nips nips2008 nips2008-108 knowledge-graph by maker-knowledge-mining

108 nips-2008-Integrating Locally Learned Causal Structures with Overlapping Variables

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Author: David Danks, Clark Glymour, Robert E. Tillman

Abstract: In many domains, data are distributed among datasets that share only some variables; other recorded variables may occur in only one dataset. While there are asymptotically correct, informative algorithms for discovering causal relationships from a single dataset, even with missing values and hidden variables, there have been no such reliable procedures for distributed data with overlapping variables. We present a novel, asymptotically correct procedure that discovers a minimal equivalence class of causal DAG structures using local independence information from distributed data of this form and evaluate its performance using synthetic and real-world data against causal discovery algorithms for single datasets and applying Structural EM, a heuristic DAG structure learning procedure for data with missing values, to the concatenated data.

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Integrating locally learned causal structures with overlapping variables Robert E. [sent-1, score-0.473]

2 edu Abstract In many domains, data are distributed among datasets that share only some variables; other recorded variables may occur in only one dataset. [sent-6, score-0.122]

3 While there are asymptotically correct, informative algorithms for discovering causal relationships from a single dataset, even with missing values and hidden variables, there have been no such reliable procedures for distributed data with overlapping variables. [sent-7, score-0.563]

4 Such predictions require knowledge of causal relationships between observed variables. [sent-10, score-0.295]

5 There are existing asymptotically correct algorithms for learning such relationships from data, possibly with missing values and hidden variables [1][2][3], but these algorithms all assume that every variable is measured in a single study. [sent-11, score-0.409]

6 However, given the increasing availability of large amounts of data, it is often possible to obtain several similar studies that individually measure subsets of the variables a researcher is interested in and together include all such variables. [sent-13, score-0.142]

7 In these cases, if each dataset has overlapping variable(s) with at least one other dataset, e. [sent-18, score-0.18]

8 if two datasets D1 and D2 , which measure variables V1 and V2 , respectively, have at least one variable in common (V1 ∩ V2 = ∅), then we should be able to learn many of the causal relationships between the observed variables using this set of datasets. [sent-20, score-0.486]

9 The existing algorithms, however, cannot in general be directly applied to such cases, since they may require joint observations for variables that are not all measured in a single dataset. [sent-21, score-0.069]

10 While this problem has been discussed in [4] and [5], there are no general, useful algorithms for learning causal relationships from data of this form. [sent-22, score-0.295]

11 A typical response is to concatenate the datasets to form a single common dataset with missing values for the variables that are not measured in each of the original datasets. [sent-23, score-0.255]

12 Statistical matching [6] or multiple imputation [7] procedures may then be used to ﬁll in the missing values by assuming an underlying model (or small class of models), estimating model parameters using the available data, and then using this model to interpolate the 1 missing values. [sent-24, score-0.202]

13 The Structural EM algorithm [8] avoids this problem by iteratively updating the assumed model using the current interpolated dataset and then reestimating values for the missing data to form a new interpolated dataset until the model converges. [sent-28, score-0.27]

14 The Structural EM algorithm is only justiﬁed, however, when missing data are missing at random (or indicator variables can be used to make them so) [8]. [sent-29, score-0.241]

15 The pattern of missing values in the concatenated datasets described above is highly structured. [sent-30, score-0.177]

16 While this may not be problematic in practice when doing prediction, it is problematic when learning causal relationships. [sent-32, score-0.224]

17 We present a novel, asymptotically correct algorithm—the Integration of Overlapping Networks (ION) algorithm—for learning causal relationships (or more properly, the complete set of possible causal DAG structures) from data of this form. [sent-34, score-0.662]

18 A directed graph G = V, E is a set of nodes V, which represent variables, and a set of directed edges E connecting distinct nodes. [sent-40, score-0.329]

19 If two nodes are connected by an edge then the nodes are adjacent. [sent-41, score-0.344]

20 For pairs of nodes {X, Y } ⊆ V, X is a parent (child) of Y , if there is a directed edge from X to Y (Y to X) in E. [sent-42, score-0.334]

21 A trail in G is a sequence of nodes such that each consecutive pair of nodes in the sequence is adjacent in G and no node appears more than once in the sequence. [sent-43, score-0.391]

22 A trail is a directed path if every edge between consecutive pairs of nodes points in the same direction. [sent-44, score-0.581]

