jmlr jmlr2010 jmlr2010-91 knowledge-graph by maker-knowledge-mining

91 jmlr-2010-Posterior Regularization for Structured Latent Variable Models

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Author: Kuzman Ganchev, João Graça, Jennifer Gillenwater, Ben Taskar

Abstract: We present posterior regularization, a probabilistic framework for structured, weakly supervised learning. Our framework efﬁciently incorporates indirect supervision via constraints on posterior distributions of probabilistic models with latent variables. Posterior regularization separates model complexity from the complexity of structural constraints it is desired to satisfy. By directly imposing decomposable regularization on the posterior moments of latent variables during learning, we retain the computational efﬁciency of the unconstrained model while ensuring desired constraints hold in expectation. We present an efﬁcient algorithm for learning with posterior regularization and illustrate its versatility on a diverse set of structural constraints such as bijectivity, symmetry and group sparsity in several large scale experiments, including multi-view learning, cross-lingual dependency grammar induction, unsupervised part-of-speech induction, and bitext word alignment.1 Keywords: posterior regularization framework, unsupervised learning, latent variables models, prior knowledge, natural language processing

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Our framework efﬁciently incorporates indirect supervision via constraints on posterior distributions of probabilistic models with latent variables. [sent-12, score-0.263]

2 By directly imposing decomposable regularization on the posterior moments of latent variables during learning, we retain the computational efﬁciency of the unconstrained model while ensuring desired constraints hold in expectation. [sent-14, score-0.272]

3 1 Keywords: posterior regularization framework, unsupervised learning, latent variables models, prior knowledge, natural language processing 1. [sent-16, score-0.278]

4 Unsupervised part-of-speech and grammar induction, and word and phrase alignment for statistical machine translation in natural language processing are examples of such aims. [sent-18, score-0.551]

5 Because of computational and statistical concerns, generative models used in practice are very simplistic models of the underlying phenomena; for example, the syntactic structure of language or the language translation process. [sent-31, score-0.379]

6 Unlike parametric regularization in a Bayesian framework, our approach incorporates data-dependent constraints that are easy to encode as information about model posteriors on the observed data, but may be difﬁcult to encode as information about model parameters through Bayesian priors. [sent-39, score-0.301]

7 • An extensive evaluation of different types of constraints in several domains: multi-view learning, cross-lingual dependency grammar induction, unsupervised part-of-speech induction, and bitext word alignment. [sent-43, score-0.506]

8 Section 2 describes the posterior regularization framework and Section 3 illustrates the range of different types of weak supervision constraints representable in our framework. [sent-48, score-0.313]

9 Sections 5-8 describe applications of PR to several problems: word alignment (§5), multi-view learning (§6), cross-lingual projection (§7) and inducing sparsity structure (§8). [sent-50, score-0.37]

10 Posterior Regularization Framework In this section we describe the posterior regularization framework, which incorporates side-information into parameter estimation in the form of linear constraints on posterior expectations. [sent-53, score-0.382]

11 Suppose we know that each sentence should have at least one verb and at least one noun, and would like our model to capture this constraint on the unlabeled sentences. [sent-61, score-0.306]

12 2 Regularization via Posterior Constraints The goal of the posterior regularization framework is to restrict the space of the model posteriors on unlabeled data as a way to guide the model towards desired behavior. [sent-84, score-0.395]

13 The constraint set deﬁned in Equation 1 is usually referred to as inequality constraints with slack, since setting ε = 0 enforces inequality constraints strictly. [sent-103, score-0.326]

14 A key advantage of using regularization on posteriors is that the learned model itself remains simple and tractable, while during learning it is driven to obey the constraints through setting appropriate parameters θ. [sent-121, score-0.301]

15 The advantage of imposing the constraints via KL-divergence from the posteriors is that the objective above can be optimized using a simple EM scheme described in Section 2. [sent-122, score-0.304]

16 Running Example In order to represent a corpus-wide constraint set Q for our POS problem, we stack the constraint features into a function from X, Y pairs (sentences, part-of-speech sequences) to R2|X| , where |X| is the number of sentences in our unlabeled corpus. [sent-141, score-0.448]

17 When there are a large number of constraints, or when the constraint features φ are deﬁned by some instance-speciﬁc process, it might not be easy to choose constraint values b and slack ε that lead to satisﬁable constraints. [sent-151, score-0.314]

18 Note that on unseen unlabeled data, the learned parameters θ might not satisfy the constraints on posteriors exactly, although typically they are fairly close if the model has enough capacity. [sent-231, score-0.347]

19 Note that f are features used to deﬁne the model and do not appear anywhere in the constraints while φ are constraint features that do not appear anywhere in the model. [sent-251, score-0.3]

