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    "text": "Reading Dependencies from Polytree-Like Bayesian Networks Pe˜na\nDivision of Computational Biology\nDepartment of Physics, Chemistry and Biology\nLink¨oping University, SE-58183 Link¨oping, Sweden",
    "paper_id": "1206.5263",
    "title": "Reading Dependencies from Polytree-Like Bayesian Networks",
    "authors": [
      "Jose M. Pena"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
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    "text": "Abstract Section 3 that assuming composition and weak transitivity is not too restrictive. We present our graphical\nWe present a graphical criterion for reading criterion in Section 4 and prove that it is sound and\ndependencies from the minimal directed in- complete. Finally, we close with some discussion in\ndependence map G of a graphoid p when G Section 5.\nis a polytree and p satisfies composition and\nweak transitivity. We prove that the crite- 2 PRELIMINARIES\nrion is sound and complete. We argue that\nassuming composition and weak transitivity The definitions and results in this section are taken\nis not too restrictive. from [Lauritzen, 1996, Pearl, 1988, Studen´y, 2005]. We use the juxtaposition XY to denote X ∪Y, and\nX to denote the singleton {X}. Let U denote a set\n1 INTRODUCTION\nof random variables.",
    "paper_id": "1206.5263",
    "title": "Reading Dependencies from Polytree-Like Bayesian Networks",
    "authors": [
      "Jose M. Pena"
    ],
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    "text": "Unless otherwise stated, all the\nindependence models and graphs in this paper are deA minimal directed independence map G of an indefined over U.\npendence model p is used to read independencies that\nhold in p. Sometimes, however, G can also be used to Let X, Y, Z and W denote four mutually disjoint\nread dependencies holding in p. For instance, if p is a subsets of U. An independence model p is a set of\ngraphoid that is faithful to G then, by definition, lack independencies of the form X is independent of Y\nof vertex separation is a sound and complete graphi- given Z. We represent that an independence is in\ncal criterion for reading dependencies from G. If p is p by X ⊥⊥Y|Z and that an independence is not in\nsimply a graphoid, then there also exists a sound and p by X ̸⊥⊥Y|Z. In the latter case, we may equicomplete graphical criterion for reading dependencies valently say that the dependence X ̸⊥⊥Y|Z is in p.\nfrom G [Bouckaert, 1995]. However, this criterion can- An independence model is a graphoid when it satisfies\nnot be applied to check whether two set of nodes X the following five properties: Symmetry X ⊥⊥Y|Z ⇒\nand Y are dependant given a third set Z unless X and Y ⊥⊥X|Z, decomposition X ⊥⊥YW|Z ⇒X ⊥⊥Y|Z,\nY are adjacent in G, i.e. unless there is an edge in weak union X ⊥⊥YW|Z ⇒X ⊥⊥Y|ZW, contraction\nG from some A ∈X to some B ∈Y. In this paper, X⊥⊥Y|ZW ∧X⊥⊥W|Z ⇒X⊥⊥YW|Z, and intersecwe try to overcome this shortcoming so that X and tion X ⊥⊥Y|ZW ∧X ⊥⊥W|ZY ⇒X ⊥⊥YW|Z. Any\nY are not required to be adjacent in G. We do so by strictly positive probability distribution is a graphoid.\nconstraining G and p. Specifically, we present a sound\nLet sep(X, Y|Z) denote that X is separated from Y\nand complete graphical criterion for reading dependengiven Z in a graph G. Specifically, sep(X, Y|Z) holds\ncies from G under the assumptions that G is a polytree\nwhen every path in G between X and Y is blocked\nand p is a graphoid that satisfies composition and weak\nby Z. If G is an undirected graph (UG), then a path\ntransitivity. We argue that assuming composition and\nin G between X and Y is blocked by Z when there\nweak transitivity is not too restrictive.",
    "paper_id": "1206.5263",
