nips nips2006 nips2006-53 knowledge-graph by maker-knowledge-mining

53 nips-2006-Combining causal and similarity-based reasoning

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Author: Charles Kemp, Patrick Shafto, Allison Berke, Joshua B. Tenenbaum

Abstract: Everyday inductive reasoning draws on many kinds of knowledge, including knowledge about relationships between properties and knowledge about relationships between objects. Previous accounts of inductive reasoning generally focus on just one kind of knowledge: models of causal reasoning often focus on relationships between properties, and models of similarity-based reasoning often focus on similarity relationships between objects. We present a Bayesian model of inductive reasoning that incorporates both kinds of knowledge, and show that it accounts well for human inferences about the properties of biological species. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Combining causal and similarity-based reasoning Charles Kemp, Patrick Shafto, Allison Berke & Joshua B. [sent-1, score-0.625]

2 edu Abstract Everyday inductive reasoning draws on many kinds of knowledge, including knowledge about relationships between properties and knowledge about relationships between objects. [sent-3, score-0.788]

3 Previous accounts of inductive reasoning generally focus on just one kind of knowledge: models of causal reasoning often focus on relationships between properties, and models of similarity-based reasoning often focus on similarity relationships between objects. [sent-4, score-1.767]

4 We present a Bayesian model of inductive reasoning that incorporates both kinds of knowledge, and show that it accounts well for human inferences about the properties of biological species. [sent-5, score-0.518]

5 Many kinds of knowledge appear to be relevant: different researchers emphasize the role of causal knowledge, similarity, category judgments, associations, analogical mappings, scripts, and intuitive theories, and each of these approaches accounts for an important subset of everyday inferences. [sent-9, score-0.628]

6 As an initial step towards this goal, we present a model of inductive reasoning that is sensitive both to causal relationships between properties and to similarity relationships between objects. [sent-12, score-1.352]

7 Previous accounts of inductive reasoning generally address some version of this problem. [sent-14, score-0.323]

8 Models of causal reasoning [2] usually focus on relationships between properties (Figure 1a): if animal A has wings, for instance, it is likely that animal A can ﬂy. [sent-15, score-0.964]

9 Similarity-based models [3, 4, 5] usually focus on relationships between objects (Figure 1b): if a duck carries gene X, a goose is probably more likely than a pig to carry the same gene. [sent-16, score-0.438]

10 Previous models, however, cannot account for inferences that rely on similarity and causality: if a duck carries gene X and gene X causes enzyme Y to be expressed, it is likely that a goose expresses enzyme Y (Figure 1c). [sent-17, score-0.594]

11 We develop a unifying model that handles inferences like this, and that subsumes previous probabilistic approaches to causal reasoning [2] and similarity-based reasoning [5, 6]. [sent-18, score-0.906]

12 Approaches that rely on causal graphical models typically assume that the feature vectors of any two objects (any two rows of the matrix in Figure 1a) are conditionally independent given a causal network over the features. [sent-20, score-1.135]

13 Suppose, for example, that the rows of the matrix correspond to people and the causal network states that smoking leads to lung cancer with probability 0. [sent-21, score-0.726]

14 Figure 1: (a) Models of causal reasoning generally assume that the rows of an object-feature matrix are conditionally independent given a causal structure over the features. [sent-69, score-1.113]

15 (b) Models of similaritybased reasoning generally assume that columns of the matrix are conditionally independent given a similarity structure over the objects. [sent-71, score-0.358]

16 (c) We develop a generative model for object-feature matrices that incorporates causal relationships between features and similarity relationships between objects. [sent-73, score-1.081]

17 variables is difﬁcult, but we suggest that knowledge about similarity between objects can help. [sent-75, score-0.295]

18 Since Tim is more similar to Tom than Zach, our model correctly predicts that Tom is more likely to have lung cancer than Zach. [sent-76, score-0.256]

19 Previous models of similarity-based reasoning [5, 6] also suffer from a restrictive assumption of conditional independence. [sent-77, score-0.207]

20 This time the assumption states that features (columns of the matrix in Figure 1b) are conditionally independent given information about the similarity between objects. [sent-78, score-0.241]

21 Empirical tests of similarity-based models often attempt to satisfy this assumption by using blank properties—subjects, for example, might be told that coyotes have property P, and asked to judge the probability that foxes have property P [3]. [sent-79, score-0.435]