20 In the statistical word alignment application described in Section 5, the goal is to identify pairs or sets of words that are direct translations of each other. [sent-275, score-0.411]

21 Set of alignment links between words in the sentences (exponentially many possibilities). [sent-279, score-0.277]

22 Bijective: number of times each source word is used in the alignment y. [sent-280, score-0.424]

23 Dependency parse tree (set of edges from one word to another, forming a tree). [sent-289, score-0.327]

24 Indexed by w, i, s; 1 if the ith occurrence of word w in the corpus is tagged with syntactic category s, and 0 otherwise. [sent-294, score-0.406]

25 This results in each rare word in a source language being aligned to 4 or 5 words in the target language. [sent-299, score-0.522]

26 To alleviate this problem, we introduce constraint features that count how many target words are aligned to each source word, and use PR to encourage models where this number is small in expectation. [sent-300, score-0.394]

27 Because the two languages have different syntactic structures, and errors in word alignment abound, using such out-of-language information requires the ability to train with missing and noisy labeling. [sent-307, score-0.52]

28 A well-known but difﬁcult to capture piece of prior knowledge for this problem is that each word type should only occur with a small number of syntactic categories, even though there are some syntactic categories that occur with many different word types. [sent-311, score-0.592]

29 For the word alignment application and the multi-view learning application we found decoding with the projected distribution improved performance. [sent-315, score-0.428]

30 By approximating this problem, we recover either the generalized expectation constraints framework of Mann and McCallum (2007), or with a further approximation we recover a special case of the posterior regularization framework presented in Section 2. [sent-324, score-0.272]

31 They then train a model to have high likelihood on the labeled data, but also match the constraint features on unlabeled data. [sent-381, score-0.345]

32 The constraint features take as input the unlabeled data set X as well as a full labeling Y, and produce some value φ(X, Y), which is never observed directly. [sent-395, score-0.284]

33 (1994) as hidden variables in probabilistic models for statistical machine translation (IBM models 1-5), describe the correspondence between words in source and target sentences. [sent-450, score-0.282]

34 A word alignment will be represented as a matrix with entries yi j indicating that target word i is a translation of source word j. [sent-464, score-1.046]

35 Although the original IBM models are no longer competitive for machine translation, the resulting word alignments are still a valuable resource. [sent-465, score-0.442]

36 Word alignments are used primarily for extracting minimal translation units for machine translation, for example, phrases in phrase-based translation systems (Koehn et al. [sent-466, score-0.413]

37 But their importance has grown far beyond machine translation: for instance, transferring annotations between languages by projecting POS taggers, NP chunkers and parsers through word alignment (Yarowsky and Ngai, 2001; Rogati et al. [sent-471, score-0.416]

38 Here we describe two types of prior knowledge that when introduced as constraints in different word alignment models signiﬁcantly boost their performance. [sent-476, score-0.482]

39 The two constraints are: (i) bijectivity: “one word should not translate to many words”; and (ii) symmetry: “directional alignments of one model should agree with those of another model”. [sent-477, score-0.61]

40 A more extensive description of these constraints applied to the task of word alignments and the quality of the resulting alignments can be found in Graça et al. [sent-478, score-0.757]

41 Both models can be expressed as: p(xt , y | xs ) = ∏ pd (y j | j, y j−1 )pt (xtj | xs j ), y j where y is the alignment and y j is the index of the hidden state (source language index) generating the target language word at index j. [sent-484, score-0.604]

42 The HMM model assumes that only the distance between the current and previous source word index is important pd (y j | j, y j−1 ) = pd (y j | y j − y j−1 ). [sent-487, score-0.293]

43 Both models are augmented by adding a special “null” word to the source sentence. [sent-488, score-0.293]

44 • Non-bijective: Multiple target words can align to a single source word with no penalty. [sent-496, score-0.373]

45 The top row of Figure 5 shows an example of the posterior distribution for the alignment between an English and a French sentence using the HMM model. [sent-498, score-0.349]

46 The left ﬁgure shows the alignment in the English to French direction where the rows are source words and columns are target words, while the right ﬁgure shows the alignment posteriors of the opposite direction. [sent-499, score-0.535]

47 Sure alignments are squares with borders; possible alignments are squares without borders. [sent-506, score-0.406]

48 That is, rare words in the source language end up aligned to too many words in the target language because the generative model has to distribute translation probability for each source word among all candidate target words. [sent-515, score-0.902]

49 Since the rare source word occurs in only a few sentences it needs to spread its probability mass over fewer competing target words. [sent-516, score-0.437]

