZjY ) 4. Weak Union: I(XjZjY [ W ) I(XjZ [ Y jW ) 5. Contraction: I(XjZjY ) I(XjZ [ Y jW ) I(XjZjY [ W ) 6. Intersection: I(XjZ[W jY [W )I(XjZ[Y jW[W ) I(XjZjY [W ) This ....

J. Pearl and A. Paz. Graphoids: a graph-based logic for reasoning about relevance relations. Technical report, Cognitive Science Laboratory, Computer Science Department, University of California, Los Angeles, 1985.

....) P ( Y ) The notion of conditional independence provides, as already mentioned, the connection to a graph representation. It has been shown in general that a notion of conditional independence satisfying certain axioms, which are known as the semigraphoid axioms [Dawid, 1979; Spohn, 1980; Pearl and Paz, 1987; Smith, 1989] can be used to define a graph structure on the set of attributes. These axioms are symmetry: X Y j Z) Y X j Z) decomposition: W [ X Y j Z) W Y j Z) X Y j Z) weak union: W [ X Y j Z) X Y j Z [ W ) contraction: W Y j Z) X Y j Z [ W ) ....

J. Pearl and A. Paz. Graphoids: A Graph Based Logic for Reasoning about Relevance Relations. In: B. D. Boulay et al., eds. Advances in Artificial Intelligence 2, pp. 357-- 363. North Holland, Amsterdam, 1987.

....of Tunisia boutheina.yaghlane, khaled.mellouli ihec.rnu.tn 2 IRIDIA, Universite Libre de Bruxelles psmets ulb.ac.be Extended Abstract. The concept of conditional independence has been extensively studied in probability theory (see, for instance, 2] 3] 6] Pearl and Paz [7] have introduced some basic properties of the conditional independence relation, called graphoid axioms . These axioms are satisfied not only by probabilistic conditional independence, but also by embedded multi valued dependency models in relational databases [8] by conditional independence in ....

J. Pearl and A. Paz, Graphoids: Graph-Based Logic for Reasoning about Relevance Relations, in Boulay, B.D., D. Hogg, and L. Steele (eds.), Advances in Artificial Intelligence - II, NorthHolland, Amsterdam, 357-363, 1987.

....measure satisfies the pairwise Markov 9 property with respect to an undirected graph G if and only if it factorizes according to G. Proof See Lauritzen (1996) 2 In fact, if (C5 ) holds, the global, local, and pairwise Markov properties coincide. This fact is stated in the theorem below, due to Pearl and Paz (1987); see also Pearl (1988) Theorem 2 (Pearl and Paz) If a probability distribution on X is such that (C5 ) holds for disjoint subsets A; B; C; D then (G) L) P) Proof See Lauritzen (1996) 2 The global Markov property (G) is important because it gives a general criterion for deciding when ....

Pearl, J. and Paz, A. (1987). Graphoids: A graph based logic for reasoning about relevancy relations. In Advances in Artificial Intelligence---II, (ed. B. D. Boulay, D. Hogg, and L. Steel), pp. 357--63. North-Holland, Amsterdam.

.... mentioned above satisfy the semi graphoid axioms which have been established as basic requirements for any reasonable concept of conditional independence in graphical models [35] Possibilistic conditional independence derived from Dempster conditioning even satisfies the graphoid axioms [36], just as probabilistic conditional independence does. If we confine ourselves to conditional possibilistic non interactivity in accordance with the interpretation of possibility distributions we preferred above, it is straightforward to define conditional possibilistic independence graphs: An ....

J. Pearl and A. Paz. Graphoids --- A Graph Based Logic for Reasoning about Relevance Relations. In: B.D. Boulay et al., eds. Advances in Artificial Intelligence 2, 357--363. North-Holland, Amsterdam, Netherlands 1991

....Z; for any X and Z. An independency model over U is a set of independency statements. A complete independency model M I of a distribution P over U is the set of all valid independency statements in P . For positive definite distributions, the following axioms called independency axioms apply [1, 7, 8]. symmetry I(X; Z; Y ) I(Y; Z; X) decomposition I(X; Z; WY ) I(X; Z; Y ) weak union I(X; Z; WY ) I(X; ZW; Y ) contraction I(X; ZW; Y ) I(X; Z; W ) I(X; Z; WY ) intersection I(X; ZW; Y ) I(X; ZY; W ) I(X; Z; WY ) With these axioms independency statements can be derived from other ....

