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15 nips-2011-A rational model of causal inference with continuous causes

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Author: Thomas L. Griffiths, Michael James

Abstract: Rational models of causal induction have been successful in accounting for people’s judgments about causal relationships. However, these models have focused on explaining inferences from discrete data of the kind that can be summarized in a 2× 2 contingency table. This severely limits the scope of these models, since the world often provides non-binary data. We develop a new rational model of causal induction using continuous dimensions, which aims to diminish the gap between empirical and theoretical approaches and real-world causal induction. This model successfully predicts human judgments from previous studies better than models of discrete causal inference, and outperforms several other plausible models of causal induction with continuous causes in accounting for people’s inferences in a new experiment. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 A rational model of causal induction with continuous causes Michael D. [sent-1, score-1.354]

2 edu Abstract Rational models of causal induction have been successful in accounting for people’s judgments about causal relationships. [sent-5, score-1.888]

3 We develop a new rational model of causal induction using continuous dimensions, which aims to diminish the gap between empirical and theoretical approaches and real-world causal induction. [sent-8, score-2.025]

4 This model successfully predicts human judgments from previous studies better than models of discrete causal inference, and outperforms several other plausible models of causal induction with continuous causes in accounting for people’s inferences in a new experiment. [sent-9, score-2.299]

5 1 Introduction The problem of causal induction is central to science, and is something at which people are remarkably skilled, especially given its apparent difﬁculty. [sent-10, score-1.092]

6 Understanding how people identify causal relationships has consequently become a challenge taken up by many research programs in cognitive science. [sent-11, score-0.938]

7 One of the most successful of these programs has used rational solutions to the abstract problem of causal induction (in the spirit of [1, 2]) as a source of explanations for people’s inferences [3, 4, 5, 6]. [sent-12, score-1.116]

8 However nearly all this research has assumed people have access to categorical information about whether or not a cause or effect is present on a given trial – the sort of information that appears in a 2 × 2 contingency table (see Figure 1(a)). [sent-13, score-0.425]

9 Such an assumption may not be valid for many of the causal relationships that we see in the world. [sent-14, score-0.795]

10 For a simple example of a situation in which a continuous cause is relevant, consider the case of drinking coffee and wakefulness. [sent-15, score-0.413]

11 Clearly, someone who drinks a beverage made by placing a single drop of coffee in a gallon of water will experience no effects of wakefulness, as an insufﬁcient amount of the cause was present. [sent-16, score-0.317]

12 How much coffee one drinks is closely linked to whether wakefulness occurs – merely knowing that some amount of coffee was drunk is insufﬁcient. [sent-18, score-0.223]

13 And this problem is not relegated to those who wish to titrate their caffeination; many causes exist along continuous dimensions, even if their effects do not (e. [sent-19, score-0.28]

14 1 The primary strategy that has been explored in previous work on causal induction from continuous causes is one in which ambiguous examples are immediately categorized as indicating either the presence or the absence of the cause. [sent-22, score-1.373]

15 This approach, taken by Marsh and Ahn [9], provides a way to 1 We will focus on the case of continuous causes with binary outcomes. [sent-23, score-0.282]

16 (b) Graphical models showing possible causal relationships between cause C, effect E, and background B. [sent-30, score-1.054]

17 In this paper, however, we argue that another approach can be fruitful – developing models that work directly with continuous values. [sent-32, score-0.196]

18 We extend the causal support model [4], originally deﬁned for binary causes, to work with continuous-valued causes. [sent-33, score-0.846]

19 We then re-analyze the results of Marsh and Ahn [9], comparing people’s causal judgments to predictions made by a number of rational models of causal induction with both discrete and continuous causes. [sent-34, score-2.271]

20 The predictions made by the continuous models for these experiments perform well, but are extremely similar, which led us to conduct a new experiment using stimuli that discriminate among the different models. [sent-35, score-0.344]

21 We show that continuous causal support provides a better account of these data than the other models we consider. [sent-36, score-1.019]

22 2 Background In this section we review previous work on rational models of causal induction, and summarize the results of Marsh and Ahn [9] that we will use to evaluate different models later in the paper. [sent-37, score-0.956]

23 1 Rational models of causal induction Rational models of causal induction have focused on the problem of determining the nature of the relationship between a cause C and an effect E. [sent-39, score-2.276]

24 One group focuses on estimating causal strength, such as ∆P [10], causal power [3] and pCI [11], which attempt to identify the degree of relationship between two variables. [sent-41, score-1.584]

25 The other group focuses on causal structure, such as causal support [4], which attempts to identify how certain one can be that a causal relationship exists at all. [sent-42, score-2.407]

26 The causal support model has proven effective in predicting human judgments in several studies [4, 5, 6], and we use it as the starting point for our model of causal induction with continuous causes. [sent-43, score-2.109]

27 The causal support model can be most easily described in the context of causal graphical models [12] (see Figure 1(b)). [sent-44, score-1.619]

28 In its most general form causal support is this calculation, described less technically P (D|G0 ) as identifying the evidence that D provides in favor of G1 over G0 [4]. [sent-51, score-0.855]

29 In the particular case of causal inference over binary variables, we have three random variables representing the unknown background causes assumed to be always present (B), the possible cause (C) and the effect (E) in question. [sent-52, score-1.129]

30 How we parameterize the relationship between B, C, and E determines the type of causal relationship we are considering. [sent-57, score-0.873]

31 In order to capture generative causal relationships we use a noisy-OR parameterization for P (e|b+ , c, w0 , w1 ; G1 ). [sent-58, score-0.873]

32 2 a similar noisy-AND-NOT parameterization can be used for preventive causes [4], but we focus on generative causes in this paper. [sent-61, score-0.266]

33 The data consists of the values of all n observed occurrences of cause and effects (i. [sent-63, score-0.197]

34 If we were concerned with estimating causal strength, we could use this likelihood to determine the estimates of w0 and w1 under G1 and G0 . [sent-70, score-0.765]

35 However, if we want to compute a measure of causal structure we need to integrate over all possible values of w0 and w1 , assuming prior distributions on w0 and w1 . [sent-71, score-0.781]

36 In the original causal support model [4], a uniform prior was used on w0 and w1 (for a more complex prior, see [6]). [sent-72, score-0.839]

37 Despite its success in modeling human judgments, this measure of causal support only works in a limited set of cases – those cases where data can be summarized in a 2×2 contingency table. [sent-73, score-0.918]

38 , [9]) have attempted to solve this by collapsing continuous data into binary form. [sent-80, score-0.188]

39 2 Previous work on continuous-valued causal induction Marsh and Ahn [9] note the insufﬁciencies of current models of causal induction that result from considering only binary variables. [sent-83, score-1.986]

40 The ﬁrst solution is that people simply ignore ambiguous information, and only deal with instances that can easily be categorized into “cause” and “not cause”. [sent-85, score-0.257]

41 They reject this solution and instead opt for the idea that learners “spontaneously categorize ambiguous evidence into one of the four types of evidence [used in contingency tables]. [sent-86, score-0.266]

42 4) To test these claims, Marsh and Ahn conducted a series of experiments in which participants observe visual stimuli (e. [sent-88, score-0.231]

43 , Figure 2 (a)) representing a particular value along a continuous dimension paired with a (binary) event either occurring or not occurring. [sent-90, score-0.225]

44 Then, participants were asked “to judge the strength between C and E on a scale from 0 (not a cause) to 100 (strongly causes)”. [sent-93, score-0.216]

45 Marsh and Ahn used this second measure to show that participants use ambiguous evidence when making causal judgments, refuting the idea that people ignore the instances which cannot be easily categorized. [sent-94, score-1.205]

46 Furthermore, they discovered that engaging in causal inference changes participants’ judgments of how many instances of each category they saw. [sent-95, score-0.915]

47 This evidence that people’s frequency ratings were altered based on whether or not the effect was paired with the ambiguous stimuli was used to dismiss the possibility that participants were learning a continuous causal relationship. [sent-99, score-1.527]

48 While Marsh and Ahn demonstrate that causal induction altered how people assigned ambiguous stimuli to categories, this does not necessarily mean that people were spontaneously categorizing these stimuli and using that categorization information to make causal judgments. [sent-100, score-2.333]

