D. Geiger and J. Pearl, Logical and algorithmic properties of conditional independence, to appear in: Proc. 2nd International Workshop on Art. Intel. and Stat. (Fort Lauderdale, Florida, 1989).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of efficient methods to only access the relevant portions of the database in query processing. A culminating result [4] is that acyclic join dependency (AJD) provides a basis for schema design as it possesses many desirable properties in database applications. Several researchers including [13], 21] 25] 40] have noticed similarities between relational databases and Bayesian networks. Here we advocate that a Bayesian network is indeed a generalized relational database. Our unified approach [42] 45] is to express the concepts used in Bayesian networks by generalizing the ....

....dependency. In 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 ....

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D. Geiger and J. Pearl, "Logical and algorithmic properties of conditional independence," Univ. California, Tech. Rep. R-97-II-L, 1989.

....This will be considered below in the definition of systems of information. We only know that from a mathematical point of view is a possible conditional information. For the definitions of dependence and conditional independence we follow the Pearl s idea of considering them primitive concepts, [14]. 11 Definition 3 Given a family of variables (X 1 ; Xn ) a dependence structure on it is a mapping D : f1; ng) Theta P (f1; ng) Theta P (f1; ng) Gamma f0; 1g where if D(I,J,K) 0 is said that (X i ) i2I is independent of (X k ) k2K given (X j ) j2J and verifying the ....

.... of variables (X 1 ; Xn ) a dependence structure on it is a mapping D : f1; ng) Theta P (f1; ng) Theta P (f1; ng) Gamma f0; 1g where if D(I,J,K) 0 is said that (X i ) i2I is independent of (X k ) k2K given (X j ) j2J and verifying the following axioms (see Geiger, Pearl,[14]) ffl Symmetry. If D(I ; J; K) 0 then D(K; J; I) 0 and viceverse. ffl Decomposition. If D(I ; J; K[L) 0 then D(I ; J; K) 0 and D(I ; J; L) 0 ffl Weak union. If D(I ; J; K [ L) 0 then D(I ; J [ L; K) 0 ffl Contraction. If D(I ; J; K) 0 and D(I ; J [ K;L) 0 then D(I ; J; K ....

D. Geiger, J. Pearl, Logical and algorithmic properties of conditional independence. Technical Report R-97, Cognitive Systems Laboratory, University of California, Los Angeles, 1988.

....Z in G then L does not linearly entail that X is independent of Y given Z. Proof. Suppose that X is not d separated from Y given Z. By Lemma 2, if X is not d separated from Y given Z in a cyclic graph G, then there is some acyclic subgraph G of G in which X is not d separated from Y given Z. Geiger and Pearl (1988) have shown that if X is not d separated fromY given Z in a directed acyclic graph, then there is some distribution represented by the directed acyclic graph in which X is not independent of Y given Z, and it has been shown (Spirtes, Glymour and Scheines, 1993) that there is in particular a linear ....

Geiger, D., and Pearl, J., (1988) Logical and Algorithmic properties of Conditional Independence. Technical Report R-97, Cognitive Systems Laboratory, University of California, Los Angeles.

.... firstly by finding a further property of stochastic CI [24] and finally by showing that the CI structures within the probabilistic framework cannot be characterized as dependency models closed under a finite number of inference rules [26] This result was lately strengthened by Geiger and Pearl [5] who showed that so called disjunctive inference rules cannot bring help. Another framework in which the concept of CI was introduced is Spohn s theory of ordinal conditional functions [22] This theory, motivated from a philosophical point of view, provides a tool for the mathematical ....

....consonant iff its marginals coincide, i.e. for each couple S; T 2 Z with S T 6= it holds that ( S ) S T = T ) S T . Supposing that N 6= is a finite set of attributes we shall say that a class ; 6= Z ae 7 Inference rules with one consequent are called in literature also Horn axioms [5]. exp N) n f0g is reduced iff each pair of its sets is incomparable 8 . Moreover, a class Z will be called solvable iff for each assignment of sets of possible states X i ; i 2 N (i.e. in any corresponding situation (S) every consonant system of NCFs f Z ; Z 2 Zg where Z is an NCF on X Z ....