    "title": "Reading Dependencies from Polytree-Like Bayesian Networks",
    "authors": [
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    "text": "Specifically,\nexists some Z ∈Z in the path. We say that a node is\nwe show that there exist important families of probaa head-to-head node in a path if it has two parents in\nbility distributions, among them Gaussian probability\nthe path. If G is a directed and acyclic graph (DAG),\ndistributions, that satisfy these two properties.\nthen a path in G between X and Y is blocked by Z\nThe rest of the paper is organized as follows. We start when there exists a node Z in the path such that eiby reviewing some concepts in Section 2. We show in ther (i) Z is not a head-to-head node in the path and Z ∈Z, or (ii) Z is a head-to-head node in the path V ∈U \\ (XYZ). We now argue that there exist imand neither Z nor any of its descendants in G is in Z. portant families of probability distributions that are\nAn independence model p is faithful to an UG or DAG CWT graphoids and, thus, that WT graphoids are\nG when X⊥⊥Y|Z iffsep(X, Y|Z). Any independence worth studying.",
    "paper_id": "1206.5263",
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    "text": "For instance, any probability distrimodel that is faithful to some UG or DAG is a gra- bution that is Gaussian or faithful to some UG or DAG\nphoid. DAG) G is an undirected (resp. is a CWT graphoid [Pearl, 1988, Studen´y, 2005]. The\ndirected) independence map of an independence model theorem below proves that the marginals and condip when X⊥⊥Y|Z if sep(X, Y|Z). Moreover, G is a mi- tionals of a probability distribution that is a CWT\nnimal undirected (resp. directed) independence map graphoid are CWT graphoids, although they may be\nof p when removing any edge from G makes it cease to neither Gaussian nor faithful to any UG or DAG. We\nbe an independence map of p. We abbreviate minimal give an example at the end of this section.",
    "paper_id": "1206.5263",
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    "text": "We refer\nundirected (resp. directed) independence map as MUI the reader to [Pe˜na et al., 2006a, Pe˜na et al., 2006b,\n(resp. If G is a MUI map of p, then two Pe˜na et al., 2007] for the proofs of the theorems in this\nnodes X and Y are adjacent in G iffX ̸⊥⊥Y |U\\(XY ). section. On the other hand, if G is a MDI map of p, then the\nparents of a node X in G, Pa(X), are the smallest sub- Theorem 1 Let p be a probability distribution that is\nset of the nodes preceding X in a given total ordering a CWT graphoid and let W ⊆U. Then, p(U \\ W) is\nof U, Pre(X), such that X ⊥⊥Pre(X)\\Pa(X)|Pa(X). a CWT graphoid. If p(U \\ W|W = w) has the same\nWe denote the children of a node X by Ch(X). independencies for all w, then p(U \\ W|W = w) for\nany w is a CWT graphoid. Let Cl denote the set of cliques of an UG G. A\nMarkov network (MN) is a pair (G, θ) where G is an\nHereinafter, we say that a probability distribution p\nUG and θ are parameters specifying a non-negative\nhas context-specific independencies if there exists some\nfunction for the random variables in each C ∈Cl,\nW ⊆U such that p(U \\ W|W = w) does not have\nφ(C). The MN represents the probability distribution Q the same independencies for all w. We now show that\np = K C∈Cl φ(C) where K is a normalizing constant. it is not too restrictive to assume, as in the theorem\nIf a probability distribution p can be represented by a\nabove, that a probability distribution is a CWT graMN with UG G, then G is an undirected independence\nphoid that has no context-specific independencies, bemap of p. When p is strictly positive, the opposite also\ncause there exist important families of probability disholds.