23 X is an ancestor (descendant) of Y if there is a directed path from X to Y (Y to X). [sent-45, score-0.074]

24 G is a directed acyclic graph (DAG) if for every pair {X, Y } ⊆ V, X is not both an ancestor and a descendent of Y (no directed cycles). [sent-46, score-0.251]

25 A collider (v-structure) is a triple of nodes X, Y, Z such that X and Z are parents of Y . [sent-47, score-0.114]

26 A trail is active given C ⊆ V if (i) for every collider X, Y, Z in the trail either Y ∈ C or some descendant of Y is in C and (ii) no other node in the trail is in C. [sent-48, score-0.668]

27 For disjoint sets of nodes X, Y, and Z, X is d-separated (d-connected) from Y given Z if and only if there are no (at least one) active trails between any X ∈ X and any Y ∈ Y given Z. [sent-49, score-0.216]

28 B is a causal Bayesian network if an edge from X to Y indicates that X is a direct cause of Y relative to V. [sent-52, score-0.4]

29 Most algorithms for causal discovery, or learning causal relationships from nonexperimental data, assume that the distribution over the observed variables P is decomposable according to a DAG G and P is faithful to G. [sent-53, score-0.611]

30 Most causal discovery algorithms return a set of possible DAGs which entail the same d-separations and d-connections, e. [sent-55, score-0.308]

31 The DAGs in this set have the same adjacencies but only some of the same directed edges. [sent-58, score-0.074]

32 The directed edges common to each DAG represent causal relationships that are learned from the data. [sent-59, score-0.417]

33 A partial ancestral graph (PAG) represents the set of DAGs in a particular Markov equivalence class when latent common causes may be present. [sent-61, score-0.172]

34 Edges are of four types: − ◦− ◦− and ◭−◮, where a ◦ indicates either an ◮ or − orientation, ◮, ◮, ◦ bidirected edges indicate the presence of a latent common cause, and fully directed edges (−◮) 2 indicate that the directed edge is present in every DAG, e. [sent-63, score-0.524]

35 For {X, Y } ⊆ V, a possibly active trail between X and Y given Z ⊆ V/{X, Y } is a trail in a PAG between X and Y such that some orientation of ◦’s on edges between consecutive nodes in the trail, to either − or ◮, makes the trail active given Z. [sent-66, score-0.904]

36 3 Integration of Overlapping Networks (ION) algorithm The ION algorithm uses conditional independence information to discover the complete set of PAGs over a set of variables V that are consistent with a set of datasets over subsets of V which have overlapping variables. [sent-67, score-0.435]

37 A standard causal discovery algorithm that checks for latent common causes, such as FCI [1] or GES [3] with latent variable postprocessing steps1 , must ﬁrst be applied to each of the original datasets to learn these PAGs that will be input to ION. [sent-69, score-0.515]

38 Input : PAGs Gi ∈ G with nodes Vi ⊆ V for i = 1, . [sent-72, score-0.084]

39 , k Output: PAGs Hi ∈ H with nodes Vi = V for i = 1, . [sent-75, score-0.084]

40 ’ from the edge between X and Y in Ai PR ← every combination of removing or not removing ‘? [sent-80, score-0.23]

41 if X and Y are not adjacent in Gi then remove the edge between X and Y , if X is directed into Y in Gi then set the endpoint at Y on the edge between X and Y to ◮. [sent-83, score-0.534]

42 Once these orientations and edge removals are made, the changes to the complete graph are propagated using the rules in [10], which provably make every change that is entailed by the current changes made to the graph. [sent-84, score-0.655]

43 Lines 4-9 ﬁnd every possibly active trail for every {X, Y } ⊆ V given Z ⊆ V/{X, Y } such that X and Y are d-separated given Z in some Gi ∈ G. [sent-85, score-0.359]

44 The constructed set PC includes all minimal hitting sets of graphical changes, e. [sent-86, score-0.134]

45 unique sets of minimal changes that are not subsets of other sets of changes, which make these paths no longer active. [sent-88, score-0.187]

46 For each minimal hitting set, a new graph is constructed by making the changes in the set and propagating these changes. [sent-89, score-0.315]

47 the graph does not imply a d-separation for some {X, Y } ⊆ V given Z ⊆ V/{X, Y } such that X and Y are d-connected in some Gi ∈ G, then this graph is added to the current set of possible graphs. [sent-92, score-0.098]