50 In this case, choosing to align the rare word to all of these target words leads to higher likelihood than correctly aligning them or aligning them to the special null word, since it increases the likelihood of this sentence without lowering the likelihood of many other sentences. [sent-517, score-0.427]

51 2 Bijectivity Constraints Bijectivity constraints are based on the observation that in most gold alignments, words are aligned one-to-one (98% for the sure alignments in the Hansard corpus). [sent-523, score-0.429]

52 We simply add inequality constraints E[φ(x, y)] ≤ 1 where we have one feature for each source word j that counts how many times it is aligned to a target word in the alignment y: Bijective Features : φ j (x, y) = ∑ 1(yi = j). [sent-527, score-0.887]

53 i For example, in the alignment at the top right of Figure 5, the posteriors over the source word schism clearly sum to more than 1. [sent-528, score-0.563]

54 Note that the bijectivity constraints only hold approximately in practice and so we expect that having a KL-based penalty for them might be better than ensuring that the model satisﬁes them, since we can violate bijectivity in order to achieve higher likelihood. [sent-532, score-0.268]

55 The average entropy for the translation probability, averaged over all source language words for EM training is 2. [sent-535, score-0.276]

56 3 Symmetry Constraints Word alignment should not depend on translation direction, but this principle is clearly violated by the directional models. [sent-542, score-0.278]

57 9 However, we show that it is much better to train two directional models concurrently, coupling their posterior distributions over alignments with constraints that force them to approximately agree. [sent-545, score-0.514]

58 The feature takes the value zero in expectation if a word pair i, j is aligned with equal probability in both directions. [sent-556, score-0.312]

59 Also, one can see that in most cases the probability mass was moved to the correct place with the exception of the word pair internal/le; this is because the word internal does not appear on the French side, but the model still has to spread around the probability mass for that word. [sent-566, score-0.478]

60 Furthermore, because the constraints for different instances do not interact, we can project the sentences one at a time and we do not need to store the posteriors for the whole corpus in memory at once. [sent-570, score-0.466]

61 We compute q(y) for each sentence pair x using projected gradient ascent for the bijective constraints and L-BFGS for the symmetric constraints. [sent-571, score-0.339]

62 Following prior work by Och and Ney (2003), we initialize the Model 1 translation table with uniform probabilities over word pairs that occur together in same sentence. [sent-574, score-0.344]

63 We use a 40 word maximum length cutoff for training sentences and train all models on 100, 000 sentences, testing precision and recall on the standard test set. [sent-579, score-0.391]

64 A comprehensive comparison of the word alignment application is presented in Graça et al. [sent-590, score-0.37]

65 The alignments are also evaluated in the task of statistical machine translation where they are used as an initial step to extract parallel phrases used for translation. [sent-594, score-0.308]

66 For example we can encourage two part-of-speech taggers with different tag sets to produce compatible parts of speech, such as noun in tag set one and singular-noun in tag set 2, as opposed to noun in tag set 1 and verb in tag set 2. [sent-659, score-0.615]

67 We use content and context views, where the content view is the current word and POS tag while the context view is the previous and next words and POS tags. [sent-882, score-0.377]

68 Test data comprises 500 sentences, with 8436 sentences divided among labeled and unlabeled train data. [sent-922, score-0.305]

69 Dependency grammars are arguably more robust to transfer than constituent grammars, since syntactic relations between aligned words of parallel sentences are better conserved in translation than phrase structure (Fox, 2002; Hwa et al. [sent-935, score-0.467]

70 The two main challenges to accurate training and evaluation from aligned bitext are: (1) errors in word alignments and source language parses, (2) unaligned words due to non-literal or distant translation. [sent-937, score-0.775]

71 (2005) proposed to learn generative dependency grammars using Collins’ parser (Collins, 1999) by constructing full target parses via projected dependencies. [sent-939, score-0.34]

72 In what follows, X will indicate parallel part-of-speech tagged sentences in a bitext corpus, along with a dependency parse of the source language. [sent-950, score-0.312]

73 ) (b) Overview of our grammar induction approach via bitext: the source (English) is parsed and word-aligned with target; after ﬁltering, projected dependencies deﬁne constraints over target parse tree space, providing weak supervision for learning a target grammar. [sent-954, score-0.383]

74 An edge from English parent p to child c is called conserved if word p aligns to word p′ in the second language, c aligns to c′ in the second language, and p′ is the parent of c′ . [sent-957, score-0.564]

75 Unfortunately the sentence in Figure 8(a) is highly unusual in its amount of dependency conservation, so we need to do more than directly transfer conserved edges to get good parsing accuracy. [sent-959, score-0.375]