22 To a ﬁrst approximation, inferences in tasks like this conform to judgments of similarity: subjects conclude, for example, that foxes are more likely to have property P than mice, since coyotes are more similar to foxes than mice. [sent-80, score-0.596]

23 It now seems that desert rats are more likely to share this property than arctic foxes, even though desert foxes are more similar in general to arctic foxes than to desert rats. [sent-83, score-0.537]

24 Our model captures inferences like this by incorporating causal relationships between properties: in this case, having sunburn-resistant skin is linked to the property of living in the desert. [sent-84, score-0.859]

25 Our model uses a distribution that is sensitive both to causal relationships between properties and to similarity relationships between objects. [sent-86, score-1.081]

26 The results suggest that people are able to combine causality with similarity, and that our model accounts well for this capacity. [sent-89, score-0.219]

27 Assume that So is an object structure: a graphical model that captures relationships between a known set of objects (Figure 1b). [sent-91, score-0.348]

28 Suppose, for instance, that the objects include a mouse, a rat, a squirrel and a sheep (o1 through o4 ). [sent-92, score-0.326]

29 So can be viewed as a graphical model that captures phylogenetic relationships, or as a formalization of the intuitive similarity between these animals. [sent-93, score-0.319]

30 For instance, if λ is low, the model (So , λ) will predict that a novel feature will probably not be found in any of the animals, but if the feature does occur in exactly two of the animals, the mouse and the rat are a more likely pair than the mouse and the sheep. [sent-98, score-0.941]

31 The mutation process can be formalized as a continuous-time Markov process with two states (off and on) and with inﬁnitesimal matrix: Q= −λ 1−λ λ −(1 − λ) We can generate object vectors from this model by imagining a binary feature spreading out over the tree from root to leaves. [sent-99, score-0.22]

32 Assume that Sf is a feature structure: a graphical model that captures relationships between a known set of features (Figure 1a). [sent-104, score-0.356]

33 The features, for instance, may correspond to enzymes, and Sf may capture the causal relationships between these enzymes. [sent-105, score-0.662]

34 One possible structure states that enzyme f1 is involved in the production of enzyme f2 , which is in turn involved in the production of enzyme f3 . [sent-106, score-0.294]

35 Given that the mouse expresses enzyme f1 , for instance, a combined model should predict that rats are more likely than squirrels to express enzyme f2 . [sent-109, score-0.618]

36 Formally, we seek a distribution P (M ), where M is an object-feature matrix, and P (M ) is sensitive to both the relationships between features and the relationships between animals. [sent-110, score-0.448]

37 Consider then the case where Sf captures causal relationships between the features (Figure 1c). [sent-113, score-0.718]

38 These causal relationships will typically depend on several hidden variables. [sent-114, score-0.662]

39 Causal relationships between enzymes, for instance, are likely to depend on other biological variables, and the causal link between smoking and lung cancer is mediated by many genetic and environmental variables. [sent-115, score-0.92]

40 t indicates whether the mechanism of causal transmission between c and e is active, and b indicates whether e is true owing to a background cause independent of c. [sent-141, score-0.485]

41 All of the root variables (c, t and b) are independent, and the remaining variables (e) are deterministically speciﬁed once the D root variables are ﬁxed. [sent-142, score-0.271]

42 (c) A graphical model created by combining Sf with a tree-structured D representation of the similarity between three objects. [sent-143, score-0.268]

43 3 Experiments When making inductive inferences, a rational agent should exploit all of the information available, including causal relationships between features and similarity relationships between objects. [sent-156, score-1.16]

44 On one hand, there are motivating examples like the case of the three smokers where it seems natural to think about causal relationships and similarity relationships at the same time. [sent-158, score-1.103]

45 On the other hand, Rehder [7] argues that causal information tends to overwhelm similarity information, and supports this conclusion with data from several tasks involving artiﬁcial categories. [sent-159, score-0.628]

46 To help resolve these competing views, we designed several tasks where subjects were required to simultaneously reason about causal relationships between enzymes and similarity relationships between animals. [sent-160, score-1.247]

47 Each subject was trained on two causal structures, each of which involved three enzymes. [sent-165, score-0.461]

48 In the chain condition, subjects were told that f3 is known to be produced by several a) Task 1 2 f1 f2 f3 r=0. [sent-167, score-0.373]