48 3 19 attempt to discover any additional PAGs that may be consistent with each Gi ∈ G after deleting edges from PAGs in the current set and propagating the changes. [sent-94, score-0.179]

49 If some pair of nodes {X, Y } ⊆ V that are adjacent in a current PAG are d-connected given ∅ in some Gi ∈ G, then we do not consider sets of edge removals which remove this edge. [sent-95, score-0.382]

50 The ION algorithm is provably sound in the sense that the output PAGs are consistent with every Gi ∈ G, e. [sent-96, score-0.191]

51 Every structure Ai constructed at line 7 provably entails every d-separation entailed by some Gi ∈ G. [sent-104, score-0.167]

52 Such structures are only added to A if they do not entail a d-separation corresponding to a d-connection in some Gi ∈ G. [sent-105, score-0.085]

53 The only changes made (other than changes resulting from propagating other changes which are provably correct by [10]) in lines 10-19 are edge removals, which can only create new d-separations. [sent-106, score-0.542]

54 The ION algorithm is provably complete in the sense that if there is some structure Hi over the variables V that is consistent with every Gi ∈ G, then Hi ∈ H. [sent-108, score-0.275]

55 Let Hi be a PAG over the variables V such that for every pair {X, Y } ⊆ V, if X and Y are d-separated (d-connected) given Z ⊆ V/{X, Y } in some Gi ∈ G, then X and Y are d-separated (d-connected) given Z in Hi . [sent-111, score-0.123]

56 Every change made at line 3 is provably necessary to ensure soundness. [sent-114, score-0.068]

57 At least one graph added to A at line 8 provably has every adjacency (possibly more) in Hi and no non-◦ endpoints on an edge found in Hi that is not also present in Hi . [sent-115, score-0.372]

58 Some sequence of edge removals will provably produce Hi at line 16 and it will be added to the output set since it is consistent with every Gi ∈ G. [sent-116, score-0.435]

59 Finding all minimal hitting sets is an NP-complete problem [11]. [sent-122, score-0.134]

60 In practice, however, we can break the minimal hitting set problem into a sequence of smaller subproblems and use a branch and bound approach that is tractable in many cases and still results in an asymptotically correct algorithm. [sent-124, score-0.259]

61 For each DAG, we then randomly chose two subsets of size 2 or 3 of the nodes in the DAG such that the union of the subsets included all 4 nodes and at least one overlapping variable between the two subsets was present. [sent-129, score-0.553]

62 samples of sizes N = 50, N = 100, N = 500, N = 1000 and N = 2500 from the DAGs for only the variables in each subset. [sent-133, score-0.114]

63 We used both FCI and GES with latent variable postprocessing to 4 5 Edge omissions 4 3 2 1 0 3 2. [sent-134, score-0.254]

64 5 Edge commissions FCI−baseline ION−FCI GES−baseline ION−GES Structural EM 2 1. [sent-135, score-0.102]

65 To evaluate the accuracy of ION, we counted the number of edge omission, edge commision, and orientation errors (◮ instead of −) for each PAG in the ION output set and averaged the results. [sent-138, score-0.595]

66 ION-FCI and ION-GES refer the the performance of ION when the input PAGs are obtained using the FCI algorithm and the GES algorithm with latent variable postprocessing, respectively. [sent-141, score-0.074]

67 For Structural EM, we took each of the datasets over subsets of the nodes in each DAG and formed a concatenated dataset, as described in section 1, which was input to the Structural EM algorithm. [sent-142, score-0.272]

68 sample of sizes N = 50, N = 100, N = 500, N = 1000 and N = 2500 for all of the variables in each DAG and used these datasets as input for the FCI and GES with latent variable postprocessing algorithms, respectively, to obtain a measure for how well these algorithms perform when no data is missing. [sent-146, score-0.343]

69 Almost no edge omission errors are made, but more edge commissions errors are made than any of the other methods and the edge commission errors do not decrease as the sample size increases. [sent-150, score-1.109]

70 When we looked at the results, we found that Structural EM always returned either the complete graph or a graph that was almost complete, indicating that Structural EM is not a reliable method for causal discovery in this scenario where there is a highly structured pattern to the missing data. [sent-151, score-0.502]

71 For the larger sample sizes (where more missing values need to be estimated at each iteration), a single run required several hours in some instances. [sent-153, score-0.165]