76 Then given a parse y, 1 the proportion of conserved edges is φ(x, y) = |Cx | ∑y∈y 1(y ∈ Cx ) and the expected proportion of conserved edges under distribution p(y | x) is E p [φ(x, y)] = 1 ∑ p(y | x). [sent-963, score-0.308]

77 |Cx | y∈Cx Consider how this constraint addresses errors in word alignment and source language parses, challenge (1) from above. [sent-964, score-0.602]

78 For each edge (i, j), xi -word is the parent word and x j word is the child word, analogously for POS tags. [sent-972, score-0.478]

79 This is possible since there is only one constraint per sentence and the constraints do not interact. [sent-980, score-0.322]

80 A parallel corpus is word-level aligned using the method described in Section 5, where we train an alignment model via PR using symmetry constraints. [sent-982, score-0.361]

81 Then, the ﬁltering stage eliminates low-conﬁdence alignments such as noun-to-verb alignments, restricts the training set to exclude possible sentence fragments, and follows the method of Klein and Manning (2004) in stripping out punctuation. [sent-985, score-0.311]

82 For both corpora, we performed word alignments with the open source PostCAT (Graça et al. [sent-1008, score-0.496]

83 Our model uses constraints of the form: the expected proportion of conserved edges in a sentence pair is at least η = 90%. [sent-1017, score-0.354]

84 We observed another desirable property of the discriminative model: While the generative model can get confused and perform poorly when the training data contains very long sentences, the discriminative parser does not appear to have this drawback. [sent-1090, score-0.267]

85 Learning algorithms typically assume that each word type can be associated with any existing tag, even though in reality each word type is only ever associated with a few tags. [sent-1100, score-0.478]

86 One well-known reason for this is that EM tends to allow each word to be generated by most POS tags some of the time. [sent-1102, score-0.29]

87 We choose the PR constraint to encourage each word to be associated with only a few parts of speech. [sent-1127, score-0.383]

88 Let the constraint feature φwti (X, Y) have value 1 whenever the ith occurrence of word w has part-of-speech tag t, and value 0 otherwise. [sent-1128, score-0.438]

89 For every word w, we would like there to be only a few POS tags t such that there are occurrences i where t has nonzero probability. [sent-1129, score-0.29]

90 This can be achieved if it “costs” a lot the ﬁrst time an occurrence of a word takes a particular tag, but afterwards future occurrences of the word can receive that same tag for free. [sent-1130, score-0.575]

91 More precisely, for each word type w, we would like the sum (ℓ1 norm), over tags t, of the maximum (ℓ∞ norm), over all occurrences wi of w, of p(wi | t), to be small; we want ∑t,w maxi p(wi | t) to be small. [sent-1131, score-0.29]

92 An advantage of using slack penalties in this case is that ℓ1 /ℓ∞ as a slack constraint in the primal would lead to a non-differentiable dual penalty term, which somewhat complicates optimization. [sent-1143, score-0.276]

93 Note that we are not requiring the posteriors to be sparse, which would be equivalent to preferring that the distribution is peaked; rather, we want a word to concentrate its tag posterior on a few tags across all instances of the word. [sent-1151, score-0.636]

94 One reason for this behavior is that this metric is very sensitive to the number of word types associated with each hidden state. [sent-1211, score-0.282]

95 VEM tends to encourage some large hidden states with many word types, which is preferable using the 1-1 mapping for large word categories such as nouns. [sent-1212, score-0.563]

96 Conclusion In this paper we have presented posterior regularization (PR), a technique for regularizing models by encoding prior knowledge in constraints on model posteriors. [sent-1227, score-0.272]

97 The applications we focus on in this paper are word alignment, multi-view learning, dependency parsing, and part-of-speech tagging. [sent-1236, score-0.31]

98 A last key extension to the current PR work is to explore the case where the constraint set Q is not easily speciﬁed using linear constraints on some constraint features φ. [sent-1244, score-0.359]

99 When performing the M-step, we use a mixture of the projected posteriors q and the model posteriors pθ (Y|X) to 2042 P OSTERIOR R EGULARIZATION FOR S TRUCTURED L ATENT VARIABLE M ODELS update the model parameters. [sent-1264, score-0.336]

100 A new string-to-dependency machine translation algorithm with a target dependency language model. [sent-1790, score-0.291]