49 97 f2 2 1 0 f3 −1 Task 2 2 1 0 −1 mouse rat sqrl sheep mouse rat sqrl sheep mouse rat sqrl sheep mouse rat sqrl sheep Human Combined Causal Similarity Figure 3: Experiment 1: Behavioral data (column 1) and predictions for three models. [sent-179, score-3.513]

50 Known test results are marked with arrows: in task 1, subjects were told only that the mouse had tested positive for f1 , and in task 2 they were told in addition that the rat had tested negative for f2 . [sent-181, score-1.156]

51 In the common-effect condition, subjects were told that f3 is known to be produced by several pathways, and that one of the most common pathways involves f1 and the other involves f2 . [sent-185, score-0.352]

52 To reinforce each causal structure, subjects were shown 20 cards representing animals from twenty different mammal species (names of the species were not supplied). [sent-186, score-0.867]

53 The cards were chosen to be representative of the distribution captured by a causal network with known structure (chain or common-effect) and known parameterization. [sent-188, score-0.564]

54 3 (f2 and f3 ), and the probability that each causal link was active was set to 0. [sent-191, score-0.461]

55 After subjects had studied the cards for as long as they liked, the cards were removed and subjects were asked several questions about the enzymes (e. [sent-193, score-0.546]

56 ”) The questions in this training phase were intended to encourage subjects to reﬂect on the causal relationships between the enzymes. [sent-196, score-0.809]

57 In both conditions, subjects were told that they would be testing the four animals (mouse, rat, sheep and squirrel) for each of the three enzymes. [sent-197, score-0.615]

58 In the chain condition, subjects were told that the mouse had tested positive for f1 , and asked to predict the outcome of each remaining test (Figure 1c). [sent-199, score-0.717]

59 Subjects were then told in addition that the rat had tested negative for f2 , and again asked to predict the outcome of each remaining test. [sent-200, score-0.543]

60 Note that this second task requires subjects to integrate causal reasoning with similarity-based reasoning: causal reasoning predicts that the mouse has f2 , and similarity-based reasoning predicts that it does not. [sent-201, score-1.904]

61 In the common-effect condition, subjects were told that the mouse had tested positive for f3 , then told in addition that the rat had tested negative for f2 . [sent-202, score-1.076]

62 Our combined model uses a tree over the four animals and a causal network over the features. [sent-209, score-0.68]

63 The smaller the path length, the more likely that all four animals have the same feature values, and the greater the path length, the more likely that distant animals in the tree (e. [sent-212, score-0.331]

64 the mouse and the sheep) will have different feature values. [sent-214, score-0.272]

65 The causal component of our model includes no free parameters, since we used the parameters of the network that generated the cards shown to subjects during the training phase. [sent-216, score-0.739]

66 Columns 3 and 4 of Figure 3 show model predictions when we remove the similarity component (column 3) or the causal component (column 4) from our combined model. [sent-218, score-0.788]

67 The model that uses the causal network alone is described by [2], among others, and the model that uses the tree alone is described by [6]. [sent-219, score-0.587]

68 In task 1 of each condition, the causal model makes identical predictions about the rat, the squirrel and the sheep: in task 1 of the chain condition, for example, it cannot use the similarity between the mouse and the rat to predict that the rat is also likely to test positive for f1 . [sent-221, score-1.795]

69 In task 1 of each condition the similarity model predicts that the unobserved features (f2 and f3 for the chain condition, and f1 and f2 for the common-effect condition) are distributed identically across the four animals. [sent-222, score-0.425]

70 In task 1 of the chain condition, for example, the similarity model does not predict that the mouse is more likely than the sheep to test positive for f2 and f3 . [sent-223, score-0.861]

71 The limitations of the causal and similarity models suggest that some combination of causality and similarity is necessary to account for our data. [sent-224, score-0.954]

72 There are likely to be approaches other than our combined model that account well for our data, but we suggest that accurate predictions will only be achieved when the causal network and the similarity information are tightly integrated. [sent-225, score-0.911]

73 2 Experiment 2 Our working hypothesis is that similarity and causality should be combined in most contexts. [sent-228, score-0.346]

74 An alternative hypothesis—the root-variables hypothesis—was suggested to us by Bob Rehder, and states that similarity relationships are used only if some of the root variables in a causal structure Sf are unobserved. [sent-229, score-0.959]