\ntributions whose all or almost all the members satisfy\nA Bayesian network (BN) is a pair (G, θ) where G such assumptions. For instance, a Gaussian probabiis a DAG and θ are parameters specifying a multi- lity distribution is a CWT graphoid [Studen´y, 2005],\nnomial probability distribution for each X ∈U gi- and has no context-specific independencies because its\nven its parents in G, p(X|Pa(X)). The BN repre- independencies are determined by its covariance masents the multinomial probability distribution p = trix [Lauritzen, 1996]. The theorems below prove that Q\nX∈U p(X|Pa(X)). A probability distribution p can this is also the case for almost all the probability disbe represented by a BN with DAG G iffG is an direc- tributions in M(G) where G is an UG or DAG.\nted independence map of p. Theorem 2 Let G be an UG.",
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    "text": "M(G) has non-zero\nGiven an UG (resp. DAG) G, we denote by M(G) Lebesgue measure wrt Rn, where n is the number of\nall the multinomial probability distributions that can MN parameters corresponding to G. The probabibe represented by a MN (resp. BN) with UG (resp. lity distributions in M(G) that are not faithful to G\nDAG) G.",
    "paper_id": "1206.5263",
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    "text": "Finally, recall that a polytree is a directed or have context-specific independencies have zero Legraph without undirected cycles. In other words, there besgue measure wrt Rn.\nexists at most one undirected path between any two\nnodes X and Y . We denote that path by X : Y if it Theorem 3 Let G be a DAG. M(G) has non-zero\nexists.",
    "paper_id": "1206.5263",
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    "text": "Recall also that a directed tree is a polytree Lebesgue measure wrt Rn, where n is the number\nwhere every node has at most one parent. of BN parameters corresponding to G. The probability distributions in M(G) that are not faithful to G\n3 CWT GRAPHOIDS or have context-specific independencies have zero Lebesgue measure wrt Rn. Let X, Y, Z and W denote four mutually disjoint\nsubsets of U. We call CWT graphoid to any gra- The theorems above imply that, for any UG or DAG\nphoid that satisfies composition X ⊥⊥Y|Z ∧X ⊥⊥ G, almost all the probability distributions in M(G) are\nW|Z ⇒X ⊥⊥YW|Z, and weak transitivity X ⊥⊥ CWT graphoids [Pearl, 1988], and that all the margiY|Z ∧X ⊥⊥Y|ZV ⇒X ⊥⊥V |Z ∨V ⊥⊥Y|Z with nals and conditionals of almost all the probability disPEÑA 305 tributions in M(G) are CWT graphoids as well due to hand side of any of the properties above holds if the\nTheorem 1. corresponding sep statement holds in G. This is the\nbest solution we can hope for because sep is sound\nFinally, we give an example of a probability distribuand complete. By definition, sep is sound in the sense\ntion that is a CWT graphoid although it is neither\nthat it only identifies independencies in p. FurtherGaussian nor faithful to any UG or DAG.",
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    "text": "Let p be\nmore, sep is complete in the sense that it identifies all\na multinomial probability distribution that is faithful\nthe independencies in p that can be identified by stuto the DAG {X →Y, Y →Z, Z →W, X →V, V →\ndying G alone because (i) there exist multinomial and\nW, A →B, C →B} and that has no context-specific\nGaussian probability distributions that are faithful to\nindependencies. Such a probability distribution exists\nG [Meek, 1995], and (ii) such probability distributions\ndue to Theorem 3 and, moreover, it is a CWT graphoid\nare CWT graphoids [Pearl, 1988], G is a MDI map\n[Pearl, 1988]. Then, p(X, Y, Z, V, A, B, C|W = w) for\nfor them, and they only have the independencies that\nany w is a CWT graphoid by Theorem 1, but it is\nsep identifies from G. Moreover, this solution does\nneither Gaussian nor faithful to any UG or DAG,\nnot require more information about p than what it is\nbecause it is multinomial and faithful to the graph\navailable, because G can be constructed from the de-\n{X −Y, Y −Z, Z −V, V −X, A →B, C →B}\npendence base of p for G. We call the CWT graphoid\n[Chickering & Meek, 2002, Pe˜na et al., 2006b].