49 An alternative account is that the boundary between the categories was ambiguous, and the evidence about the relationship between cause and effect inﬂuenced where people placed this boundary. [sent-101, score-0.458]

50 Previous research suggests that category structures should not always be thought of as ﬁxed [13] and that causal information can be used when learning category structures and meanings [14]. [sent-102, score-0.807]

51 Our focus here is on investigating how people might induce causal relationships that involve continuous variables, rather than understanding their inﬂuence on categorization. [sent-103, score-1.086]

52 However, the existence of a plausible alternative account of Marsh and Ahn’s results raises the possibility that we can understand their data without assuming that people spontaneously categorize ambiguous stimuli in order to make causal judgments. [sent-104, score-1.212]

53 We will explore this possibility after introducing our rational model of causal induction. [sent-105, score-0.927]

54 3 Deﬁning causal support for continuous causes Our goal in this section is to extend the rational analysis used to deﬁne the causal support model [4] to causes with continuous values. [sent-110, score-2.293]

55 Following the original model, we take causal support to be the log likelihood ratio in favor of G1 over G0 , and assume that the causes combine in a noisy-OR. [sent-111, score-0.917]

56 For any such function fλ (·) : R → [0, 1], with parameters λ, we then have the parameterization P (e+ |b+ , c, w0 , λ; G1 ) = 1 − (1 − w0 )(1 − fλ (c)) (3) where c is the (continuous) value of the cause C. [sent-113, score-0.235]

57 It is straightforward to show that the original causal support model corresponds to a special case of this model when C only takes on a single value when it is present. [sent-119, score-0.823]

58 3 In our continuous model, we assume the cause is always present but with varying strength. [sent-132, score-0.341]

59 If we allow for the possibility that the cause is absent, and that it has no inﬂuence on the effect in such a situation, then we obtain P (e+ |b+ , c− , w0 , θ, γ; G1 ) = w0 , as required. [sent-133, score-0.261]

60 Vertical bars in Ex2 contingencies separate the three possible strategies (1|2|3) proposed in [9] for assimilating ambiguous stimuli. [sent-146, score-0.257]

61 We developed this rational model in order to be able to investigate how people engage in causal inference in the case of continuous causes. [sent-147, score-1.204]

62 First, in order to demonstrate the usefulness of considering any model of continuous causal inference, we reanalyzed the causal ratings provided by participants in Marsh and Ahn’s [9] Experiments 1 and 2. [sent-149, score-1.902]

63 Second, in order to better identify which model best predicts human judgments among the continuous causal models, we conducted a new experiment designed to distinguish between the various rational models. [sent-150, score-1.233]

64 Following [4], we evaluated ﬁve models of causal induction from discrete data: ∆P [10], causal power [3], pCI [11], (discrete) causal support [4], and the χ2 statistic. [sent-155, score-2.622]

65 Because there are three potential sets of true event counts under the assimilation hypothesis for Experiment 2, in order to analyze the assimilation hypothesis under the best possible case, we will run the discrete models under all three possible methods of assimilation. [sent-159, score-0.182]

66 These three possible ways of assimilating the ambiguous cases are represented in Table 1, as contingencies separated by vertical bars (“|”). [sent-160, score-0.274]

67 We also evaluated several models that consider the causal variable to be continuously valued. [sent-162, score-0.796]

68 This includes the causal support model described in the previous section, as well as several traditional models for statistical inference in cases where there is a relationship between continuous and binary variables. [sent-163, score-1.115]

69 Because they are usually used for hypothesis tests about whether or not there is a relationship between a continuous and a binary variable, the two tests we use are probit regression and a two-sample Student t-test. [sent-164, score-0.427]

70 The former tests whether there is a relationship between a continuous valued variable mapped to a binary variable, while the latter tests whether there is a relationship between a binary variable mapped to a continuous variable. [sent-165, score-0.584]

71 Both continuous causal support and the discrete models have the property that with more evidence there is for a cause the larger the positive score produced by the model. [sent-166, score-1.264]

72 132 sample t-test as a case of linear regression (with an indicator variable for whether or not the effect occurred as the regressor), we obtain β values for both the probit model and the t-test model. [sent-197, score-0.204]

73 We can treat these t values as alternative measures of causal structure. [sent-200, score-0.781]

74 We ran the model over each set of stimulus values, and averaged these four predictions to obtain the ﬁnal general predictions the means of which were compared to the mean human judgments. [sent-205, score-0.201]

75 This was the transformation y = sign(x) ∗ abs(x)α , where α was chosen to maximize the correlation (r) between the mean human ratings and mean model predictions across the conditions. [sent-208, score-0.19]

76 The re-analysis supports the idea that people were using continuous values in their causal judgments. [sent-210, score-1.056]

77 The best possible correlation achieved by any discrete model was discrete causal support and χ2 , r = . [sent-211, score-0.918]

78 On the other hand, the models of continuous causal inference successfully captured much of the variation in responses, with all the continuous models performing well (all r > . [sent-213, score-1.176]

79 984, with Continuous causal support and the t-test models not far behind, with r = . [sent-216, score-0.854]

80 5 Distinguishing between the continuous models In the previous section, all of the models for continuous causal induction performed well. [sent-219, score-1.358]

81 However, the continuous models all made very similar predictions to one another. [sent-220, score-0.264]

82 As a result, it is difﬁcult to distinguish which model of continuous causal induction people might be using. [sent-221, score-1.257]

83 In order to better determine which of these models most accurately captures human causal induction over continuous dimensions, we need to construct data sets that will result in divergent predictions across the various models. [sent-222, score-1.293]

84 Because of the noisy-OR parameterization of the generative model, (discrete) causal support predictions are sensitive to the base rate of occurrence while standard statistical tests (e. [sent-223, score-1.016]

85 The continuous causal support model also uses a noisy-OR parameterization, meaning that it will also be sensitive 6 Data Set:3 1 Effect Value Data Set:2 1 Effect Value Effect Value Effect Value Data Set:1 1 Data Set:4 1 0. [sent-226, score-0.988]

86 In order to account for any participants who did not read the instructions and consider the data, we eliminated any participants who took less than sixty seconds to complete the study. [sent-267, score-0.323]

87 4 Because of this constraint, twelve participants were removed, leaving 135 participants for analysis. [sent-268, score-0.302]

88 Participants were told that they would be assisting a scientist in identifying “whether or not different levels of a chemical cause a type of bacteria to produce a protein”. [sent-271, score-0.284]

89 01 60 40 20 1 2 3 4 5 6 7 8 9 Data Set 0 1 2 3 4 5 6 7 8 9 Data Set Figure 4: Experimental results, showing human judgments (error bars are one standard error), together with unscaled model predictions and corresponding correlations. [sent-283, score-0.244]

90 This scale was designed to obtain scalar estimates of degrees of belief in causal structure [6]. [sent-285, score-0.784]

91 The optimized correlation between the mean human responses and mean model predictions for the probit |β| model and the t-test |t| model were r = . [sent-289, score-0.259]

92 The results are striking in that, though all the models performed well at predicting people’s judgments in the Marsh and Ahn studies, all but the continuous causal support model perform poorly here. [sent-302, score-1.129]

93 Continuous causal support outperforms every other model of continuous causal inference (r = . [sent-303, score-1.772]

94 Still, it does seem to underestimate human causal ratings in data sets 8 and 9 (see Figure 4), which suggests further investigation of this phenomenon is needed. [sent-306, score-0.901]

95 6 Conclusion We have proposed a new rational model of causal induction using continuous dimensions, continuous causal support, which aims to be a ﬁrst step towards ﬁlling the gap between existing models of causal induction and real-world cases of causal learning. [sent-307, score-3.952]

96 This model successfully predicts human judgments found in previous work, and outperforms several other plausible models of causal induction with continuous causes. [sent-308, score-1.335]

97 Future work will hopefully continue to bring our models of causal induction ever closer to addressing the problem of real-world causal induction, for example by looking at how temporal information is used in conjunction with traditional statistical information. [sent-309, score-1.762]

98 Consistent with a continuous view of causal induction, we suspect that more work in each of these directions is likely to produce positive results. [sent-310, score-0.949]