Geiger, D., and Pearl, J., Logical and algorithmic properties of conditional independence and graphical models, to appear in Annals of Statistics.

....under 4 concerete inference rules) This hypothesis was supported by several partial results, in that some substructures of CI were characterized in this way. Independently Mat us [12] and Geiger, Paz, Pearl [3] characterized ordinary (unconditional) stochastic independence; Geiger and Pearl [4] and Malvestuto [10] independently found an axiomatization for the class of so called fixed context CI statements. Nevertheless, the original conjecture 1 This work has been supported by the internal grant n. 275105 of the Academy of Sciences of the Czech Republic Conditional independence ....

....This result has an analogy both in the stochastic case [26] and in the case of EMVD models in the theory of database relations [18] Firstly we give a construction allowing the generation of CI models in the NCF theory very simply. The result also has an analogy in the probabilistic case see [4], 26] Proposition 1 Having fixed a nonempty finite set of attributes N , the intersection of two CI models induced by NCFs is a CI model induced by an NCF. Proof: Suppose that two NCFs over N are given: 1 : XN f0; 1; g and 2 : YN f0; 1; g where XN = Q i2N X i ; YN ....

Geiger, D., and Pearl, J., Logical and algorithmic properties of conditional independence, in Proceedings of 2nd International Workshop on AI and Statistics, January 4--7, 1989, Fort Lauderdale, Florida.

....we omit definitions of usual graphical models, that is structural models induced by undirected graphs, acyclic directed graphs or chain graphs. Let us remark that all mentioned graphical models are positive probabilistic models [12] A basic construction of discrete probability distribution (see [4], Theorem 6) allows to show the following facts. For every pair of probability distributions P; Q over N there exists a distribution R over N such that A B j C [R] iff f A B j C [P ] A B j C [Q] g for every hA; BjCi 2 T (N ) If both P and Q is positive, then R can be chosen positive ....

Geiger D., and Pearl J., Logical and algorithmic properties of conditional independence and their application to Bayesian networks, Annals of Mathematics and Artificial Intelligence 2 (1990), pp. 165-178.

....Y are not d separated given C we say that X and Y are d connected given C. The definition of d separation of two nodes can be easily extended to the d separation of two disjoint sets of nodes. However, because it is not used in our algorithms, we do not give the definition here. It is proven in [Geiger and Pearl, 1988] that the concept of d separation can reveal all the conditional independence relations that are encoded in a Bayesian network. In other words, no other criterion can do better. When learning Bayesian networks from data, we use information theoretic measures to detect conditional independence ....

Geiger, D. and Pearl, J., Logical and algorithmic properties of conditional independence, Technical Report R-97, Cognitive Systems Laboratory, UCLA, 1988.

....model fhA; BjCi 2 T (N ) A B j C (P ) g called the independency model induced by P . An independency model is then called a (consistent) CI model over N if it is induced by some probability distribution over N . CI models have the following important property for proof see [21] or [5]. Lemma 3 The intersection of two CI models over N (that is set of CI statements belonging to both CI models) is also a CI model. As mentioned in the Introduction the purpose of the presented approach is to describe CI models in terms of so called inference rules. Definition 4 (inference rule, ....

D. Geiger and J. Pearl, Logical and algorithmic properties of conditional independence and their application to Bayesian networks, Ann. Math. Artif. Intell., 2 (1990) 165--178.

....is the model of CI structure of P iff I is exactly the set of triplets obeyed by P . This terminology emphasizes the presented view on dependency models. Note that authors dealing with dependency models have used also various another phrases: I is induced by P in [26] P is perfect for I in [4], I is conditional independence relation corresponding to P in [22] Owing to well known properties of CI (treated by Dawid [3] resp. Spohn [21] resp. Smith [19] some of dependency models cannot serve as (complete) models of CIstructures. Therefore Pearl and Paz [15] introduced the concept of ....