\nclosure of the dependence base of p for G to the set of\ndependencies that are in the dependence base of p for\n4 READING DEPENDENCIES G plus those that can be derived from it by applying\nthe CWT graphoid properties. In this section, we propose a sound and complete criWe now introduce our criterion for reading dependen-terion for reading dependencies from a polytree-like\ncies from a polytree-like MDI map G of a CWT gra-MDI map of a CWT graphoid. If G is a MDI map\nphoid p.of a CWT graphoid p then we know, by construction\nof G, that X ̸⊥⊥(Pre(X) \\ Pa(X))Y |Pa(X) \\ Y for\nDefinition 1 Let X, Y and Z denote three mutuallyall X ∈U and Y ∈Pa(X). We call these dependisjoint subsets of U. We say that dep(X, Y|Z) holdsdencies the dependence base of p for G. Further deif there exist some A ∈X and B ∈Y such that (i)pendencies in p can be derived from this dependence\nsep(A, B|Z) does not hold, and (ii) for every head-base via the CWT graphoid properties. For this purto-head node C in A : B, either Z contains C or Zpose, we rephrase the CWT graphoid properties as\ncontains exactly one descendant of C that is not a des-follows. Let X, Y, Z and W denote four mutually\ncendant of another descendant of C that is in Z.disjoint subsets of U. Symmetry Y ̸⊥⊥X|Z ⇒X ̸⊥⊥\nY|Z. Decomposition X ̸⊥⊥Y|Z ⇒X ̸⊥⊥YW|Z. We now prove that dep is sound, i.e. it only iden-Weak union X ̸⊥⊥Y|ZW ⇒X ̸⊥⊥YW|Z.",
    "paper_id": "1206.5263",
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    "text": "Contractifies dependencies in p. Actually, it only identifies\ntion X ̸⊥⊥YW|Z ⇒X ̸⊥⊥Y|ZW ∨X ̸⊥⊥W|Z is\ndependencies in the CWT graphoid closure of the deproblematic for deriving new dependencies because\npendence base of p for G.\nit contains a disjunction in the right-hand side and,\nthus, it should be split into two properties: ContracTheorem 4 Let G be a MDI map of a CWT graphoid\ntion1 X ̸⊥⊥YW|Z ∧X ⊥⊥Y|ZW ⇒X ̸⊥⊥W|Z, and\np. If G is a polytree then, if dep(X, Y|Z) then X ̸⊥⊥\ncontraction2 X ̸⊥⊥YW|Z ∧X ⊥⊥W|Z ⇒X ̸⊥⊥Y|ZW. Y|Z is in the CWT graphoid closure of the dependence\nLikewise, intersection X ̸⊥⊥YW|Z ⇒X ̸⊥⊥Y|ZW∨\nbase of p for G. X̸⊥⊥W|ZY gives rise to intersection1 X̸⊥⊥YW|Z ∧X\n⊥⊥Y|ZW ⇒X ̸⊥⊥W|ZY, and intersection2 X ̸⊥⊥\nProof: Let us assume that every head-to-head node\nYW|Z ∧X⊥⊥W|ZY ⇒X̸⊥⊥Y|ZW. Note that interin A : B is in Z. We prove that dep(A, B|Z) implies\nsection1 and intersection2 are equivalent and, thus, we\nA̸⊥⊥B|Z. We prove it by induction over the length of\nrefer to them simply as intersection. Likewise, compoA : B.\nsition X ̸⊥⊥YW|Z ⇒X ̸⊥⊥Y|Z ∨X ̸⊥⊥W|Z gives rise\nto composition1 X̸⊥⊥YW|Z ∧X ⊥⊥Y|Z ⇒X̸⊥⊥W|Z, We first prove the result for length one, i.e. A : B is\nand composition2 X̸⊥⊥YW|Z∧X⊥⊥W|Z ⇒X̸⊥⊥Y|Z. Assume without loss of generality\nSince composition1 and composition2 are equivalent, that A : B is A →B. Let ZA denote the nodes in Z\nwe refer to them simply as composition.",