99 Spontaneous assimilation of continuous values and temporal information in causal induction. [sent-381, score-0.968]

100 Making causal judgments from the proportion of conﬁrming instances: the pCI rule. [sent-393, score-0.875]

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For example, animals that “live in water” typically “can swim,” and animals that have “no legs” cannot “jump far.” Matrix D can be used to formulate problems where a reasoner observes one or two features of a new object (i.e. animal o130 ) and must make inferences about the remaining features of the animal. The next two sections describe graphical models that can be used to address this problem. The ﬁrst graphical model O captures relationships between objects, and the second model F captures relationships between features. We then discuss how these models can be combined, and introduce a family of exemplar-style models that will be compared with our graphical models. A graphical model over objects Many accounts of inductive reasoning focus on similarity relationships between objects [6, 8]. Here we describe a tree-structured graphical model O that captures these relationships. The tree was constructed from matrix D using average linkage clustering and the Jaccard similarity measure, and part of the resulting structure is shown in Figure 2a. The subtree in Figure 2a includes clusters 2 alligator caiman crocodile monitor lizard dinosaur blindworm boa cobra python snake viper chameleon iguana gecko lizard salamander frog toad tortoise turtle anchovy herring sardine cod sole salmon trout carp pike stickleback eel flatfish ray plaice piranha sperm whale squid swordfish goldfish dolphin orca whale shark bat fox wolf beaver hedgehog hamster squirrel mouse rabbit bison elephant hippopotamus rhinoceros lion tiger polar bear deer dromedary llama giraffe zebra kangaroo monkey cat dog cow horse donkey pig sheep (a) (b) can swim lives in water eats fish eats nuts eats grain eats grass has gills can jump far has two legs has no legs has six legs has four legs can fly can be ridden has sharp teeth nocturnal has wings strong predator can see in dark eats berries lives in the sea lives in the desert crawls lives in the woods has mane lives in trees can climb well lives underground has feathers has scales slow has fur heavy Figure 2: Graph structures used to deﬁne graphical models O and F. (a) A tree that captures similarity relationships between animals. The full tree includes 129 animals, and only part of the tree is shown here. The grey points along the branches indicate locations where a novel animal o130 could be attached to the tree. (b) A network capturing pairwise dependency relationships between features. The edges capture both positive and negative dependencies. All edges in the network are shown, and the network also includes 20 isolated nodes for the following features: is black, is blue, is green, is grey, is pink, is red, is white, is yellow, is a pet, has a beak, stings, stinks, has a long neck, has feelers, sucks blood, lays eggs, makes a web, has a hump, has a trunk, and is cold-blooded. corresponding to amphibians and reptiles, aquatic creatures, and land mammals, and the subtree omitted for space includes clusters for insects and birds. We assume that the features in matrix D (i.e. the columns) are generated independently over O: P (f i |O, π i , λi ). P (D|O, π, λ) = i i i i The distribution P (f |O, π , λ ) is based on the intuition that nearby nodes in O tend to have the same value of f i . Previous researchers [8, 18] have used a directed graphical model where the distribution at the root node is based on the baserate π i , and any other node v with parent u has the following conditional probability distribution: i P (v = 1|u) = π i + (1 − π i )e−λ l , if u = 1 i π i − π i e−λ l , if u = 0 (1) where l is the length of the branch joining node u to node v. The variability parameter λi captures the extent to which feature f i is expected to vary over the tree. Note, for example, that any node v must take the same value as its parent u when λ = 0. To avoid free parameters, the feature baserates π i and variability parameters λi are set to their maximum likelihood values given the observed values of the features {f i } in the data matrix D. The conditional distributions in Equation 1 induce a joint distribution over all of the nodes in graph O, and the distribution P (f i |O, π i , λi ) is computed by marginalizing out the values of the internal nodes. Although we described O as a directed graphical model, the model can be converted into an equivalent undirected model with a potential for each edge in the tree and a potential for the root node. Here we use the undirected version of the model, which is a natural counterpart to the undirected model F described in the next section. The full version of structure O in Figure 2a includes 129 familiar animals, and our task requires inferences about a novel animal o130 that must be slotted into the structure. Let D′ be an expanded version of D that includes a row for o130 , and let O′ be an expanded version of O that includes a node for o130 . The edges in Figure 2a are marked with evenly spaced gray points, and we use a 3 uniform prior P (O′ ) over all trees that can be created by attaching o130 to one of these points. Some of these trees have identical topologies, since some edges in Figure 2a have multiple gray points. Predictions about o130 can be computed using: P (D′ |D) = P (D′ |O′ , D)P (O′ |D) ∝ O′ P (D′ |O′ , D)P (D|O′ )P (O′ ). (2) O′ Equation 2 captures the basic intuition that the distribution of features for o130 is expected to be consistent with the distribution observed for previous animals. For example, if o130 is known to ﬂy then the trees with high posterior probability P (O′ |D) will be those where o130 is near other ﬂying creatures (Figure 1a), and since these creatures have wings Equation 2 predicts that o130 probably also has wings. As this example suggests, model O captures dependency relationships between features implicitly, and therefore stands in contrast to models like F that rely on explicit representations of relationships between features. A graphical model over features Model F is an undirected graphical model deﬁned over features. The graph shown in Figure 2b was created by identifying pairs where one feature depends directly on another. The author and a research assistant both independently identiﬁed candidate sets of pairwise dependencies, and Figure 2b was created by merging these sets and reaching agreement about how to handle any discrepancies. As previous researchers have suggested [13, 15], feature dependencies can capture several kinds of relationships. For example, wings enable ﬂying, living in the sea leads to eating ﬁsh, and having no legs rules out jumping far. We work with an undirected graph because some pairs of features depend on each other but there is no clear direction of causal inﬂuence. For example, there is clearly a dependency relationship between being nocturnal and seeing in the dark, but no obvious sense in which one of these features causes the other. We assume that the rows of the object-feature matrix D are generated independently from an undirected graphical model F deﬁned over the feature structure in Figure 2b: P (oi |F). P (D|F) = i Model F includes potential functions for each node and for each edge in the graph. These potentials were learned from matrix D using the UGM toolbox for undirected graphical models [19]. The learned potentials capture both positive and negative relationships: for example, animals that live in the sea tend to eat ﬁsh, and tend not to eat berries. Some pairs of feature values never occur together in matrix D (there are no creatures that ﬂy but do not have wings). We therefore chose to compute maximum a posteriori values of the potential functions rather than maximum likelihood values, and used a diffuse Gaussian prior with a variance of 100 on the entries in each potential. After learning the potentials for model F, we can make predictions about a new object o130 using the distribution P (o130 |F). For example, if o130 is known to ﬂy (Figure 1a), model F predicts that o130 probably has wings because the learned potentials capture a positive dependency between ﬂying and having wings. Combining object and feature relationships There are two simple ways to combine models O and F in order to develop an approach that incorporates both relationships between features and relationships between objects. The