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Abstract: For many supervised learning tasks it may be infeasible (or very expensive) to obtain objective and reliable labels. Instead, we can collect subjective (possibly noisy) labels from multiple experts or annotators. In practice, there is a substantial amount of disagreement among the annotators, and hence it is of great practical interest to address conventional supervised learning problems in this scenario. In this paper we describe a probabilistic approach for supervised learning when we have multiple annotators providing (possibly noisy) labels but no absolute gold standard. The proposed algorithm evaluates the different experts and also gives an estimate of the actual hidden labels. Experimental results indicate that the proposed method is superior to the commonly used majority voting baseline. Keywords: multiple annotators, multiple experts, multiple teachers, crowdsourcing 1. Supervised Learning From Multiple Annotators/Experts A typical supervised learning scenario consists of a training set D = {(xi , yi )}N containing N i=1 instances, where xi ∈ X is an instance (typically a d-dimensional feature vector) and yi ∈ Y is the corresponding known label. The task is to learn a function f : X → Y which generalizes well on unseen data. Speciﬁcally for binary classiﬁcation the supervision is from the set Y = {0, 1}, for multi-class classiﬁcation Y = {1, . . . , K}, for ordinal regression Y = {1, . . . , K} (with an ordering 1 < . . . < K), and Y = R for regression. c 2010 Vikas C. Raykar, Shipeng Yu, Linda H. Zhao, Gerardo H. Valadez, Charles Florin, Luca Bogoni and Linda Moy. R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY However, for many real life tasks, it may not be possible, or may be too expensive (or tedious) to acquire the actual label yi for training—which we refer to as the gold standard or the objective ground truth. Instead, we may have multiple (possibly noisy) labels y1 , . . . , yR provided by R i i different experts or annotators. In practice, there is a substantial amount of disagreement among the experts, and hence it is of great practical interest to address conventional supervised learning algorithms in this scenario. Our motivation for this work comes from the area of computer-aided diagnosis1 (CAD), where the task is to build a classiﬁer to predict whether a suspicious region on a medical image (like a X-ray, CT scan, or MRI) is malignant (cancerous) or benign. In order to train such a classiﬁer, a set of images is collected from hospitals. The actual gold standard (whether it is cancer or not) can only be obtained from a biopsy of the tissue. Since it is an expensive, invasive, and potentially dangerous process, often CAD systems are built from labels assigned by multiple radiologists who identify the locations of malignant lesions. Each radiologist visually examines the medical images and provides a subjective (possibly noisy) version of the gold standard.2 The radiologist also annotates various descriptors of the potentially malignant lesion, like the size (a regression problem), shape (a multiclass classiﬁcation problem), and also degree of malignancy (an ordinal regression problem). The radiologists come from a diverse pool including luminaries, experts, residents, and novices. Very often there is lot of disagreement among the annotations. For a lot of tasks the labels provided by the annotators are inherently subjective and there will be substantial variation among different annotators. The domain of text classiﬁcation offers such a scenario. In this context the task is to predict the category for a token of text. The labels for training are assigned by human annotators who read the text and attribute their subjective category. With the advent of crowdsourcing (Howe, 2008) services like Amazon’s Mechanical Turk,3 Games with a Purpose,4 and reCAPTCHA5 it is quite inexpensive to acquire labels from a large number of annotators (possibly thousands) in a short time (Sheng et al., 2008; Snow et al., 2008; Sorokin and Forsyth, 2008). Websites such as Galaxy Zoo6 allow the public to label astronomical images over the internet. In situations like these, the performance of different annotators can vary widely (some may even be malicious), and without the actual gold standard, it may not be possible to evaluate the annotators. In this work, we provide principled probabilistic solutions to the following questions: 1. How to adapt conventional supervised learning algorithms when we have multiple annotators providing subjective labels but no objective gold standard? 2. How to evaluate systems when we do not have absolute gold-standard? 3. A closely related problem—particularly relevant when there are a large number of annotators— is to estimate how reliable/trustworthy is each annotator. 1. See Fung et al. (2009) for an overview of the data mining issues in this area. 2. Sometimes even a biopsy cannot conﬁrm whether it is cancer or not and hence all we can hope to get is subjective ground truth. 