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    "text": "Finally, weak that are in Pa(A) or connected to A by an undirected\ntransitivity X ̸⊥⊥V |Z ∧V ̸⊥⊥Y|Z ⇒X ̸⊥⊥Y|Z ∨X ̸⊥⊥ path that passes through Pa(A). Let ZA denote the\nY|ZV with V ∈U \\ (XYZ) gives rise to weak transi- nodes in Z that are in Ch(A) \\ B or connected to A\ntivity1 X̸⊥⊥V |Z ∧V ̸⊥⊥Y|Z ∧X⊥⊥Y|Z ⇒X̸⊥⊥Y|ZV , by an undirected path that passes through Ch(A)\\B.\nand weak transitivity2 X ̸⊥⊥V |Z ∧V ̸⊥⊥Y|Z ∧X Let ZB denote the nodes in Z that are in Pa(B) \\ A\n⊥⊥Y|ZV ⇒X ̸⊥⊥Y|Z. The independence in the left- or connected to B by an undirected path that passes Let ZB denote the nodes in Z (24) B ̸⊥⊥C|Z \\ (ZCC) by the induction hypothesis\nthat are in Ch(B) or connected to B by an undirected\n(25) A⊥⊥B|Z \\ (ZCC) by sep(A, B|Z \\ (ZCC)) path that passes through Ch(B). Note that the nodes\nin Z \\ (ZAZAZBZB) are not connected to A or B by (26) A̸⊥⊥B|Z \\ ZC by weak transitivity1 on (23-25).\nany undirected path. Then,\n(27) A̸⊥⊥BZC|Z \\ ZC by decomposition on (26)\n(1) A(Pre(B) \\ Pa(B)) ̸⊥⊥B|Pa(B) \\ A from the de-\n(28) A⊥⊥ZC|Z \\ ZC by sep(A, ZC|Z \\ ZC) pendence base of p for G\n(29) A̸⊥⊥B|Z by contraction2 on (27) and (28).\n(2) Pre(B) \\ Pa(B) ⊥⊥B|Pa(B) by sep(Pre(B) \\\nPa(B), B|Pa(B)) Let us now assume that for every head-to-head node\nC in A : B, either Z contains C or Z contains exactly\n(3) A̸⊥⊥B|Pa(B) \\ A by contraction1 on (1) and (2)\none descendant of C that is not a descendant of ano-\n(4) A̸⊥⊥B(Pa(B) \\ A)|∅by weak union on (3) ther descendant of C that is in Z. We prove that\ndep(A, B|Z) implies A ̸⊥⊥B|Z. We prove it by induc-\n(5) A⊥⊥Pa(B) \\ A|∅by sep(A, Pa(B) \\ A|∅)\ntion over the number of head-to-head nodes in A : B\n(6) A̸⊥⊥B|∅by composition on (4) and (5) that are not in Z.",
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    "text": "As proven above, the result holds\nwhen this number is zero. Assume as induction hypo-\n(7) AZAZA ̸⊥⊥B|∅by decomposition on (6) thesis that the result holds when this number is smaller\n(8) ZAZA ⊥⊥B|A by sep(ZAZA, B|A) than l. We now prove the result for l.",
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    "text": "Let C be any\nhead-to-head node in A : B that is not in Z. Then,\n(9) A̸⊥⊥B|ZAZA by intersection on (7) and (8) Z must contain exactly one descendant of C, say D,\n(10) A̸⊥⊥BZB|ZAZA by decomposition on (9) that is not a descendant of another descendant of C\nthat is in Z. Let ZD denote the nodes in Z that are in (11) A⊥⊥ZB|ZAZA by sep(A, ZB|ZAZA) Ch(D) or connected to D by an undirected path that\n(12) A̸⊥⊥B|ZAZAZB by contraction2 on (10) and (11) passes through Ch(D). Then,\n(13) A̸⊥⊥BZB|ZAZAZB by decomposition on (12) (30) A̸⊥⊥D|Z \\ (ZDD) by the induction hypothesis\n(14) A⊥⊥ZB|ZAZAZBB by sep(A, ZB|ZAZAZBB) (31) B ̸⊥⊥D|Z \\ (ZDD) by the induction hypothesis\n(15) A ̸⊥⊥B|ZAZAZBZB by intersection on (13) and (32) A⊥⊥B|Z \\ (ZDD) by sep(A, B|Z \\ (ZDD))\n(14)\n(33) A̸⊥⊥B|Z \\ ZD by weak transitivity1 on (30-32)\n(16) A ̸⊥⊥BZ \\ (ZAZAZBZB)|ZAZAZBZB by decom-\n(34) A̸⊥⊥BZD|Z \\ ZD by decomposition on (33) position on (15)\n(35) A⊥⊥ZD|Z \\ ZD by sep(A, ZD|Z \\ ZD) (17) A⊥⊥Z\\(ZAZAZBZB)|ZAZAZBZB by sep(A, Z\\\n(ZAZAZBZB)|ZAZAZBZB) (36) A̸⊥⊥B|Z by contraction2 on (34) and (35). (18) A̸⊥⊥B|Z by contraction2 on (16) and (17). Therefore, we have proven that dep(A, B|Z) implies\nA ̸⊥⊥B|Z which, in turn, implies X ̸⊥⊥Y|Z by deAssume as induction hypothesis that the result holds\ncomposition. Since dep(X, Y|Z) implies dep(A, B|Z),\nwhen the length of A : B is smaller than l. We now\ndep(X, Y|Z) implies X̸⊥⊥Y|Z. Moreover, this last deprove the result for length l.",
    "paper_id": "1206.5263",
    "title": "Reading Dependencies from Polytree-Like Bayesian Networks",
    "authors": [
      "Jose M. Pena"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
    "arxiv_url": "http://arxiv.org/abs/1206.5263v1",
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    "chunk_id": "612ef201-cf3c-4ff3-b3e0-1713b190b01f",