output combination model computes the predictions of both models in isolation, then combines these predictions using a weighted sum. The resulting model is similar to a mixture-of-experts model, and to avoid free parameters we use a mixing weight of 0.5. The structure combination model combines the graph structures used by the two models and relies on a set of potentials deﬁned over the resulting graph product. An example of a graph product is shown in Figure 1b, and the potential functions for this graph are inherited from the component models in the natural way. Kemp et al. [11] use a similar approach to combine a functional causal model with an object model O, but note that our structure combination model uses an undirected model F rather than a functional causal model over features. Both combination models capture the intuition that inductive inferences rely on relationships between features and relationships between objects. The output combination model has the virtue of 4 simplicity, and the structure combination model is appealing because it relies on a single integrated representation that captures both relationships between features and relationships between objects. To preview our results, our data suggest that the combination models perform better overall than either O or F in isolation, and that both combination models perform about equally well. Exemplar models We will compare the family of graphical models already described with a family of exemplar models. The key difference between these model families is that the exemplar models do not rely on explicit representations of relationships between objects and relationships between features. Comparing the model families can therefore help to establish whether human inferences rely on representations of this sort. Consider ﬁrst a problem where a reasoner must predict whether object o130 has feature k after observing that it has feature i. An exemplar model addresses the problem by retrieving all previouslyobserved objects with feature i and computing the proportion that have feature k: P (ok = 1|oi = 1) = |f k & f i | |f i | (3) where |f k | is the number of objects in matrix D that have feature k, and |f k & f i | is the number that have both feature k and feature i. Note that we have streamlined our notation by using ok instead of o130 to refer to the kth feature value for object o130 . k Suppose now that the reasoner observes that object o130 has features i and j. The natural generalization of Equation 3 is: P (ok = 1|oi = 1, oj = 1) = |f k & f i & f j | |f i & f j | (4) Because we focus on chimeras, |f i & f j | = 0 and Equation 4 is not well deﬁned. We therefore evaluate an exemplar model that computes predictions for the two observed features separately then computes the weighted sum of these predictions: P (ok = 1|oi = 1, oj = 1) = wi |f k & f i | |f k & f j | + wj . i| |f |f j | (5) where the weights wi and wj must sum to one. We consider four ways in which the weights could be set. The ﬁrst strategy sets wi = wj = 0.5. The second strategy sets wi ∝ |f i |, and is consistent with an approach where the reasoner retrieves all exemplars in D that are most similar to the novel animal and reports the proportion of these exemplars that have feature k. The third strategy sets wi ∝ |f1i | , and captures the idea that features should be weighted by their distinctiveness [20]. The ﬁnal strategy sets weights according to the coherence of each feature [21]. A feature is coherent if objects with that feature tend to resemble each other overall, and we deﬁne the coherence of feature i as the expected Jaccard similarity between two randomly chosen objects from matrix D that both have feature i. Note that the ﬁnal three strategies are all consistent with previous proposals from the psychological literature, and each one might be expected to perform well. Because exemplar models and prototype models are often compared, it is natural to consider a prototype model [22] as an additional baseline. A standard prototype model would partition the 129 animals into categories and would use summary statistics for these categories to make predictions about the novel animal o130 . We will not evaluate this model because it corresponds to a coarser version of model O, which organizes the animals into a hierarchy of categories. The key characteristic shared by both models is that they explicitly capture relationships between objects but not features. 2 Experiment 1: Chimeras Our ﬁrst experiment explores how people make inferences about chimeras, or novel animals with features that have not previously been observed to co-occur. Inferences about chimeras raise challenges for exemplar models, and therefore help to establish whether humans rely on explicit representations of relationships between features. Each argument can be represented as f i , f j → f k 5 exemplar r = 0.42 7 feature F exemplar (wi = |f i |) (wi = 0.5) r = 0.44 7 object O r = 0.69 7 output combination r = 0.31 7 structure combination r = 0.59 7 r = 0.60 7 5 5 5 5 5 3 3 3 3 3 3 all 5 1 1 0 1 r = 0.06 7 conflict 0.5 1 1 0 0.5 1 r = 0.71 7 1 0 0.5 1 r = −0.02 7 1 0 0.5 1 r = 0.49 7 0 5 5 5 5 3 3 3 3 1 5 3 0.5 r = 0.57 7 5 3 1 0 0.5 1 r = 0.51 7 edge 0.5 r = 0.17 7 1 1 0 0.5 1 r = 0.64 7 1 0 0.5 1 r = 0.83 7 1 0 0.5 1 r = 0.45 7 1 0 0.5 1 r = 0.76 7 0 5 5 5 5 3 3 3 3 1 5 3 0.5 r = 0.79 7 5 3 1 1 0 0.5 1 r = 0.26 7 other 1 0 1 0 0.5 1 r = 0.25 7 1 0 0.5 1 r = 0.19 7 1 0 0.5 1 r = 0.25 7 1 0 0.5 1 r = 0.24 7 0 7 5 5 5 5 5 3 3 3 3 1 5 3 0.5 r = 0.33 3 1 1 0 0.5 1 1 0 0.5 1 1 0 0.5 1 1 0 0.5 1 1 0 0.5 1 0 0.5 1 Figure 3: Argument ratings for Experiment 1 plotted against the predictions of six models. The y-axis in each panel shows human ratings on a seven point scale, and the x-axis shows probabilities according to one of the models. Correlation coefﬁcients are shown for each plot. where f i and f k are the premises (e.g. “has no legs” and “can ﬂy”) and f k is the conclusion (e.g. “has wings”). We are especially interested in conﬂict cases where the premises f i and f j lead to opposite conclusions when taken individually: for example, most animals with no legs do not have wings, but most animals that ﬂy do have wings. Our models that incorporate feature structure F can resolve this conﬂict since F includes a dependency between “wings” and “can ﬂy” but not between “wings” and “has no legs.” Our models that do not include F cannot resolve the conﬂict and predict that humans will be uncertain about whether the novel animal has wings. Materials. The object-feature matrix D includes 447 feature pairs {f i , f j } such that none of the 129 animals has both f i and f j . We selected 40 pairs (see the supporting material) and created 400 arguments in total by choosing 10 conclusion features for each pair. The arguments can be assigned to three categories. Conﬂict cases are arguments f i , f j → f k such that the single-premise arguments f i → f k and f j → f k lead to incompatible predictions. For our purposes, two singlepremise arguments with the same conclusion are deemed incompatible if one leads to a probability greater than 0.9 according to Equation 3, and the other leads to a probability less than 0.1. Edge cases are arguments f i , f j → f k such that the feature network in Figure 2b includes an edge between f k and either f i or f j . Note that some arguments are both conﬂict cases and edge cases. All arguments that do not fall into either one of these categories will be referred to as other cases. The 400 arguments for the experiment include 154 conﬂict cases, 153 edge cases, and 120 other cases. 34 arguments are both conﬂict cases and edge cases. We chose these arguments based on three criteria. First, we avoided premise pairs that did not co-occur in matrix D but that co-occur in familiar animals that do not belong to D. For example, “is pink” and “has wings” do not co-occur in D but “ﬂamingo” is a familiar animal that has both features. Second, we avoided premise pairs that speciﬁed two different numbers of legs—for example, {“has four legs,” “has six legs”}. Finally, we aimed to include roughly equal numbers of conﬂict cases, edge cases, and other cases. Method. 