3. Mechanical Turk found at https://www.mturk.com. 4. Games with a Purpose found at http://www.gwap.com. 5. reCAPTCHA found at http://recaptcha.net/. 6. Galaxy Zoo found at http://galaxyzoo.org. 1298 L EARNING F ROM C ROWDS 1.1 The Problem With Majority Voting When we have multiple labels a commonly used strategy is to use the labels on which the majority of them agree (or average for regression problem) as an estimate of the actual gold standard. For binary classiﬁcation problems this amounts to using the majority label,7 that is, j yi = ˆ 1 if (1/R) ∑R yi > 0.5 j=1 , j 0 if (1/R) ∑R yi < 0.5 j=1 as an estimate of the hidden true label and use this estimate to learn and evaluate classiﬁers/annotators. Another strategy is that of considering every pair (instance, label) provided by each expert as a separate example. Note that this amounts to using a soft probabilistic estimate of the actual ground truth to learn the classiﬁer, that is, R Pr[yi = 1|y1 , . . . , yR ] = (1/R) ∑ yi . i i j j=1 Majority voting assumes all experts are equally good. However, for example, if there is only one true expert and the majority are novices, and if novices give the same incorrect label to a speciﬁc instance, then the majority voting method would favor the novices since they are in a majority. One could address this problem by introducing a weight capturing how good each expert is. But how would one measure the performance of an expert when there is no gold standard available? 1.2 Proposed Approach and Organization To address the apparent chicken-and-egg problem, we present a maximum-likelihood estimator that jointly learns the classiﬁer/regressor, the annotator accuracy, and the actual true label. For ease of exposition we start with binary classiﬁcation problem in § 2. The performance of each annotator is measured in terms of the sensitivity and speciﬁcity with respect to the unknown gold standard (§ 2.1). The proposed algorithm automatically discovers the best experts and assigns a higher weight to them. In order to incorporate prior knowledge about each annotator, we impose a beta prior on the sensitivity and speciﬁcity and derive the maximum-a-posteriori estimate (§ 2.6). The ﬁnal estimation is performed by an Expectation Maximization (EM) algorithm that iteratively establishes a particular gold standard, measures the performance of the experts given that gold standard, and reﬁnes the gold standard based on the performance measures. While the proposed approach is described using logistic regression as the base classiﬁer (§ 2.2), it is quite general, and can be used with any black-box classiﬁer (§ 2.7), and can also handle missing labels (that is, each expert is not required to label all the instances). Furthermore, we extend the proposed algorithm to handle categorical (§ 3), ordinal (§ 4), and regression problems (§ 5). In § 6 section we extensively validate our approach using both simulated data and real data from different domains. 1.3 Related Work and Novel Contributions We ﬁrst summarize the novel contributions of this work in context of other related work in this emerging new area. There has been a long line of work in the biostatistics and epidemiology literature on latent variable models where the task is to get an estimate of the observer error rates based 7. When there is no clear majority among the multiple experts (that is, yi = 0.5) in CAD domain the ﬁnal decision is ˆ often made by an adjudicator or a super-expert. When there is no adjudicator a fair coin toss is used. 1299 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY on the results from multiple diagnostic tests without a gold standard (see Dawid and Skene, 1979, Hui and Walter, 1980, Hui and Zhou, 1998, Albert and Dodd, 2004 and references therein). In the machine learning community Smyth et al. (1995) ﬁrst addressed the same problem in the context of labeling volcanoes in satellite images of Venus. We differ from this previous body of work in the following aspects: 1. Unlike Dawid and Skene (1979) and Smyth et al. (1995) which just focused on estimating the ground truth from multiple noisy labels, we speciﬁcally address the issue of learning a classiﬁer. Estimating the ground truth and the annotator/classifer performance is a byproduct of our proposed algorithm. 2. In order to learn a classiﬁer Smyth (1995) proposed to ﬁrst estimate the ground truth (without using the features) and then use the probabilistic ground truth to learn a classiﬁer. In contrast, our proposed algorithm learns the classiﬁer and the ground truth jointly. Our experiments (§ 6.1.1) show that the classiﬁer learnt and ground truth obtained by the proposed algorithm is superior to that obtained by other procedures which ﬁrst estimates the ground truth and then learns the classiﬁer. 3. Our solution is more general and can be easily extended to categorical(§ 3), ordinal(§ 4), and continuous data(§ 5). It can also be used in conjunction with any supervised learning algorithm. A preliminary version of this paper (Raykar et al., 2009) mainly discussed the binary classiﬁcation problem. 