    "text": "Let C be any node in\npendence must be in the CWT graphoid closure of the\nA : B except A and B. If C is not a head-to-head\ndependence base of p for G, because we have only used\nnode in A : B, then\nin the proof the dependence base of p for G and the\n(19) A̸⊥⊥C|Z by the induction hypothesis CWT graphoid properties. (20) B ̸⊥⊥C|Z by the induction hypothesis We now prove that dep is complete for reading dependencies from a polytree-like MDI map G of a CWT\n(21) A⊥⊥B|ZC by sep(A, B|ZC)\ngraphoid p, in the sense that it identifies all the depen-\n(22) A̸⊥⊥B|Z by weak transitivity2 on (19-21). dencies in p that follow from the information about p\nthat is available, namely the dependence base of p for\nOn the other hand, if C is a head-to-head node in\nG and the fact that p is a CWT graphoid. A : B, then C must be in Z due to the assumption\nmade at the beginning of this proof. Let ZC denote\nthe nodes in Z that are in Ch(C) or connected to C by Theorem 5 Let G be a MDI map of a CWT graphoid\nan undirected path that passes through Ch(C). If G is a polytree, then if X̸⊥⊥Y|Z is in the CWT\ngraphoid closure of the dependence base of p for G then\n(23) A̸⊥⊥C|Z \\ (ZCC) by the induction hypothesis dep(X, Y|Z). Proof: It suffices to prove (i) that all the dependen- • Weak transitivity2 dep(X, V |Z) ∧dep(V, Y|Z) ∧\ncies in the dependence base of p for G are identified by sep(X, Y|ZV ) ⇒dep(X, Y|Z) with V ∈U \\\ndep, and (ii) that dep satisfies the CWT graphoid pro- (XYZ). Let A : V and V : B denote the paths in\nperties. Since the first point is trivial, we only prove the first and second, respectively, dep statements\nthe second point.",
    "paper_id": "1206.5263",
    "title": "Reading Dependencies from Polytree-Like Bayesian Networks",
    "authors": [
      "Jose M. Pena"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
    "arxiv_url": "http://arxiv.org/abs/1206.5263v1",
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    "text": "Let X, Y, Z and W denote four in the left-hand side. Note that V must be a nonmutually disjoint subsets of U. head-to-head node in A : B for sep(X, Y|ZV ) to\nhold. Then, A : B satisfies the right-hand side.\n• Symmetry dep(Y, X|Z) ⇒dep(X, Y|Z). The\npath A : B in the left-hand side also satisfies the\nright-hand side.",
    "paper_id": "1206.5263",
    "title": "Reading Dependencies from Polytree-Like Bayesian Networks",
    "authors": [
      "Jose M. Pena"
    ],
    "published_date": "2012-06-20",
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    "text": "Finally, the soundness of dep allows us to prove the\n• Decomposition dep(X, Y|Z) ⇒dep(X, YW|Z). following theorem. The path A : B in the left-hand side also satisfies\nTheorem 6 Let G be a MDI map of a CWT graphoid the right-hand side.\np. If G is a directed tree, then p is faithful to it.\n• Weak union dep(X, Y|ZW) ⇒dep(X, YW|Z). The path A : B in the left-hand side also satisfies Proof: Any independence in p for which the corthe right-hand side unless there exists some head- responding separation statement does not hold in G\nto-head node C in A : B such that neither C contradicts Theorem 4.\nnor any of its descendants is in Z (if several such The theorem above has been proven by\nnodes exist, let C be the closest to A). However, [Becker et al., 2000] for the case where G is a tree-like\nC or some descendant D of C must be in W for MUI map and p is either a Gaussian probability\ndep(X, Y|ZW) to hold. Then, A : C or A : D distribution or a multinomial probability distribution\nsatisfies the right-hand side. over binary random variables. In [Pe˜na et al., 2006a],\nwe have proven the theorem above for the case where • Contraction1 dep(X, YW|Z)∧sep(X, Y|ZW) ⇒\nG is a tree-like MUI map and p is a graphoid that dep(X, W|Z). Let C denote the closest node to\nsatisfies weak transitivity.",
    "paper_id": "1206.5263",
    "title": "Reading Dependencies from Polytree-Like Bayesian Networks",
    "authors": [
      "Jose M. Pena"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