75 For instance, similarity might have inﬂuenced inferences in the chain condition of Experiment 1 only because the root variable f1 was never observed for all four animals. [sent-230, score-0.442]

76 The root-variables hypothesis should be correct in cases where all root variables in the true causal structure are known. [sent-231, score-0.594]

77 In Figure 2c, for instance, similarity no longer plays a role once the root variables are observed, since the remaining variables are deterministically speciﬁed. [sent-232, score-0.332]

78 We are interested, however, in cases where Sf may not contain all of the causally relevant variables, and where similarity can help to make predictions about the effects of unobserved variables. [sent-233, score-0.324]

79 Even though the root variable is observed for Tim, Tom and Zach (all three are smokers), we still believe that Tom is more likely to suffer from lung cancer than Zach having discovered that Tim has lung cancer. [sent-235, score-0.366]

80 In the ﬁrst task for each condition, subjects were told only that the mouse had tested positive for f1 . [sent-239, score-0.64]

81 In the second task, subjects were told in addition that the rat, the squirrel and the sheep had tested positive for f1 , and that the mouse had a) Task 1 2 f1 f2 f3 r=0. [sent-240, score-0.884]

82 55 0 −1 −2 b) f1 Task 1 2 f2 1 f3 0 −1 Task 2 1 0 −1 −2 mouse rat sqrl sheep mouse rat sqrl sheep mouse rat sqrl sheep mouse rat sqrl sheep Human Combined Causal Similarity Figure 4: Experiment 2: Behavioral data and predictions for three models. [sent-252, score-3.513]

83 In task 2 of each condition, the root variable in the causal network (f1 ) is observed for all four animals. [sent-253, score-0.609]

84 The judgments for the ﬁrst task in each condition replicate the ﬁnding from Experiment 1 that subjects combine causality and similarity when just one of the 12 animal-feature pairs is observed. [sent-259, score-0.536]

85 In the chain condition, for example, the causal model predicts that the rat and the sheep are equally likely to test positive for f2 . [sent-261, score-1.136]

86 Subjects predict that the rat is less likely than the sheep to test positive for f2 , and our combined model accounts for this prediction. [sent-262, score-0.738]

87 4 Discussion We developed a model of inductive reasoning that is sensitive to causal relationships between features and similarity relationships between objects, and demonstrated in two experiments that it provides a good account of human reasoning. [sent-263, score-1.408]

88 First, it provides an integrated view of two inductive problems—causal reasoning and similarity-based reasoning—that are usually considered separately. [sent-265, score-0.271]

89 Second, unlike previous accounts of causal reasoning, it acknowledges the importance of unknown but causally relevant variables, and uses similarity to constrain inferences about the effects of these variables. [sent-266, score-0.823]

90 For expository convenience we have emphasized the distinction between causality and similarity, but the notion of similarity needed by our approach will often have a causal interpretation. [sent-268, score-0.721]

91 A treestructured taxonomy, for example, is a simple representation of the causal process that generated biological species—the process of evolution. [sent-269, score-0.461]

92 Our combined model can therefore be seen as a causal model that takes both relationships between features and evolutionary relationships between species into account. [sent-270, score-1.044]

93 More generally, our framework can be seen as a method for building sophisticated causal models, and our experiments suggest that these kinds of models will be needed to account for the complexity and subtlety of human causal reasoning. [sent-271, score-1.066]

94 In particular, the product of experts approach [12] should lead to predictions that are qualitatively similar to the predictions of our combined model. [sent-273, score-0.205]

95 Neither of our experiments explored inferences about interventions, but an adequate causal model should be able to handle inferences of this sort. [sent-275, score-0.667]

96 Causal knowledge and similarity are just two of the many varieties of knowledge that support inductive reasoning. [sent-276, score-0.344]

97 a causal network) may be a relatively general method for combining probabilistic knowledge representations. [sent-286, score-0.525]

98 Our combined model may ﬁnd applications in computational biology: predicting whether an organism expresses a certain gene, for example, should rely on phylogenetic relationships between organisms and causal relationships between genes. [sent-288, score-1.02]

99 [13] develop an approach to protein function prediction that combines phylogenetic relationships between proteins with relationships between protein functions, and several authors have explored models that combine phylogenies with hidden Markov models. [sent-290, score-0.492]

100 When similarity and causality compete in category-based property generalization. [sent-338, score-0.284]

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