16 undergraduates participated for course credit. The experiment was carried out using a custom-built computer interface, and one argument was presented on screen at a time. Participants 6 rated the probability of the conclusion on seven point scale where the endpoints were labeled “very unlikely” and “very likely.” The ten arguments for each pair of premises were presented in a block, but the order of these blocks and the order of the arguments within these blocks were randomized across participants. Results. Figure 3 shows average human judgments plotted against the predictions of six models. The plots in the ﬁrst row include all 400 arguments in the experiment, and the remaining rows show results for conﬂict cases, edge cases, and other cases. The previous section described four exemplar models, and the two shown in Figure 3 are the best performers overall. Even though the graphical models include more numerical parameters than the exemplar models, recall that these parameters are learned from matrix D rather than ﬁt to the experimental data. Matrix D also serves as the basis for the exemplar models, which means that all of the models can be compared on equal terms. The ﬁrst row of Figure 3 suggests that the three models which include feature structure F perform better than the alternatives. The output combination model is the worst of the three models that incorporate F, and the correlation achieved by this model is signiﬁcantly greater than the correlation achieved by the best exemplar model (p < 0.001, using the Fisher transformation to convert correlation coefﬁcients to z scores). Our data therefore suggest that explicit representations of relationships between features are needed to account for inductive inferences about chimeras. The model that includes the feature structure F alone performs better than the two models that combine F with the object structure O, which may not be surprising since Experiment 1 focuses speciﬁcally on novel animals that do not slot naturally into structure O. Rows two through four suggest that the conﬂict arguments in particular raise challenges for the models which do not include feature structure F. Since these conﬂict cases are arguments f i , f j → f k where f i → f k has strength greater than 0.9 and f j → f k has strength less than 0.1, the ﬁrst exemplar model averages these strengths and assigns an overall strength of around 0.5 to each argument. The second exemplar model is better able to differentiate between the conﬂict arguments, but still performs substantially worse than the three models that include structure F. The exemplar models perform better on the edge arguments, but are outperformed by the models that include F. Finally, all models achieve roughly the same level of performance on the other arguments. Although the feature model F performs best overall, the predictions of this model still leave room for improvement. The two most obvious outliers in the third plot in the top row represent the arguments {is blue, lives in desert → lives in woods} and {is pink, lives in desert → lives in woods}. Our participants sensibly infer that any animal which lives in the desert cannot simultaneously live in the woods. In contrast, the Leuven database indicates that eight of the twelve animals that live in the desert also live in the woods, and the edge in Figure 2b between “lives in the desert” and “lives in the woods” therefore represents a positive dependency relationship according to model F. This discrepancy between model and participants reﬂects the fact that participants made inferences about individual animals but the Leuven database is based on features of animal categories. Note, for example, that any individual animal is unlikely to live in the desert and the woods, but that some animal categories (including snakes, salamanders, and lizards) are found in both environments. 3 Experiment 2: Single-premise arguments Our results so far suggest that inferences about chimeras rely on explicit representations of relationships between features but provide no evidence that relationships between objects are important. It would be a mistake, however, to conclude that relationships between objects play no role in inductive reasoning. Previous studies have used object structures like the example in Figure 2a to account for inferences about novel features [11]—for example, given that alligators have enzyme Y132 in their blood, it seems likely that crocodiles also have this enzyme. Inferences about novel objects can also draw on relationships between objects rather than relationships between features. For example, given that a novel animal has a beak you will probably predict that it has feathers, not because there is any direct dependency between these two features, but because the beaked animals that you know tend to have feathers. Our second experiment explores inferences of this kind. Materials and Method. 32 undergraduates participated for course credit. The task was identical to Experiment 1 with the following exceptions. Each two-premise argument f i , f j → f k from Experiment 1 was converted into two one-premise arguments f i → f k and f j → f k , and these 7 feature F exemplar r = 0.78 7 object O r = 0.54 7 output combination r = 0.75 7 structure combination r = 0.75 7 all 5 5 5 5 5 3 3 3 3 3 1 1 0 edge 0.5 1 r = 0.87 7 1 0 0.5 1 r = 0.87 7 1 0 0.5 1 r = 0.84 7 1 0 0.5 1 r = 0.86 7 0 5 5 5 3 3 3 1 5 3 0.5 r = 0.85 7 5 3 1 1 0 0.5 1 r = 0.79 7 other r = 0.77 7 1 0 0.5 1 r = 0.21 7 1 0 0.5 1 r = 0.74 7 1 0 0.5 1 r = 0.66 7 0 5 5 5 5 3 3 3 3 1 r = 0.73 7 5 0.5 3 1 1 0 0.5 1 1 0 0.5 1 1 0 0.5 1 1 0 0.5 1 0 0.5 1 Figure 4: Argument ratings and model predictions for Experiment 2. one-premise arguments were randomly assigned to two sets. 16 participants rated the 400 arguments in the ﬁrst set, and the other 16 rated the 400 arguments in the second set. Results. Figure 4 shows average human ratings for the 800 arguments plotted against the predictions of ﬁve models. Unlike Figure 3, Figure 4 includes a single exemplar model since there is no need to consider different feature weightings in this case. Unlike Experiment 1, the feature model F performs worse than the other alternatives (p < 0.001 in all cases). Not surprisingly, this model performs relatively well for edge cases f j → f k where f j and f k are linked in Figure 2b, but the ﬁnal row shows that the model performs poorly across the remaining set of arguments. Taken together, Experiments 1 and 2 suggest that relationships between objects and relationships between features are both needed to account for human inferences. Experiment 1 rules out an exemplar approach but models that combine graph structures over objects and features perform relatively well in both experiments. We considered two methods for combining these structures and both performed equally well. Combining the knowledge captured by these structures appears to be important, and future studies can explore in detail how humans achieve this combination. 4 Conclusion This paper proposed that graphical models are useful for capturing knowledge about animals and their features and showed that a graphical model over features can account for human inferences about chimeras. A family of exemplar models and a graphical model deﬁned over objects were unable to account for our data, which suggests that humans rely on mental representations that explicitly capture dependency relationships between features. Psychologists have previously used graphical models to capture relationships between features, but our work is the ﬁrst to focus on chimeras and to explore models deﬁned over a large set of familiar features. Although a simple undirected model accounted relatively well for our data, this model is only a starting point. The model incorporates dependency relationships between features, but people know about many speciﬁc kinds of dependencies, including cases where one feature causes, enables, prevents, or is inconsistent with another. An undirected graph with only one class of edges cannot capture this knowledge in full, and richer representations will ultimately be needed in order to provide a more complete account of human reasoning. Acknowledgments I thank Madeleine Clute for assisting with this research. This work was supported in part by the Pittsburgh Life Sciences Greenhouse Opportunity Fund and by NSF grant CDI-0835797. 8 References [1] R. N. Shepard. Towards a universal law of generalization for psychological science. Science, 237:1317– 1323, 1987. [2] J. R. Anderson. The adaptive nature of human categorization. Psychological Review, 98(3):409–429, 1991. [3] E. Heit. A Bayesian analysis of some forms of inductive reasoning. In M. Oaksford and N. 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4 0.11412493 90 nips-2011-Evaluating the inverse decision-making approach to preference learning