    "arxiv_url": "http://arxiv.org/abs/1206.5263v1",
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    "text": "A that is in both W and the path A : B in\nthe left-hand side. Such a node must exist for\nsep(X, Y|ZW) to hold. For the same reason, no 5 DISCUSSION\nnode in A : C can be in Y. Then, A : C satisfies\nthe right-hand side.",
    "paper_id": "1206.5263",
    "title": "Reading Dependencies from Polytree-Like Bayesian Networks",
    "authors": [
      "Jose M. Pena"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
    "arxiv_url": "http://arxiv.org/abs/1206.5263v1",
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    "chunk_id": "903f54b1-e0ca-4940-b732-3238d7c880d0",
    "text": "We have presented a sound and complete graphical\ncriterion for reading dependencies from a polytree-\n• Contraction2 dep(X, YW|Z) ∧sep(X, W|Z) ⇒\nlike MDI map G of a CWT graphoid. We have\ndep(X, Y|ZW). Let C denote the closest node\nshown that there exist important families of probato A that is in both Y and the path A : B in\nbility distributions, among them Gaussian probabithe left-hand side. Such a node must exist for\nlity distributions, that are CWT graphoids. We think\nsep(X, W|Z) to hold. For the same reason, no\nthat this work complements previous works addresnode in A : C can be in W. Then, A : C satisfies\nsing the same question, e.g. [Bouckaert, 1995] which\nthe right-hand side.\nproposes sound and complete graphical criteria for\n• Intersection dep(X, YW|Z) ∧sep(X, Y|ZW) ⇒ reading dependencies from a MUI or MDI map of a\ndep(X, W|ZY). Consider the same reasoning as graphoid, and [Pe˜na et al., 2006a, Pe˜na et al., 2006b,\nin contraction1. Pe˜na et al., 2007] which propose a sound and complete\ngraphical criterion for reading dependencies from the\n• Composition dep(X, YW|Z) ∧sep(X, W|Z) ⇒ MUI map of graphoid that satisfies weak transitivity.\ndep(X, Y|Z). Consider the same reasoning as in See also [Pe˜na et al., 2007] for a real-world application\ncontraction2. of the graphical criterion developed in that paper to\nread biologically meaningful gene dependencies.\n• Weak transitivity1 dep(X, V |Z) ∧dep(V, Y|Z) ∧\nsep(X, Y|Z) ⇒dep(X, Y|ZV ) with V ∈U \\ Our end-goal is to apply the results in this paper\n(XYZ). Let A : V and V : B denote the paths in to our project on atherosclerosis gene expression\nthe first and second, respectively, dep statements data analysis in order to learn dependencies between\nin the left-hand side. For sep(X, Y|Z) to hold, genes. We believe that it is not unrealistic to assume\nV must be a head-to-head node in A : B or the that the probability distribution underlying our data\ndescendant of a head-to-head node in A : B that is a CWT graphoid. We base this belief on the\nneither is in Z nor has any descendant in Z. Then, following argument.",
    "paper_id": "1206.5263",
    "title": "Reading Dependencies from Polytree-Like Bayesian Networks",
    "authors": [
      "Jose M. Pena"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
    "arxiv_url": "http://arxiv.org/abs/1206.5263v1",
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    "text": "The cell is the functional unit\nA : B satisfies the right-hand side. of all the organisms and includes all the information necessary to regulate its function. This information mous referees for their comments.\nis encoded in the DNA of the cell, which is divided\ninto a set of genes, each coding for one or more",
    "paper_id": "1206.5263",
    "title": "Reading Dependencies from Polytree-Like Bayesian Networks",
    "authors": [
      "Jose M. Pena"
    ],
    "published_date": "2012-06-20",
    "primary_category": "cs.AI",
    "arxiv_url": "http://arxiv.org/abs/1206.5263v1",
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