4. Our proposed algorithm is also Bayesian—we impose a prior on the experts. The priors can potential capture the skill of different annotators. In this paper we refrain from doing a full Bayesian inference and use the mode of the posterior as a point estimate. A recent complete Bayesian generalization of these kind of models has been developed by Carpenter (2008). 5. The EM approach used in this paper is similar to that proposed by Jin and Ghahramani (2003). However their motivation is somewhat different. In their setting, each training example is annotated with a set of possible labels, only one of which is correct. There has been recent interest in the natural language processing (Sheng et al., 2008; Snow et al., 2008) and computer vision (Sorokin and Forsyth, 2008) communities where they use Amazon’s Mechanical Turk to collect annotations from many people. They show that it can be potentially as good as that provided by an expert. Sheng et al. (2008) analyzed when it is worthwhile to acquire new labels for some of the training examples. There is also some theoretical work (see Lugosi, 1992 and Dekel and Shamir, 2009a) dealing with multiple experts. Recently Dekel and Shamir (2009b) presented an algorithm which does not resort to repeated labeling, that is, each example does not have to be labeled by multiple teachers. Donmez et al. (2009) address the issue of active learning in this scenario—How to jointly learn the accuracy of labeling sources and obtain the most informative labels for the active learning task? There has also been some work in the medical imaging community (Warﬁeld et al., 2004; Cholleti et al., 2008). 2. Binary Classiﬁcation We ﬁrst describe our proposed noise model for the annotators. The performance of each annotator is measured in terms of the sensitivity and speciﬁcity with respect to the unknown gold standard. 1300 L EARNING F ROM C ROWDS 2.1 A Two-coin Model for Annotators Let y j ∈ {0, 1} be the label assigned to the instance x by the jth annotator/expert. Let y be the actual (unobserved) label for this instance. Each annotator provides a version of this hidden true label based on two biased coins. If the true label is one, she ﬂips a coin with bias α j (sensitivity). If the true label is zero, she ﬂips a coin with bias β j (speciﬁcity). In each case, if she gets heads she keeps the original label, otherwise she ﬂips the label. If the true label is one, the sensitivity (true positive rate) for the jth annotator is deﬁned as the probability that she labels it as one. α j := Pr[y j = 1|y = 1]. (1) On the other hand, if the true label is zero, the speciﬁcity (1−false positive rate) is deﬁned as the probability that she labels it as zero. β j := Pr[y j = 0|y = 0]. (2) The assumption introduced is that α j and β j do not depend on the instance x. For example, in the CAD domain, this means that the radiologist’s performance is consistent across different sub-groups of data.8 2.2 Classiﬁcation Model While the proposed method can be used for any classiﬁer, for ease of exposition, we consider the family of linear discriminating functions: F = { fw}, where for any x, w ∈ Rd , fw(x) = w⊤ x. The ﬁnal classiﬁer can be written in the following form: y = 1 if w⊤ x ≥ γ and 0 otherwise. The ˆ threshold γ determines the operating point of the classiﬁer. The Receiver Operating Characteristic (ROC) curve is obtained as γ is swept from −∞ to ∞. The probability for the positive class is modeled as a logistic sigmoid acting on fw, that is, Pr[y = 1|x, w] = σ(w⊤ x), where the logistic sigmoid function is deﬁned as σ(z) = 1/(1 + e−z ). This classiﬁcation model is known as logistic regression. 2.3 Estimation/Learning Problem Given the training data D consisting of N instances with annotations from R annotators, that is, D = {xi , y1 , . . . , yR }N , the task is to estimate the weight vector w and also the sensitivity α = i i i=1 [α1 , . . . , αR ] and the speciﬁcity β = [β1 , . . . , βR ] of the R annotators. It is also of interest to get an estimate of the unknown gold standard y1 , . . . , yN . 2.4 Maximum Likelihood Estimator Assuming the training instances are independently sampled, the likelihood function of the parameters θ = {w, α, β} given the observations D can be factored as N Pr[D |θ] = ∏ Pr[y1 , . . . , yR |xi , θ]. i i i=1 8. While this is a reasonable assumption, it is not entirely true. It is known that some radiologists are good at detecting certain kinds of malignant lesions based on their training and experience. 