J. Pearl and A. Paz. Graphoids: a graph based logic for reasoning about relevance relations. In Proceedings ECAI, 1986.

....KEYWORDS: natural conditional function, conditional independence, axiomatic characterization, marginal problem, running intersection property. INTRODUCTION Several recent works in AI have dealt with the concept of irrelevance, in particular conditional irrelevance among attributes. Pearl and Paz [15] introduced the concept of a dependency model to describe such conditional irrelevance structures within various frameworks (undirected graphs, directed acyclic graphs, probability theory) In the probabilistic framework (we bear probabilistic reasoning in expert systems in mind) the conditional ....

.... independence (CI) among random variables (describing attributes) Although the concept of CI has been studied in probability theory and statistics for more than fifteen years [2, 21, 13, 17] its importance for probabilistic expert systems was highlighted relatively recently [14] Pearl and Paz [15] proposed describing CI structures in an axiomatic way, i.e. by means of a simple deductive mechanism handling information about the CI structure. They conjectured that the CI structures for strictly positive measures coincide with a special type of dependency models, namely graphoids (which ....

[Article contains additional citation context not shown here]

Pearl, J., and Paz, A., Graphoids: a graph--based logic for reasoning about relevance relations, in Advances in Artificial Intelligence -- II (B. Du Boulay et al., Eds.), North-- Holland, Amsterdam, 357--363, 1987.

....by Dawid [3] Some technical aspects were addressed in [5] Similar ideas were introduced at about the same time by Spohn [26] motivated by the problem of explicating probabilistic causation. More recently there has been an explosion of interest, following the demonstration by Pearl and Paz [21] of the connexions between abstract conditional independence and graphical models, with particular application to probabilistic expert systems [20] PROBABILISTIC CONDITIONAL INDEPENDENCE For random variables X, Y , Z on a probability space( Omega ; F ) and P a distribution on( Omega ; F ) ....

....then, with r jk denoting the sample correlation between X j and X k , we have that r 12 , r 13 and r 23 are pairwise (though not mutually) independent. INCOMPLETENESS OF THE AXIOMS . For a time it appeared plausible that all pure properties of probabilistic CI could be derived from P1 P5 alone [21]. However, Studen y [29] proved the following pure result for probabilistic conditional independence. Theorem 2 If X Y j(Z; W ) Z W jX, Z W jY and X Y ; then Z W j(X; Y ) X Y jZ, X Y jW , and Z W . The proof of Theorem 2 makes essential use of properties of probability distributions ....

Pearl, J. and Paz, A. (1987). Graphoids: a graph-based logic for reasoning about relevance relations. Advances in Artificial Intelligence II (B. du Boulay, D. Hogg and L. Steels, Eds.), 357--363. Amsterdam: NorthHolland. Introduces principles of graph-based reasoning with CI.

....ZY;W ) I(X; Z; Y W ) 1.5 (Intersection) I(X; ZY;W ) I(X; ZW;Y ) I(X; Z; Y W ) Intersection requires a strictly positive probability distribution. Figure 3: The graphoid axioms. These axioms, a special form of which was introduced in [Dawid, 1979] and [Spohn, 1980] were rediscovered by [Pearl and Paz, 1987] who conjectured them to be complete. The conjecture has been refuted by [Studeny, 1990] who also proved that conditional independence in probability theory has no finite axiomatization. Nevertheless, the graphoid axioms capture the most important features of informational relevance, Learning ....

J. Pearl and A. Paz. Graphoids: A graph-based logic for reasoning about relevance relations. In B. Du Boulay et. al., editor, Advances in Artificial Intelligence-II, pages 357--363. North-Holland Publishing Co., 1987.