Author: Alan Jern, Christopher G. Lucas, Charles Kemp

Abstract: Psychologists have recently begun to develop computational accounts of how people infer others’ preferences from their behavior. The inverse decision-making approach proposes that people infer preferences by inverting a generative model of decision-making. Existing data sets, however, do not provide sufﬁcient resolution to thoroughly evaluate this approach. We introduce a new preference learning task that provides a benchmark for evaluating computational accounts and use it to compare the inverse decision-making approach to a feature-based approach, which relies on a discriminative combination of decision features. Our data support the inverse decision-making approach to preference learning. A basic principle of decision-making is that knowing people’s preferences allows us to predict how they will behave: if you know your friend likes comedies and hates horror ﬁlms, you can probably guess which of these options she will choose when she goes to the theater. Often, however, we do not know what other people like and we can only infer their preferences from their behavior. If you know that a different friend saw a comedy today, does that mean that he likes comedies in general? The conclusion you draw will likely depend on what else was playing and what movie choices he has made in the past. A goal for social cognition research is to develop a computational account of people’s ability to infer others’ preferences. One computational approach is based on inverse decision-making. This approach begins with a model of how someone’s preferences lead to a decision. Then, this model is inverted to determine the most likely preferences that motivated an observed decision. An alternative approach might simply learn a functional mapping between features of an observed decision and the preferences that motivated it. For instance, in your friend’s decision to see a comedy, perhaps the more movie options he turned down, the more likely it is that he has a true preference for comedies. The difference between the inverse decision-making approach and the feature-based approach maps onto the standard dichotomy between generative and discriminative models. Economists have developed an instance of the inverse decision-making approach known as the multinomial logit model [1] that has been widely used to infer consumer’s preferences from their choices. This model has recently been explored as a psychological model [2, 3, 4], but there are few behavioral data sets for evaluating it as a model of how people learn others’ preferences. Additionally, the data sets that do exist tend to be drawn from the developmental literature, which focuses on simple tasks that collect only one or two judgments from children [5, 6, 7]. The limitations of these data sets make it difﬁcult to evaluate the multinomial logit model with respect to alternative accounts of preference learning like the feature-based approach. In this paper, we use data from a new experimental task that elicits a detailed set of preference judgments from a single participant in order to evaluate the predictions of several preference learning models from both the inverse decision-making and feature-based classes. Our task requires each participant to sort a large number of observed decisions on the basis of how strongly they indicate 1 (a) (b) (c) d c c (d) b b a a x d x d c b a x 1. Number of chosen effects (−/+) 2. Number of forgone effects (+/+) 3. Number of forgone options (+/+) 4. Number of forgone options containing x (−/−) 5. Max/min number of effects in a forgone option (+/−) 6. Is x in every option? (−/−) 7. Chose only option with x? (+/+) 8. Is x the only difference between options? (+/+) 9. Do all options have same number of effects? (+/+) 10. Chose option with max/min number of effects? (−/−) Figure 1: (a)–(c) Examples of the decisions used in the experiments. Each column represents one option and the boxes represent different effects. The chosen option is indicated by the black rectangle. (d) Features used by the weighted feature and ranked feature models. Features 5 and 10 involved maxima in Experiment 1, which focused on all positive effects, and minima in Experiment 2, which focused on all negative effects. The signs in parentheses indicate the direction of the feature that suggests a stronger preference in Experiment 1 / Experiment 2. a preference for a chosen item. Because the number of decisions is large and these decisions vary on multiple dimensions, predicting how people will order them offers a challenging benchmark on which to compare computational models of preference learning. Data sets from these sorts of detailed tasks have proved fruitful in other domains. For example, data reported by Shepard, Hovland, and Jenkins [8]; Osherson, Smith, Wilkie, L´ pez, and Shaﬁr [9]; and Wasserman, Elek, Chatlosh, o and Baker [10] have motivated much subsequent research on category learning, inductive reasoning, and causal reasoning, respectively. We ﬁrst describe our preference learning task in detail. We then present several inverse decisionmaking and feature-based models of preference learning and compare these models’ predictions to people’s judgments in two experiments. The data are well predicted by models that follow the inverse decision-making approach, suggesting that this computational approach may help explain how people learn others’ preferences. 1 Multi-attribute decisions and revealed preferences We designed a task that can be used to elicit a large number of preference judgments from a single participant. The task involves a set of observed multi-attribute decisions, some examples of which are represented visually in Figure 1. Each decision is among a set of options and each option produces a set of effects. Figure 1 shows several decisions involving a total of ﬁve effects distributed among up to ﬁve options. The differently colored boxes represent different effects and the chosen option is marked by a black rectangle. For example, 1a shows a choice between an option with four effects and an option with a single effect; here, the decision maker chose the second option. In our task, people are asked to rank a large number of these decisions by how strongly they suggest that the decision maker had a preference for a particular effect (e.g., effect x in Figure 1). By imposing some minimal constraints, the space of unique multi-attribute decisions is ﬁnite and we can obtain rankings for every decision in the space. For example, Figure 2c shows a complete list of 47 unique decisions involving up to ﬁve effects, subject to several constraints described later. Three of these decisions are shown in Figure 1. If all the effects are positive—pieces of candy, for example—the ﬁrst decision (1a) suggests a strong preference for candy x, because the decision maker turned down four pieces in favor of one. The second decision (1b), however, offers much weaker evidence because nearly everyone would choose four pieces of candy over one, even without a speciﬁc preference for x. The third decision (1c) provides evidence that is strong but perhaps not quite as strong as the ﬁrst decision. When all effects are negative—like electric shocks at different body locations—decision makers may still ﬁnd some effects more tolerable than others, but different inferences are sometimes supported. For example, for negative effects, 1a provides weak evidence that x is relatively tolerable because nearly everyone would choose one shock over four. 2 A computational account of preference learning We now describe a simple computational model for learning a person’s preferences after observing that person make a decision like the ones in Figure 1. We assume that there are n available options 2 {o1 , . . . , on }, each of which produces one or more effects from the set {f1 , f2 , ..., fm }. For simplicity, we assume that effects are binary. Let ui denote the utility the decision maker assigns to effect fi . We begin by specifying a model of decision-making that makes the standard assumptions that decision makers tend to choose things with greater utility and that utilities are additive. That is, if fj is a binary vector indicating the effects produced by option oj and u is a vector of utilities assigned to each of the m effects, then the total utility associated with option oj can be expressed as Uj = fj T u. We complete the speciﬁcation of the model by applying the Luce choice rule [11], a common psychological model of choice behavior, as the function that chooses among the options: p(c = oj |u, f ) = exp(Uj ) = exp(Uk ) n k=1 exp(fj T u) n T k=1 exp(fk u) (1) where c denotes the choice made. This model can predict the choice someone will make among a speciﬁed set of options, given the utilities that person assigns to the effects in each option. To obtain estimates of someone’s utilities, we invert this model by applying Bayes’ rule: p(u|c, F) = p(c|u, F)p(u) p(c|F) (2) where F = {f1 , . . . , fn } speciﬁes the available options and their corresponding effects. This is the multinomial logit model [1], a standard econometric model. In order to apply Equation 2 we must specify a prior p(u) on the utilities. We adopt a standard approach that places independent Gaussian priors on the utilities: ui ∼ N (µ, σ 2 ). For decisions where effects are positive—like candies—we set µ = 2σ, which corresponds to a prior distribution that places approximately 2% of the probability mass below zero. Similarly, for negative effects—like electric shocks—we set µ = −2σ. 2.1 Ordering a set of observed decisions Equation 2 speciﬁes a posterior probability distribution over utilities for a single observed decision but does not provide a way to compare the inferences drawn from multiple decisions for the purposes of ordering them. Suppose we are interested in a decision maker’s preference for effect x and we wish to order a set of decisions by how strongly they support this preference. Two criteria for ordering the decisions are as follows: Absolute utility Relative utility p(c|ux , F)p(ux ) p(c|F) p(c|∀j ux ≥ uj , F)p(∀j ux ≥ uj ) p(∀j ux ≥ uj |c, F) = p(c|F) E(ux |c, F) = Eux The absolute utility model orders decisions by the mean posterior utility for effect x. This criterion is perhaps the most natural way to assess how much a decision indicates a preference for x, but it requires an inference about the utility of x in isolation, and research suggests that people often think about the utility of an effect only in relation to other salient possibilities [12]. The relative utility model applies this idea to preference learning by ordering decisions based on how strongly they suggest that x has a greater utility than all other effects. The decisions in Figures 1b and 1c are cases where the two models lead to different predictions. If the effects are all negative (e.g., electric shocks), the absolute utility model predicts that 1b provides stronger evidence for a tolerance for x because the decision maker chose to receive four shocks instead of just one. The relative utility model predicts that 1c provides stronger evidence because 1b offers no way to determine the relative tolerance of the four chosen effects with respect to one another. Like all generative models, the absolute and relative models incorporate three qualitatively different components: the likelihood term p(c|u, F), the prior p(u), and the reciprocal of the marginal likelihood 1/p(c|F). We assume that the total number of effects is ﬁxed in advance and, as a result, the prior term will be the same for all decisions that we consider. The two other components, however, will vary across decisions. The inverse decision-making approach predicts that both components should inﬂuence preference judgments, and we will test this prediction by comparing our 3 two inverse decision-making models to two alternatives that rely only one of these components as an ordering criterion: p(c|∀j ux ≥ uj , F) 1/p(c|F) Representativeness Surprise The representativeness model captures how likely the observed decision would be if the utility for x were high, and previous research has shown that people sometimes rely on a representativeness computation of this kind [13]. The surprise model captures how unexpected the observed decision is overall; surprising decisions may be best explained in terms of a strong preference for x, but unsurprising decisions provide little information about x in particular. 