.... from [16] that semigraphoids coincide with the (complete) models of CI structures appeared untrue [22] Later, we even found that models of CI structures cannot be described as dependency models closed under finite number of inference rules [23] This was strengthened by Geiger and Pearl [4] who showed that disjunctive inference rules cannot bring help. These results led us to an attempt to develop an alternative way to description of CIstructures, namely by means of faces and imsets [25] The aim of this paper is to give an equivalent view on imsetal models of CI structures ....

D.Geiger and J.Pearl, Logical and algorithmic properties of conditional independence and graphical models, to appear in Annals of Statistics (1993?).

....a certain kind of formalized qualitative knowledge on the domain of interest. These impacts, together with a striving for meaningful mathematical models of conditional thinking in arti cial intelligence, have resulted in a considerable eoeort devoted to the study of CI structures, see [8] 10] [12], 13] 18] 21] 22] 23] 27] 25] 28] 32] 37] 38] 39] 41] 42] 43] and [46] In a very wide sense, CI relations are for us ternary relations M consisting of triples (I; J; K) of subsets I, J and K of a nite set N . The prevailing language for study of CI relations has been ....

....the global and local Markov properties of random elds over undirected graphs, see [7] 13] 18] 22] 32] 36] and [46] The material will be presented directly by graph theoretical notions. At the end we deal with the conditional independence relations of acyclic directed graphs, see [10] [12], 20] and [30] From now on we are going to work with local relations. With a little abuse of language we say that L ae R(N) is a semigraphoid if it satis es h (i; jjkL) 2 L and (i; kjL) 2 L i ) h (i; kjjL) 2 L and (i; jjL) 2 L i and a pseudographoid if it satis es h (i; jjkL) 2 L and ....

[Article contains additional citation context not shown here]

D. Geiger and J. Pearl (1988) Logical and algorithmic properties of conditional independence. Technical Report R-97, Cognitive Systems Laboratory, University of California, Los Angeles.

....changing, but feel free to grab a copy. If you have additions or corrections, please let me know. RG: This has NOT been updated since around Dec 96. 1 Network Representations Defined 1.1 General Properties of Conditional Independence Graphical Axioms of C.I. Daw79] Daw80] Spo80] GP88] Pea88b] Smi89] Smi90] Critical Graph Theoretical Properties: LSV84] Spe79] Hyper Markov laws: DL93b] HB94b] Mathematical properties of Markov Fields: Ish81] Lau82] Causal Independence: HB94b] 1.2 Bayesian Networks General overviews: Cha91] HH88] HBH91] HBH88] Pea87a] ....

Dan Geiger and Judea Pearl. Logical and algorithmic properties of conditional independence. Technical Report R-97, Cognitive Systems Laboratory, U.C.L.A., 1988.

.... (a special class of dependency models introduced by Pearl and Paz [1985] was refuted in [Studen y, 1989a] Later I even found that models of CI structures cannot be characterized as dependency models closed under finite number of inference rules [Studen y, 1992a] This was lately strengthened by Geiger and Pearl [1993] who showed that even so called disjunctive inference rules do not help. These results inspired me to attempt to develop an alternative mathematical description of CI structures for finite number of random variables, namely by means of so called structural faces. This theory, presented as a series ....

....[ i ] i2N iff it is exactly the set of triplets obeyed by [ i ] i2N . Remark Various another phrases were used in literature to say that I is the model of CI structure (P denotes the distribution of the vector [ i ] i2N ) I is induced by P in [Ur and Paz, 1991] P is perfect for I in [Geiger and Pearl, 1993], I is the conditional independence relation corresponding to P in [Studen y, 1992a] I is probabilistically representable by P in [Mat us, 1991] It is well known that every model of CI strusture is a semigraphoid (see for example [Dawid, 1979] but the converse does not hold [Studen y, ....

Geiger, D. and Pearl, J. [1993], "Logical and algorithmic properties of conditional independence and graphical models.", in draft.