72 Random “chains” of nodes were used as the initial models. [sent-155, score-0.084]

73 5 Edge commissions FCI−baseline ION−FCI GES−baseline ION−GES 7 0. [sent-159, score-0.102]

74 1 0 N=50 N=100 N=500 N=1000N=2500 Sample size (a) 3 2 1 1 0 4 0 N=50 N=100 N=500 N=1000N=2500 Sample size (b) N=50 N=100 N=500 N=1000N=2500 Sample size (c) Figure 2: (a) edge omissions, (b) edge comissions, and (c) orientation errors 8 5 4 3 2 5 0. [sent-162, score-0.661]

75 5 Edge commissions FCI−baseline ION−FCI GES−baseline ION−GES 7 0. [sent-164, score-0.102]

76 The ION-FCI and ION-GES methods performed similarly to the FCI-baseline and GESbaseline methods but made slightly more errors and showed slower convergence (due to the missing data). [sent-169, score-0.185]

77 Slightly more edge omission errors were made, but these errors decrease as the sample size increases. [sent-171, score-0.515]

78 Some edge orientation errors were made even for the larger sample sizes. [sent-172, score-0.42]

79 Even if the correct equivalence class is discovered, errors result after comparing the ground truth DAG to every DAG in the equivalence class and averaging. [sent-174, score-0.287]

80 We also note that there are fewer orientation errors for the GES-baseline and ION-GES methods on the two smallest sample sizes than all of the other sample sizes. [sent-175, score-0.323]

81 While this may seem surprising, it is simply a result of the fact that more edge omission errors are made for these cases. [sent-176, score-0.349]

82 samples were generated for random subsets of sizes 2-5 with only 1 variable that is not overlapping between the two subsets; (ii) two i. [sent-180, score-0.251]

83 samples were generated for random subsets of sizes 2-5 with only 2 variables that are not overlapping between the two subsets; (iii) three i. [sent-183, score-0.32]

84 samples were generated for random subsets of sizes 2-5 with only 1 variable that is not overlapping between any pair of subsets. [sent-186, score-0.251]

85 Figures 2, 3, and 4 show edge omission, edge commission, and orientation errors for each of these cases, respectively. [sent-187, score-0.562]

86 We also tested the performance of ION-FCI using a real world dataset measuring IQ and various neuroanatomical and other traits [14]. [sent-189, score-0.108]

87 Figures 5a and 5b show the FCI output PAGs when only the data for each of these subsets of the variables is provided as input, respectively. [sent-191, score-0.175]

88 We also ran FCI on the complete dataset to have a comparison. [sent-193, score-0.095]

89 4 Orientation errors 5 Edge commissions Edge omissions 6 6 0. [sent-196, score-0.337]

90 We note that the graphical structure of the complete PAG (ﬁgures 5c and 5d) is the union of the structures shown in ﬁgures 5a and 5b. [sent-202, score-0.095]

91 5 Conclusions In practice, researchers are often unable to ﬁnd or construct a single, complete dataset containing every variable they may be interested in (or doing so is very costly). [sent-204, score-0.149]

92 We thus need some way of integrating information about causal relationships that can be discovered from a collection of datasets with related variables [5]. [sent-205, score-0.444]

93 Standard causal discovery algorithms cannot be used, since they take only a single dataset as input. [sent-206, score-0.317]

94 To address this open problem, we proposed the ION algorithm, an asymptotically correct algorithm for discovering the complete set of causal DAG structures that are consistent with such data. [sent-207, score-0.45]

95 Probably the most signiﬁcant problem is resolving contradictory information among overlapping variables in different 7 input PAGs, i. [sent-209, score-0.25]

96 X is a parent of Y in one PAG and a child of Y in another PAG, resulting from statistical errors or when the input samples are not identiﬁcally distributed. [sent-211, score-0.123]

97 In some of such cases, ION will not discover any PAGs which entail the correct d-separations and d-connections. [sent-215, score-0.107]

98 When performing conditional independence tests or evaluating score functions, statistical errors occur more frequently as the dimensionality of a dataset increases, unless the sample size also increases at an exponential rate (resulting from the so-called curse of dimensionality). [sent-217, score-0.213]

99 Furthermore, while the branch and bound approach described in section 3 is a signiﬁcant improvement over other methods we tested for computing minimal hitting sets, its memory requirements are still considerable in some instances. [sent-219, score-0.164]

100 A characterization of markov equivalence classes for causal models with latent variables. [sent-274, score-0.318]

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