2.2 Feature-based models We also consider a class of feature-based models that use surface features to order decisions. The ten features that we consider are shown in Figure 1d, where x is the effect of interest. As an example, the ﬁrst feature speciﬁes the number of effects chosen; because x is always among the chosen effects, decisions where few or no other effects belong to the chosen option suggest the strongest preference for x (when all effects are positive). This and the second feature were previously identiﬁed by Newtson [14]; we included the eight additional features shown in Figure 1d in an attempt to include all possible features that seemed both simple and relevant. We consider two methods for combining this set of features to order a set of decisions by how strongly they suggest a preference for x. The ﬁrst model is a standard linear regression model, which we refer to as the weighted feature model. The model learns a weight for each feature, and the rank of a given decision is determined by a weighted sum of its features. The second model is a ranked feature model that sorts the observed decisions with respect to a strict ranking of the features. The top-ranked feature corresponds to the primary sort key, the second-ranked feature to the secondary sort key, and so on. For example, suppose that the top-ranked feature is the number of chosen effects and the second-ranked feature is the number of forgone options. Sorting the three decisions in Figure 1 according to this criterion produces the following ordering: 1a,1c,1b. This notion of sorting items on the basis of ranked features has been applied before to decision-making [15, 16] and other domains of psychology [17], but we are not aware of any previous applications to preference learning. Although our inverse decision-making and feature-based models represent two very different approaches, both may turn out to be valuable. An inverse decision-making approach may be the appropriate account of preference learning at Marr’s [18] computational level, and a feature-based approach may capture the psychological processes by which the computational-level account is implemented. Our goal, therefore, is not necessarily to accept one of these approaches and dismiss the other. Instead, we entertain three distinct possibilities. First, both approaches may account well for the data, which would support the idea that they are valid accounts operating at different levels of analysis. Second, the inverse decision-making approach may offer a better account, suggesting that process-level accounts other than the feature-based approach should be explored. Finally, the feature-based approach may offer a better account, suggesting that inverse decision-making does not constitute an appropriate computational-level account of preference learning. 3 Experiment 1: Positive effects Our ﬁrst experiment focuses on decisions involving only positive effects. The full set of 47 decisions we used is shown in Figure 2c. This set includes every possible unique decision with up to ﬁve different effects, subject to the following constraints: (1) one of the effects (effect x) must always appear in the chosen option, (2) there are no repeated options, (3) each effect may appear in an option at most once, (4) only effects in the chosen option may be repeated in other options, and (5) when effects appear in multiple options, the number of effects is held constant across options. The ﬁrst constraint is necessary for the sorting task, the second two constraints create a ﬁnite space of decisions, and the ﬁnal two constraints limit attention to what we deemed the most interesting cases. Method 43 Carnegie Mellon undergraduates participated for course credit. Each participant was given a set of cards, with one decision printed on each card. The decisions were represented visually 4 (a) (c) Decisions 42 40 45 Mean human rankings 38 30 23 20 22 17 13 12 11 10 9 8 7 6 19 18 31 34 28 21 26 36 35 33 37 27 29 32 25 24 16 15 14 5 4 3 2 1 Absolute utility model rankings (b) Mean human rankings (Experiment 1) 47 43 44 46 45 38 37 36 34 35 30 32 33 31 29 28 24 26 27 25 21 19 22 20 18 16 17 12 13 7 6 11 5 9 4 10 8 1 2 3 42 40 41 39 47 46 44 41 43 39 23 15 14 Mean human rankings (Experiment 2) 1. dcbax 2. cbax 3. bax 4. ax 5. x 6. dcax | bcax 7. dx | cx | bx | ax 8. cax | bax 9. bdx | bcx | bax 10. dcx | bax 11. bx | ax 12. bdx | cax | bax 13. cx | bx | ax 14. d | cbax 15. c | bax 16. b | ax 17. d | c | bax 18. dc | bax 19. c | b | ax 20. dc | bx | ax 21. bdc | bax 22. ad | cx | bx | ax 23. d | c | b | ax 24. bad | bcx | bax 25. ac | bx | ax 26. cb | ax 27. cbad | cbax 28. dc | b | ax 29. ad | ac | bx | ax 30. ab | ax 31. bad | bax 32. dc | ab | ax 33. dcb | ax 34. a | x 35. bad | bac | bax 36. ac | ab | ax 37. ad | ac | ab | ax 38. b | a | x 39. ba | x 40. c | b | a | x 41. cb | a | x 42. d | c | b | a | x 43. cba | x 44. dc | ba | x 45. dc | b | a | x 46. dcb | a | x 47. dcba | x Figure 2: (a) Comparison between the absolute utility model rankings and the mean human rankings for Experiment 1. Each point represents one decision, numbered with respect to the list in panel c. (b) Comparison between the mean human rankings in Experiments 1 and 2. In both scatter plots, the solid diagonal lines indicate a perfect correspondence between the two sets of rankings. (c) The complete set of decisions, ordered by the mean human rankings from Experiment 1. Options are separated by vertical bars and the chosen option is always at the far right. Participants were always asked about a preference for effect x. as in Figure 1 but without the letter labels. Participants were told that the effects were different types of candy and each option was a bag containing one or more pieces of candy. They were asked to sort the cards by how strongly each decision suggested that the decision maker liked a particular target candy, labeled x in Figure 2c. They sorted the cards freely on a table but reported their ﬁnal rankings by writing them on a sheet of paper, from weakest to strongest evidence. They were instructed to order the cards as completely as possible, but were told that they could assign the same ranking to a set of cards if they believed those cards provided equal evidence. 3.1 Results Two participants were excluded as outliers based on the criterion that their rankings for at least ﬁve decisions were at least three standard deviations from the mean rankings. We performed a hierarchical clustering analysis of the remaining 41 participants’ rankings using rank correlation as a similarity metric. Participants’ rankings were highly correlated: cutting the resulting dendrogram at 0.2 resulted in one cluster that included 33 participants and the second largest cluster included 5 Surprise MAE = 17.8 MAE = 7.0 MAE = 4.3 MAE = 17.3 MAE = 9.5 Human rankings Experiment 2 Negative effects Representativeness MAE = 2.3 MAE = 6.7 Experiment 1 Positive effects Relative utility MAE = 2.3 Human rankings Absolute utility Model rankings Model rankings Model rankings Model rankings Figure 3: Comparison between human rankings in both experiments and predicted rankings from four models. The solid diagonal lines indicate a perfect correspondence between human and model rankings. only 3 participants. Thus, we grouped all participants together and analyzed their mean rankings. The 0.2 threshold was chosen because it produced the most informative clustering in Experiment 2. Inverse decision-making models We implemented the inverse decision-making models using importance sampling with 5 million samples drawn from the prior distribution p(u). Because all the effects were positive, we used a prior on utilities that placed nearly all probability mass above zero (µ = 4, σ = 2). The mean human rankings are compared with the absolute utility model rankings in Figure 2a, and the mean human rankings are listed in order in 2c. Fractional rankings were used for both the human data and the model predictions. The human rankings in the ﬁgure are the means of participants’ fractional rankings. The ﬁrst row of Figure 3 contains similar plots that allow comparison of the four models we considered. In these plots, the solid diagonal lines indicate a perfect correspondence between model and human rankings. Thus, the largest deviations from this line represent the largest deviations in the data from the model’s predictions. Figure 3 shows that the absolute and relative utility models make virtually identical predictions and both models provide a strong account of the human rankings as measured by mean absolute error (MAE = 2.3 in both cases). Moreover, both models correctly predict the highest ranked decision and the set of lowest ranked decisions. The only clear discrepancy between the model predictions and the data is the cluster of points at the lower left, labeled as Decisions 6–13 in Figure 2a. These are all cases in which effect x appears in all options and therefore these decisions provide no information about a decision maker’s preference for x. Consequently, the models assign the same ranking to this group as to the group of decisions in which there is only a single option (Decisions 1–5). Although people appeared to treat these groups somewhat differently, the models still correctly predict that the entire group of decisions 1–13 is ranked lower than all other decisions. The surprise and representativeness models do not perform nearly as well (MAE = 7.0 and 17.8, respectively). Although the surprise model captures some of the general trends in the human rankings, it makes several major errors. For example, consider Decision 7: dx|cx|bx|ax. This decision provides no information about a preference for x because it appears in every option. The decision is surprising, however, because a decision maker choosing at random from these options would make the observed choice only 1/4 of the time. The representativeness model performs even worse, primarily because it does not take into account alternative explanations for why an option was chosen, such as the fact that no other options were available (e.g., Decision 1 in Figure 2c). The failure of these models to adequately account for the data suggests that both the likelihood p(c|u, F) and marginal likelihood p(c|F) are important components of the absolute and relative utility models. Feature-based models We compared the performance of the absolute and relative utility models to our two feature-based models: the weighted feature and ranked feature models. For each participant, 6 (b) Ranked feature 10 10 5 Figure 4: Results of the feature-based model analysis from Experiment 1 for (a) the weighted feature models and (b) the ranked feature models. The histograms show the minimum number of features needed to match the accuracy (measured by MAE) of the absolute utility model for each participant. 15 5 1 2 3 4 5 6 >6 15 1 2 3 4 5 6 7 8 9 10 >10 Number of participants (a) Weighted feature Number of features needed we considered every subset of features1 in Figure 1d in order to determine the minimum number of features needed by the two models to achieve the same level of accuracy as the absolute utility model, as measured by mean absolute error. The results of these analyses are shown in Figure 4. For the majority of participants, at least four features were needed by both models to match the accuracy of the absolute utility model. For the weighted feature model, 14 participants could not be ﬁt as well as the absolute utility model even when all ten features were considered. These results indicate that a feature-based account of people’s inferences in our task must be supplied with a relatively large number of features. By contrast, the inverse decision-making approach provides a relatively parsimonious account of the data. 4 Experiment 2: Negative effects Experiment 2 focused on a setting in which all effects are negative, motivated by the fact that the inverse decision-making models predict several major differences in orderings when effects are negative rather than positive. For instance, the absolute utility model’s relative rankings of the decisions in Figures 1a and 1b are reversed when all effects are negative rather than positive. Method 42 Carnegie Mellon undergraduates participated for course credit. The experimental design was identical to Experiment 1 except that participants were told that the effects were electric shocks at different body locations. They were asked to sort the cards on the basis of how strongly each decision suggested that the decision maker ﬁnds shocks at the target location relatively tolerable. The model predictions were derived in the same way as for Experiment 1, but with a prior distribution on utilities that placed nearly all probability mass below zero (µ = −4, σ = 2) to reﬂect the fact that effects were all negative. 4.1 Results Three participants were excluded as outliers by the same criterion applied in Experiment 1. The resulting mean rankings are compared with the corresponding rankings from Experiment 1 in Figure 2b. The ﬁgure shows that responses based on positive and negative effects were substantially different in a number of cases. Figure 3 shows how the mean rankings compare to the predictions of the four models we considered. Although the relative utility model is fairly accurate, no model achieves the same level of accuracy as the absolute and relative utility models in Experiment 1. In addition, the relative utility model provides a poor account of the responses of many individual participants. To better understand responses at the individual level, we repeated the hierarchical clustering analysis described in Experiment 1, which revealed that 29 participants could be grouped into one of four clusters, with the remaining participants each in their own clusters. We analyzed these four clusters independently, excluding the 10 participants that could not be naturally grouped. We compared the mean rankings of each cluster to the absolute and relative utility models, as well as all one- and two-feature weighted feature and ranked feature models. Figure 5 shows that the mean rankings of participants in Cluster 1 (N = 8) were best ﬁt by the absolute utility model, the mean rankings of participants in Cluster 2 (N = 12) were best ﬁt by the relative utility model, and the mean rankings of participants in Clusters 3 (N = 3) and 4 (N = 6) were better ﬁt by feature-based models than by either the absolute or relative utility models. 1 A maximum of six features was considered for the ranked feature model because considering more features was computationally intractable. 7 Cluster 4 N =6 MAE = 4.9 MAE = 14.0 MAE = 7.9 MAE = 5.3 MAE = 2.6 MAE = 13.0 MAE = 6.2 Human rankings Relative utility Cluster 3 N =3 MAE = 2.6 Absolute utility Cluster 2 N = 12 Human rankings Cluster 1 N =8 Factors: 1,3 Factors: 1,8 MAE = 2.3 MAE = 5.2 Model rankings Best−fitting weighted feature Factors: 6,7 MAE = 4.0 Model rankings Model rankings Model rankings Human rankings Factors: 3,8 MAE = 4.8 Figure 5: Comparison between human rankings for four clusters of participants identiﬁed in Experiment 2 and predicted rankings from three models. Each point in the plots corresponds to one decision and the solid diagonal lines indicate a perfect correspondence between human and model rankings. The third row shows the predictions of the best-ﬁtting two-factor weighted feature model for each cluster. The two factors listed refer to Figure 1d. To examine how well the models accounted for individuals’ rankings within each cluster, we compared the predictions of the inverse decision-making models to the best-ﬁtting two-factor featurebased model for each participant. In Cluster 1, 7 out of 8 participants were best ﬁt by the absolute utility model; in Cluster 2, 8 out of 12 participants were best ﬁt by the relative utility model; in Clusters 3 and 4, all participants were better ﬁt by feature-based models. No single feature-based model provided the best ﬁt for more than two participants, suggesting that participants not ﬁt well by the inverse decision-making models were not using a single alternative strategy. Applying the feature-based model analysis from Experiment 1 to the current results revealed that the weighted feature model required an average of 6.0 features to match the performance of the absolute utility model for participants in Cluster 1, and an average of 3.9 features to match the performance of the relative utility model for participants in Cluster 2. Thus, although a single model did not ﬁt all participants well in the current experiment, many participants were ﬁt well by one of the two inverse decision-making models, suggesting that this general approach is useful for explaining how people reason about negative effects as well as positive effects. 5 Conclusion In two experiments, we found that an inverse decision-making approach offered a good computational account of how people make judgments about others’ preferences. Although this approach is conceptually simple, our analyses indicated that it captures the inﬂuence of a fairly large number of relevant decision features. Indeed, the feature-based models that we considered as potential process models of preference learning could only match the performance of the inverse decision-making approach when supplied with a relatively large number of features. We feel that this result rules out the feature-based approach as psychologically implausible, meaning that alternative process-level accounts will need to be explored. One possibility is sampling, which has been proposed as a psychological mechanism for approximating probabilistic inferences [19, 20]. However, even if process models that use large numbers of features are considered plausible, the inverse decision-making approach provides a valuable computational-level account that helps to explain which decision features are informative. Acknowledgments This work was supported in part by the Pittsburgh Life Sciences Greenhouse Opportunity Fund and by NSF grant CDI-0835797. 8 References [1] D. McFadden. Conditional logit analysis of qualitative choice behavior. In P. Zarembka, editor, Frontiers in Econometrics. Amademic Press, New York, 1973. [2] C. G. Lucas, T. L. Grifﬁths, F. Xu, and C. Fawcett. A rational model of preference learning and choice prediction by children. In Proceedings of Neural Information Processing Systems 21, 2009. [3] L. Bergen, O. R. Evans, and J. B. Tenenbaum. Learning structured preferences. In Proceedings of the 32nd Annual Conference of the Cognitive Science Society, 2010. [4] A. Jern and C. Kemp. Decision factors that support preference learning. In Proceedings of the 33rd Annual Conference of the Cognitive Science Society, 2011. [5] T. Kushnir, F. Xu, and H. M. Wellman. Young children use statistical sampling to infer the preferences of other people. Psychological Science, 21(8):1134–1140, 2010. [6] L. Ma and F. Xu. Young children’s use of statistical sampling evidence to infer the subjectivity of preferences. Cognition, in press. [7] M. J. Doherty. Theory of Mind: How Children Understand Others’ Thoughts and Feelings. Psychology Press, New York, 2009. [8] R. N. Shepard, C. I. Hovland, and H. M. Jenkins. Learning and memorization of classiﬁcations. Psychological Monographs, 75, Whole No. 517, 1961. [9] D. N. Osherson, E. E. Smith, O. Wilkie, A. L´ pez, and E. Shaﬁr. Category-based induction. Psychological o Review, 97(2):185–200, 1990. [10] E. A. Wasserman, S. M. Elek, D. L. Chatlosh, and A. G. Baker. Rating causal relations: Role of probability in judgments of response-outcome contingency. Journal of Experimental Psychology: Learning, Memory, and Cognition, 19(1):174–188, 1993. [11] R. D. Luce. Individual choice behavior. John Wiley, 1959. [12] D. Ariely, G. Loewenstein, and D. Prelec. Tom Sawyer and the construction of value. Journal of Economic Behavior & Organization, 60:1–10, 2006. [13] D. Kahneman and A. Tversky. Subjective probability: A judgment of representativeness. Cognitive Psychology, 3(3):430–454, 1972. [14] D. Newtson. Dispositional inference from effects of actions: Effects chosen and effects forgone. Journal of Experimental Social Psychology, 10:489–496, 1974. [15] P. C. Fishburn. Lexicographic orders, utilities and decision rules: A survey. Management Science, 20(11):1442–1471, 1974. [16] G. Gigerenzer and P. M. Todd. Fast and frugal heuristics: The adaptive toolbox. Oxford University Press, New York, 1999. [17] A. Prince and P. Smolensky. Optimality Theory: Constraint Interaction in Generative Grammar. WileyBlackwell, 2004. [18] D. Marr. Vision. W. H. Freeman, San Francisco, 1982. [19] A. N. Sanborn, T. L. Grifﬁths, and D. J. Navarro. Rational approximations to rational models: Alternative algorithms for category learning. Psychological Review, 117:1144–1167, 2010. [20] L. Shi and T. L. Grifﬁths. Neural implementation of Bayesian inference by importance sampling. In Proceedings of Neural Information Processing Systems 22, 2009. 9

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