....in Lemma 3.1. It follows from Frydenberg s analysis of chain graphs, which applies to strictly positive distributions. The more direct analysis of Verma and Pearl [11] renders the criterion applicable to arbitrary distributions, as well as to non probabilistic dependencies of the graphoid type [9]. 2 1.1 Problem Given a list M of conditional independence statements 2 ranging over a set of variables U it is required to decide whether there exists a directed acyclic graph (dag) D that is consistent with M . 1.2 Definitions A dependency model is a list of conditional independence ....

Judea Pearl and Azaria Paz. Graphoids: A graph-based logic for reasoning about relevance relations. In B. Du Boulay et al., editor, Advances in Artificial Intelligence-II, pages 357--363. North Holland, Amsterdam, 1986.

....when Z is known if P (x yjz) P (xjz) Delta P (yjz) for all possible values x; y; z of random variables X; Y; Z. We will say that a relation I T (V ) is induced by a distribution P over V if a triplet (X; Z; Y ) is in I if and only if X; Z and Y satisfy the above relation. The well known fact [4] is that any relation I induced by a probability distribution satisfies the properties (0) 4) above, and if the distribution P is strictly positive then the induced 4 Paz et al. Annotated graphs i i i i y z x w Gamma Gamma Gamma Gamma Gamma Gamma Gamma ....

....x and z and therefore the triplet (x; yw; z) is represented as well. No other triplet (except for symmetrical images of these two triplets and trivial triplets) is represented in the graph. Thus I(G) f (y; xz; w) x; yw; z) their symmetrical images trivial triplets g : Pearl and Paz [4] gave a characterization of the properties of ternary relations induced by UGs by means of properties of graphoid type. A relation can be represented by an UG if and only if it satisfies the following mutually independent axioms. Paz et al. Annotated graphs 5 (0) I( Z; Y ) Trivial Property ....

[Article contains additional citation context not shown here]

Pearl, J. and Paz, A.: "Graphoids: a graph-based logic for reasoning about relevance relations". B. du Boulay, D. Hoggs and L. Steels (eds.), Advances in Artificial Intelligence - volume II, North Holland, Amsterdam, 1987, 357--363.

....follows from Frydenberg s analysis of chain graphs, which applies to strictly positive distributions. The more direct analysis of Verma and Pearl [Verma and Pearl, 1990] renders the criterion applicable to arbitrary distributions, as well as to non probabilistic dependencies of the graphoid type [Pearl and Paz, 1986]. 1.2 Definitions A dependency model is a list of conditional independence statements of the form I(A; BjC) where A, B and C are disjoint subsets of some set of variables U . A dag D is consistent with a dependency model M if every statement in M and no statement outside M follows from the ....

Pearl, J. and Paz, A. (1986). Graphoids: A graph-based logic for reasoning about relevance relations. In et al., B. D. B., editor, Advances in Artificial Intelligence-II, pages 357--363. North Holland, Amsterdam.

No context found.

Pearl, J. and Paz, A. (1987) Graphoids: a graph based logic for reasoning about relevancy relations. In Advance in Artificial Intelligence-II, 357-363. North-Holland.

No context found.

Pearl, J. and Paz, A. (1987) Graphoids: a graph based logic for reasoning about relevancy relations. In Advance in Artificial Intelligence-II, 357-363. North-Holland.

No context found.

Pearl, J. and Paz, A. (1987) Graphoids: a graph based logic for reasoning about relevancy relations. In Advance in Artificial Intelligence-II, 357-363. North-Holland.

No context found.

J. Pearl and A. Paz. Graphoids: Graph-based logic for reasoning about relevance relations. Technical Report R-53-L, University of California, 1985.

No context found.

J. Pearl and A. Paz, Graphoids: a graph-based logic for reasoning about relevance relations. Technical Report. CSD850038. Cognitive Science Laboratory, Computer Science Department, University of California, Los Angeles, 1985.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC