nips nips2011 nips2011-139 knowledge-graph by maker-knowledge-mining

139 nips-2011-Kernel Bayes' Rule

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Author: Kenji Fukumizu, Le Song, Arthur Gretton

Abstract: A nonparametric kernel-based method for realizing Bayes’ rule is proposed, based on kernel representations of probabilities in reproducing kernel Hilbert spaces. The prior and conditional probabilities are expressed as empirical kernel mean and covariance operators, respectively, and the kernel mean of the posterior distribution is computed in the form of a weighted sample. The kernel Bayes’ rule can be applied to a wide variety of Bayesian inference problems: we demonstrate Bayesian computation without likelihood, and ﬁltering with a nonparametric statespace model. A consistency rate for the posterior estimate is established. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 com Abstract A nonparametric kernel-based method for realizing Bayes’ rule is proposed, based on kernel representations of probabilities in reproducing kernel Hilbert spaces. [sent-7, score-0.583]

2 The prior and conditional probabilities are expressed as empirical kernel mean and covariance operators, respectively, and the kernel mean of the posterior distribution is computed in the form of a weighted sample. [sent-8, score-0.716]

3 The kernel Bayes’ rule can be applied to a wide variety of Bayesian inference problems: we demonstrate Bayesian computation without likelihood, and ﬁltering with a nonparametric statespace model. [sent-9, score-0.34]

4 A consistency rate for the posterior estimate is established. [sent-10, score-0.106]

5 1 Introduction Kernel methods have long provided powerful tools for generalizing linear statistical approaches to nonlinear settings, through an embedding of the sample to a high dimensional feature space, namely a reproducing kernel Hilbert space (RKHS) [16]. [sent-11, score-0.285]

6 The inner product between feature mappings need never be computed explicitly, but is given by a positive deﬁnite kernel function, which permits efﬁcient computation without the need to deal explicitly with the feature representation. [sent-12, score-0.203]

7 More recently, the mean of the RKHS feature map has been used to represent probability distributions, rather than mapping single points: we will refer to these representations of probability distributions as kernel means. [sent-13, score-0.238]

8 Kernel means in characteristic RKHSs have been applied successfully in a number of statistical tasks, including the two sample problem [9], independence tests [10], and conditional independence tests [8]. [sent-15, score-0.096]

9 An advantage of the kernel approach is that these tests apply immediately to any domain on which kernels may be deﬁned. [sent-16, score-0.235]

10 We propose a general nonparametric framework for Bayesian inference, expressed entirely in terms of kernel means. [sent-17, score-0.294]

11 The goal of Bayesian inference is to ﬁnd the posterior of x given observation y; p(y|x)π(x) q(x|y) = , qY (y) = p(y|x)π(x)dµX (x), (1) qY (y) where π(x) and p(y|x) are respectively the density function of the prior, and the conditional density or likelihood of y given x. [sent-18, score-0.254]

12 In our framework, the posterior, prior, and likelihood are all expressed as kernel means: the update from prior to posterior is called the Kernel Bayes’ Rule (KBR). [sent-19, score-0.375]

13 To implement KBR, the kernel means are learned nonparametrically from training data: the prior and likelihood means are expressed in terms of samples from the prior and joint probabilities, and the posterior as a kernel mean of a weighted sample. [sent-20, score-0.691]

14 This leads to the main advantage of the KBR approach: in the absence of a speciﬁc parametric model or an analytic form for the prior and likelihood densities, we can still perform Bayesian inference by making sufﬁcient observations on the system. [sent-22, score-0.122]

15 We further 1 establish the rate of consistency of the estimated posterior kernel mean to the true posterior, as a function of training sample size. [sent-25, score-0.399]

16 The proposed kernel realization of Bayes’ rule is an extension of the approach used in [20] for state-space models. [sent-26, score-0.243]

17 This earlier work applies a heuristic, however, in which the kernel mean of the previous hidden state and the observation are assumed to combine additively to update the hidden state estimate. [sent-27, score-0.332]

18 More recently, a method for belief propagation using kernel means was proposed [18, 19]: unlike the present work, this directly estimates conditional densities assuming the prior to be uniform. [sent-28, score-0.317]

19 An alternative to kernel means would be to use nonparametric density estimates. [sent-29, score-0.31]

20 Classical approaches include ﬁnite distribution estimates on a partitioned domain or kernel density estimation, which perform poorly on high dimensional data. [sent-30, score-0.267]

21 Alternatively, direct estimates of the density ratio may be used in estimating the conditional p. [sent-31, score-0.094]

22 By contrast with density estimation approaches, KBR makes it easy to compute posterior expectations (as an RKHS inner product) and to perform conditioning and marginalization, without requiring numerical integration. [sent-35, score-0.12]

23 1 Kernel expression of Bayes’ rule Positive deﬁnite kernel and probabilities We begin with a review of some basic concepts and tools concerning statistics on RKHS [1, 3, 6, 7]. [sent-37, score-0.289]

24 Given a set Ω, a (R-valued) positive deﬁnite kernel k on Ω is a symmetric kernel k : Ω×Ω → R such n that i,j=1 ci cj k(xi , xj ) ≥ 0 for arbitrary points x1 , . [sent-38, score-0.426]

25 It is known [1] that a positive deﬁnite kernel on Ω uniquely deﬁnes a Hilbert space H (RKHS) consisting of functions on Ω, where f, k(·, x) = f (x) for any x ∈ Ω and f ∈ H (reproducing property). [sent-45, score-0.221]

26 Throughout this paper, it is assumed that positive deﬁnite kernels on the measurable spaces are measurable and bounded, where boundedness is deﬁned as supx∈Ω k(x, x) < ∞. [sent-47, score-0.078]

27 Let kX be a positive deﬁnite kernel on a measurable space (X , BX ), with RKHS HX . [sent-48, score-0.226]

28 The kernel mean mX of X on HX is deﬁned by the mean of the HX -valued random variable kX (·, X), namely mX = kX (·, x)dPX (x). [sent-49, score-0.273]

29 Since the kernel mean depends only on the distribution of X (and the kernel), it may also be written mPX ; we will use whichever of these equivalent notations is clearest in each context. [sent-51, score-0.238]

30 g, CY X f HY = E[f (X)g(Y )] ( = g ⊗ f, m(Y X) HY ⊗HX ) It should be noted that CY X is identiﬁed with the mean m(Y X) in the tensor product space HY ⊗HX , which is given by the product kernel kY kX [1]. [sent-55, score-0.256]

31 A bounded measurable positive deﬁnite kernel k is called characteristic if EX∼P [k(·, X)] = EX ′ ∼Q [k(·, X ′ )] implies P = Q: probabilities are uniquely determined by their kernel means [7, 22]. [sent-59, score-0.504]

32 With this property, problems of statistical inference can be cast in terms of inference on the kernel means. [sent-60, score-0.269]

33 A widely used characteristic kernel on Rm is the Gaussian kernel, exp(− x − y 2 /(2σ 2 )). [sent-61, score-0.234]

34 Empirical estimates of the kernel mean and covariance operator are straightforward to obtain. [sent-62, score-0.305]

35 , (Xn , Yn ) with law P , the empirical kernel mean and covariance operator are respectively (n) mX = 1 n n kX (·, Xi ), (n) CY X = i=1 (n) 1 n n i=1 kY (·, Yi ) ⊗ kX (·, Xi ), where CY X is written in the tensor product form. [sent-69, score-0.327]

36 2 Kernel Bayes’ rule We now derive the kernel mean implementation of Bayes’ rule. [sent-72, score-0.278]

37 Our goal is to obtain an estimator of the kernel mean of posterior mQX |y = kX (·, x)q(x|y)dµX (x). [sent-82, score-0.35]

38 (4), we can obtain an expression for the kernel mean of QY . [sent-88, score-0.258]

39 Assume CXX is injective, and let mΠ and mQY be the kernel means of Π in HX and QY in HY , respectively. [sent-90, score-0.203]

40 (5) −1 As discussed in [20], the operator CY X CXX implements forward ﬁltering of the prior π with the conditional density p(y|x), as in Eq. [sent-92, score-0.137]

41 In the following, we nonetheless derive a population expression of Bayes’ rule under these strong assumptions, use it as a prototype for an empirical estimator expressed in terms of Gram matrices, and prove its consistency subject to appropriate smoothness conditions on the distributions. [sent-95, score-0.151]

42 In deriving kernel realization of Bayes’ rule, we will also use Theorem 2 to obtain a kernel mean representation of the joint probability Q: −1 mQ = C(Y X)X CXX mΠ ∈ HY ⊗ HX . [sent-96, score-0.441]

43 (5), we have a population expression (7) E[kY (·, Y )|X = x] = CY X CXX −1 kX (·, x), which was used by [20, 18, 19] as the kernel mean of the conditional probability p(y|x). [sent-103, score-0.288]

44 (8) This is exactly the kernel mean of the posterior which we want to obtain. [sent-107, score-0.315]

45 Recalling that the mean mQ = m(ZW ) ∈ HX ⊗ HY can be identiﬁed with the covariance operator CZW : HY → HX , and m(W W ) ∈ HY ⊗ HY with CW W , we use Eq. [sent-110, score-0.081]

46 (8), and thus the kernel mean expression of Bayes’ rule. [sent-112, score-0.258]

47 The above argument can be rigorously implemented for empirical estimates of the kernel means and covariances. [sent-113, score-0.246]

48 sample with law P , and assume a consistent estimator for mΠ given by ℓ (ℓ) γj kX (·, Uj ), mΠ = j=1 where U1 , . [sent-120, score-0.095]

49 , Uℓ is the sample that deﬁnes the estimator (which need not be generated by Π), and γj are the weights. [sent-123, score-0.07]

50 (ii) {(Uj , γj )}ℓ : weighted sample to express the kernel i=1 j=1 mean of the prior mΠ . [sent-132, score-0.331]

51 Given conditioning value y, the kernel mean of the posterior q(x|y) is estimated by the weighted sample {(Xi , wi )}n with w = RX|Y kY (y), where kY (y) = (kY (Yi , y))n . [sent-141, score-0.368]

52 , the expression of Bayes’ rule entirely in terms of kernel means. [sent-160, score-0.263]

53 In using a weighted sample to represent the posterior, KBR has some similarity to Monte Carlo methods such as importance sampling and sequential Monte Carlo ([4]). [sent-168, score-0.079]

54 3 Consistency of KBR estimator We now demonstrate the consistency of the KBR estimator in Eq. [sent-174, score-0.099]

55 We assume that the sample size ℓ = ℓn for the prior goes to inﬁnity as the (ℓ ) sample size n for the likelihood goes to inﬁnity, and that mΠ n is nα -consistent. [sent-177, score-0.138]

56 If ℓn = n, and if mΠ is a direct empirical kernel mean with an i. [sent-192, score-0.238]

57 3 Bayesian inference with Kernel Bayes’ Rule In Bayesian inference, tasks of interest include ﬁnding properties of the posterior (MAP value, moments), and computing the expectation of a function under the posterior. [sent-197, score-0.11]

58 We now demonstrate the use of the kernel mean obtained via KBR in solving these problems. [sent-198, score-0.238]

59 First, we have already seen from Theorem 5 that we may obtain a consistent estimator under the posterior for the expectation of some f ∈ HX . [sent-199, score-0.112]

60 Next, regarding a point estimate of x, [20] proposes to use the preimage x = arg minx kX (·, x) − kT RX|Y kY (y) 2 X , which represents the posterior mean most effectively by one point. [sent-203, score-0.112]

61 In KBR the prior and likelihood are expressed in terms of samples. [sent-206, score-0.095]

62 f of the prior and/or likelihood is hard to obtain explicitly, but sampling is possible: (a) In population genetics, branching processes are used for the likelihood to model the split of species, for which the explicit density is hard to obtain. [sent-214, score-0.139]

63 Using KBR, the expectation of the posterior given y is obtained simply by the inner product as in Eq. [sent-224, score-0.077]

64 The KBR approach nonetheless has a weakness common to other nonparametric methods: if a new data point appears far from the training sample, the reliability of the output will be low. [sent-226, score-0.084]

65 Kernel choice or model selection is key to the effectiveness of KBR, as in other kernel methods. [sent-231, score-0.203]

66 KBR involves three model parameters: the kernel (or its parameters), and the regularization parameters εn and δn . [sent-232, score-0.203]

67 The strategy for parameter selection depends on how the posterior is to be used in the inference problem. [sent-233, score-0.11]

68 The posterior density q(x|y) averaged with PY is then equal to the marginal density pX . [sent-237, score-0.163]

69 We are then able to compare the discrepancy of the kernel mean of PX and the average of the estimators QX |y=Yi over Yi . [sent-238, score-0.238]

70 , n} into K disjoint subsets {Ta }K , let mQX |y be the kernel mean a=1 [−a] of posterior estimated with data {(Xi , Yi )}i∈Ta , and the prior mean mX with data {Xi }i∈Ta . [sent-243, score-0.39]

71 j X 5 Application to nonparametric state-space model. [sent-245, score-0.064]

72 We do not assume the conditional probabilities p(Yt |Xt ) and q(Xt+1 |Xt ) to be known explicitly, nor do we estimate them with simple parametric models. [sent-247, score-0.077]

73 , (XT +1 , YT +1 ) is given for both the observable and hidden variables in the training phase. [sent-251, score-0.074]

74 In ﬁltering, we wish to estimate the current hidden state xt , given observations y1 , . [sent-259, score-0.12]

75 , yt ) can be derived using KBR (we give only a sketch below; see y ˜ Supplementary material for the detailed derivation). [sent-266, score-0.105]

76 Suppose we already have an estimator of the kernel mean of p(xt |˜1 , . [sent-267, score-0.273]

77 By applying Theorem 2 twice, the y ˜ (t+1) T kY (·, Yi ), where kernel mean of p(yt+1 |˜1 , . [sent-277, score-0.238]

78 With the notation kernel Bayes’ rule yields α(t+1) = Λ(t+1) GY (Λ(t+1) GY )2 + δT IT −1 Λ(t+1) kY (˜t+1 ). [sent-288, score-0.243]

79 When the likelihood and/or prior is not obtained in an analytic form but sampling is possible, the ABC approach [25, 12, 17] is popular for Bayesian computation. [sent-297, score-0.068]

80 The ABC rejection method generates a sample from q(X|Y = y) as follows: (1) generate Xt from the prior Π, (2) generate Yt from p(y|Xt ), (3) if D(y, Yt ) < ρ, accept Xt ; otherwise reject, (4) go to (1). [sent-298, score-0.075]

81 , Xn from the prior Π, (ii) generate a sample Yt from p(y|Xt ) (t = 1, . [sent-303, score-0.075]

82 The distribution of a sample given by ABC approaches the true posterior if ρ → 0, while the empirical posterior estimate of KBR converges to the true one as n → ∞. [sent-310, score-0.189]

83 Finally, ABC generates a sample, which allows any statistic of the posterior to be approximated. [sent-312, score-0.077]

84 In the case of KBR, certain statistics of the posterior (such as conﬁdence intervals) can be harder to obtain, since consistency is guaranteed only for expectations of RKHS functions. [sent-313, score-0.106]

85 1 Experiments Nonparametric inference of posterior First we compare KBR and the standard kernel density estimation (KDE). [sent-318, score-0.356]

86 sample {Uj }ℓ j=1 Y X from the prior Π, the posterior q(x|y) is represented by the weighted sample (Ui , wi ) with wi = ℓ p(y|Ui )/ j=1 p(y|Uj ) as importance weight (IW). [sent-333, score-0.205]

87 We can thus expect that the posterior mean can be estimated effectively. [sent-339, score-0.112]

88 For the KBR method, the sample sizes n of the likelihood and prior are varied. [sent-375, score-0.103]

89 There are some works on nonparametric state-space model or HMM which use nonparametric estimation of conditional p. [sent-384, score-0.158]

90 such as KDE or partitions [27, 26] and, more recently, kernel method [20, 21]. [sent-387, score-0.203]

91 KBR has the regularization parameters εT , δT , and kernel parameters for kX and kY (e. [sent-389, score-0.203]

92 To reduce the search space, we set δT = 2εT and use the Gaussian kernel deviation βσX and βσY , where σX and σY are the median of pairwise distances among the training samples ([10]), leaving only two parameters β and εT to be tuned. [sent-393, score-0.223]

93 Next, we applied the KBR ﬁlter to the camera rotation problem used in [20]1 , where the angle of a camera is the hidden variable and the movie frames of a room taken by the camera are observed. [sent-443, score-0.145]

94 In the second setting, we exploit the fact Xt ∈ SO(3): for the Kalman ﬁlter, Xt is represented by a quanternion, and for the KBR ﬁlter the kernel k(A, B) = Tr[AB T ] is used for Xt . [sent-450, score-0.203]

95 5 Conclusion We have proposed a general, novel framework for implementing Bayesian inference, where the prior, likelihood, and posterior are expressed as kernel means in reproducing kernel Hilbert spaces. [sent-453, score-0.557]

96 The model is expressed in terms of a set of training samples, and inference consists of a small number of straightforward matrix operations. [sent-454, score-0.08]

97 We have addressed two applications: Bayesian inference without likelihood, and sequential ﬁltering with nonparametric state-space model. [sent-456, score-0.123]

98 Future studies could include more comparisons with sampling approaches like advanced Monte Carlo, and applications to various inference problems such as nonparametric Bayesian models and Bayesian reinforcement learning. [sent-457, score-0.116]

99 Dimensionality reduction for supervised learning with reproducing kernel Hilbert spaces. [sent-501, score-0.25]

100 Monte carlo hidden markov models: Learning non-parametric models of partially observable stochastic processes. [sent-666, score-0.083]

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1 0.96173942 130 nips-2011-Inductive reasoning about chimeric creatures

Author: Charles Kemp

Abstract: Given one feature of a novel animal, humans readily make inferences about other features of the animal. For example, winged creatures often ﬂy, and creatures that eat ﬁsh often live in the water. We explore the knowledge that supports these inferences and compare two approaches. The ﬁrst approach proposes that humans rely on abstract representations of dependency relationships between features, and is formalized here as a graphical model. The second approach proposes that humans rely on speciﬁc knowledge of previously encountered animals, and is formalized here as a family of exemplar models. We evaluate these models using a task where participants reason about chimeras, or animals with pairs of features that have not previously been observed to co-occur. The results support the hypothesis that humans rely on explicit representations of relationships between features. Suppose that an eighteenth-century naturalist learns about a new kind of animal that has fur and a duck’s bill. Even though the naturalist has never encountered an animal with this pair of features, he should be able to make predictions about other features of the animal—for example, the animal could well live in water but probably does not have feathers. Although the platypus exists in reality, from a eighteenth-century perspective it qualiﬁes as a chimera, or an animal that combines two or more features that have not previously been observed to co-occur. Here we describe a probabilistic account of inductive reasoning and use it to account for human inferences about chimeras. The inductive problems we consider are special cases of the more general problem in Figure 1a where a reasoner is given a partially observed matrix of animals by features then asked to infer the values of the missing entries. This general problem has been previously studied and is addressed by computational models of property induction, categorization, and generalization [1–7]. A challenge faced by all of these models is to capture the background knowledge that guides inductive inferences. Some accounts rely on similarity relationships between animals [6, 8], others rely on causal relationships between features [9, 10], and others incorporate relationships between animals and relationships between features [11]. We will evaluate graphical models that capture both kinds of relationships (Figure 1a), but will focus in particular on relationships between features. Psychologists have previously suggested that humans rely on explicit mental representations of relationships between features [12–16]. Often these representations are described as theories—for example, theories that specify a causal relationship between having wings and ﬂying, or living in the sea and eating ﬁsh. Relationships between features may take several forms: for example, one feature may cause, enable, prevent, be inconsistent with, or be a special case of another feature. For simplicity, we will treat all of these relationships as instances of dependency relationships between features, and will capture them using an undirected graphical model. Previous studies have used graphical models to account for human inferences about features but typically these studies consider toy problems involving a handful of novel features such as “has gene X14” or “has enzyme Y132” [9, 11]. Participants might be told, for example, that gene X14 leads to the production of enzyme Y132, then asked to use this information when reasoning about novel animals. Here we explore whether a graphical model approach can account for inferences 1 (a) slow heavy flies (b) wings hippo 1 1 0 0 rhino 1 1 0 0 sparrow 0 0 1 1 robin 0 0 1 1 new ? ? 1 ? o Figure 1: Inductive reasoning about animals and features. (a) Inferences about the features of a new animal onew that ﬂies may draw on similarity relationships between animals (the new animal is similar to sparrows and robins but not hippos and rhinos), and on dependency relationships between features (ﬂying and having wings are linked). (b) A graph product produced by combining the two graph structures in (a). about familiar features. Working with familiar features raises a methodological challenge since participants have a substantial amount of knowledge about these features and can reason about them in multiple ways. Suppose, for example, that you learn that a novel animal can ﬂy (Figure 1a). To conclude that the animal probably has wings, you might consult a mental representation similar to the graph at the top of Figure 1a that speciﬁes a dependency relationship between ﬂying and having wings. On the other hand, you might reach the same conclusion by thinking about ﬂying creatures that you have previously encountered (e.g. sparrows and robins) and noticing that these creatures have wings. Since the same conclusion can be reached in two different ways, judgments about arguments of this kind provide little evidence about the mental representations involved. The challenge of working with familiar features directly motivates our focus on chimeras. Inferences about chimeras draw on rich background knowledge but require the reasoner to go beyond past experience in a fundamental way. For example, if you learn that an animal ﬂies and has no legs, you cannot make predictions about the animal by thinking of ﬂying, no-legged creatures that you have previously encountered. You may, however, still be able to infer that the novel animal has wings if you understand the relationship between ﬂying and having wings. We propose that graphical models over features can help to explain how humans make inferences of this kind, and evaluate our approach by comparing it to a family of exemplar models. The next section introduces these models, and we then describe two experiments designed to distinguish between the models. 1 Reasoning about objects and features Our models make use of a binary matrix D where the rows {o1 , . . . , o129 } correspond to objects, and the columns {f 1 , . . . , f 56 } correspond to features. A subset of the objects is shown in Figure 2a, and the full set of features is shown in Figure 2b and its caption. Matrix D was extracted from the Leuven natural concept database [17], which includes 129 animals and 757 features in total. We chose a subset of these features that includes a mix of perceptual and behavioral features, and that includes many pairs of features that depend on each other. 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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