....inference rules. INTRODUCTION Although CI ( conditional independence) was studied in modern statistics many years ago [Dawid, 1979] Mouchart and Rolin, 1984] its importance for probabilistic expert systems was explicitly discerned and highlighted relatively lately [Pearl, 1986] Smith, 1989] [Geiger and Pearl, 1989], Spiegelhalter and Lauritzen, 1990] Pearl and Paz [1985] proposed the concept of dependency model to describe structures of CI for finite number of random variables. Unfortunately, their original hypothesis (see also [Pearl, 1986] that models of CI structures coincide with semigraphoids (a ....

....class of all subsets of N will be denoted by exp N . Review the symbols for number sets : Z integers Z = Z h0; 1) nonnegative integers N strictly positive integers (natural numbers) x1 BASIC CONCEPTS Let s begin with the concept of dependency model and semigraphoid. The definition from [Geiger and Pearl, 1989] is slightly modified here. Def 1 (dependency model, semigraphoid) Denote by T (N) the set of ordered triplets hA; B; Ci where A; B; C are pairwise disjoint subsets of N and A; B nonempty. By dependency model over N we will understand every subset of T (N ) A dependency model over N is called ....

Geiger, D. and Pearl, J. [1989], "Logical and algorithmic properties of conditional independence." Proceedings of 2nd International Workshop on Artificial Intelligence and Statistics, January 4-7, 1989, Fort Lauderdale, Florida.

....case of general valuations. This process relys on the consideration of independence relationships among variables as primitive concepts, that may be associated to different uncertainty representations, and that may be known prior to any numerical relatioships among the variables in consideration, [5, 9, 15]. Here we give some of the basic ideas of the construction of a global valuation from elementary valuations. Let us assume that (X 1 ; X n ) is an n dimensional variable and that oe is a permutation on the set f1; ng, then a global valuation can be built by means of the combination ....

D. Geiger, J. Pearl, Logical and algorithmic properties of conditional independence. Technical Report R-97, Cognitive Systems Laboratory, University of California, Los Angeles, 1988.

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D. Geiger and J. Pearl, Logical and algorithmic properties of conditional independence, to appear in: Proc. 2nd International Workshop on Art. Intel. and Stat. (Fort Lauderdale, Florida, 1989).

No context found.

Geiger, D. and Pearl, J. (1993). Logical and Algorithmic Properties of Conditional Independence and Graphical Models. The Annals of Statistics, 21(4):2001-2021.

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Geiger, D. and Pearl, J. (1993). Logical and Algorithmic Properties of Conditional Independence and Graphical Models. The Annals of Statistics, 21(4):2001-2021.

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Geiger, D. and Pearl, J. (1993). Logical and Algorithmic Properties of Conditional Independence and Graphical Models. The Annals of Statistics, 21(4):2001-2021.

No context found.

D. Geiger and J. Pearl. Logical and algorithmic properties of conditional independence and graphical models. The Annals of Statistics, 21(4):2001.

No context found.

D. Geiger and J. Pearl. Logical and algorithmic properties of conditional independence. Technical Report R-97-II-L, University of California, 1989.

No context found.

D. Geiger and J. Pearl. Logical and algorithmic properties of conditional independence and graphical models. The Annals of Statistics, 21(4):2001.

No context found.

D. Geiger and J. Pearl. Logical and algorithmic properties of conditional independence. Technical Report R-97-II-L, University of California, 1989.

No context found.

Geiger, D. and Pearl, J., Logical and algorithmic properties of conditional independence, Technical Report R-97, Cognitive Systems Laboratory, UCLA, 1988.

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D. Geiger and J. Pearl, Logical and algorithmic properties of conditional independence, to appear in: Proc. 2nd International Workshop on Art. Intel. and Stat. (Fort Lauderdale, Florida, 1989).

No context found.

Geiger, D., and Pearl, J., 1988, Logical and Algorithmic properties of Conditional Independence. Technical Report R-97, Cognitive Systems Laboratory, University of California, Los Angeles.

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