1301 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY j Conditioning on the true label yi , and also using the assumption yi is conditionally independent (of everything else) given α j , β j and yi , the likelihood can be decomposed as N Pr[D |θ] = ∏ Pr[y1 , . . . , yR |yi = 1, α]Pr[yi = 1|xi, w] i i i=1 + Pr[y1 , . . . , yR |yi = 0, β]Pr[yi = 0|xi, w] . i i Given the true label yi , we assume that y1 , . . . , yR are independent, that is, the annotators make their i i decisions independently.9 Hence, R R Pr[y1 , . . . , yR |yi = 1, α] = ∏ Pr[yi |yi = 1, α j ] = ∏ [α j ]yi [1 − α j ]1−yi . i i j j=1 j j j=1 Similarly, we have R Pr[y1 , . . . , yR |yi = 0, β] = ∏ [β j ]1−yi [1 − β j ]yi . i i j j j=1 Hence the likelihood can be written as N Pr[D |θ] = ∏ ai pi + bi (1 − pi ) , i=1 where we have deﬁned pi := σ(w⊤ xi ). R ai := ∏[α j ]y [1 − α j ]1−y . j i j i j=1 R bi := ∏[β j ]1−y [1 − β j ]y . j i j i j=1 The maximum-likelihood estimator is found by maximizing the log-likelihood, that is, ˆ θML = {α, β, w} = arg max{ln Pr[D |θ]}. ˆ ˆ ˆ θ 2.5 The EM Algorithm This maximization problem can be simpliﬁed a lot if we use the Expectation-Maximization (EM) algorithm (Dempster et al., 1977). The EM algorithm is an efﬁcient iterative procedure to compute the maximum-likelihood solution in presence of missing/hidden data. We will use the unknown hidden true label yi as the missing data. If we know the missing data y = [y1 , . . . , yN ] then the complete likelihood can be written as N ln Pr[D , y|θ] = ∑ yi ln pi ai + (1 − yi ) ln(1 − pi )bi . i=1 9. This assumption is not true in general and there is some correlations among the labels assigned by multiple annotators. For example in the CAD domain if the cancer is in advanced stage (which is very easy to detect) almost all the radiologists assign the same label. 1302 L EARNING F ROM C ROWDS Each iteration of the EM algorithm consists of two steps: an Expectation(E)-step and a Maximization(M)step. The M-step involves maximization of a lower bound on the log-likelihood that is reﬁned in each iteration by the E-step. 1. E-step. Given the observation D and the current estimate of the model parameters θ, the conditional expectation (which is a lower bound on the true likelihood) is computed as N E {ln Pr[D , y|θ]} = ∑ µi ln pi ai + (1 − µi ) ln(1 − pi )bi , (3) i=1 where the expectation is with respect to Pr[y|D , θ], and µi = Pr[yi = 1|y1 , . . . , yR , xi , θ]. Using i i Bayes’ theorem we can compute µi ∝ Pr[y1 , . . . , yR |yi = 1, θ] · Pr[yi = 1|xi, θ] i i ai pi = . ai pi + bi (1 − pi ) 2. M-step. Based on the current estimate µi and the observations D , the model parameters θ are then estimated by maximizing the conditional expectation. By equating the gradient of (3) to zero we obtain the following estimates for the sensitivity and speciﬁcity: j j αj = ∑N µi yi i=1 , ∑N µi i=1 βj = ∑N (1 − µi )(1 − yi ) i=1 . ∑N (1 − µi ) i=1 Due to the non-linearity of the sigmoid, we do not have a closed form solution for w and we have to use gradient ascent based optimization methods. We use the Newton-Raphson update given by wt+1 = wt − ηH −1 g, where g is the gradient vector, H is the Hessian matrix, and η is the step length. The gradient vector is given by N g(w) = ∑ µi − σ(w⊤ xi ) xi . i=1 The Hessian matrix is given by N H(w) = − ∑ σ(w⊤ xi ) 1 − σ(w⊤ xi ) xi x⊤ . i i=1 Essentially, we are estimating a logistic regression model with probabilistic labels µi . These two steps (the E- and the M-step) can be iterated till convergence. The log-likelihood increases monotonically after every iteration, which in practice implies convergence to a local maximum. The EM algorithm is only guaranteed to converge to a local maximum. In practice multiple restarts with different initializations can potentially mitigate the local maximum problem. In this j paper we use majority voting µi = 1/R ∑R yi as the initialization for µi to start the EM-algorithm. j=1 1303 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY 2.6 A Bayesian Approach In some applications we may want to trust a particular expert more than the others. One way to achieve this is by imposing priors on the sensitivity and speciﬁcity of the experts. Since α j and β j represent the probability of a binary event, a natural choice of prior is the beta prior. The beta prior is also conjugate to the binomial distribution. For any a > 0, b > 0, and δ ∈ [0, 1] the beta distribution is given by δa−1 (1 − δ)b−1 , Beta(δ|a, b) = B(a, b) where B(a, b) = 01 δa−1 (1 − δ)b−1 dδ is the beta function.We assume a beta prior10 for both the sensitivity and the speciﬁcity as R j j j j j j j j Pr[α j |a1 , a2 ] = Beta(α j |a1 , a2 ). Pr[β j |b1 , b2 ] = Beta(β j |b1 , b2 ). For sake of completeness we also assume a zero mean Gaussian prior on the weights w with inverse covariance matrix Γ, that is, Pr[w] = N (w|0, Γ−1 ). Assuming that {α j }, {β j }, and w have independent priors, the maximum-a-posteriori (MAP) estimator is found by maximizing the logposterior, that is, ˆ θMAP = arg max{ln Pr[D |θ] + ln Pr[θ]}. θ An EM algorithm can be derived in a similar fashion for MAP estimation by relying on the interpretation of Neal and Hinton (1998). The ﬁnal algorithm is summarized below: j 1. Initialize µi = (1/R) ∑R yi based on majority voting. j=1 2. Given µi , estimate the sensitivity and speciﬁcity of each annotator/expert as follows. j αj = j a1 − 1 + ∑N µi yi i=1 j j a1 + a2 − 2 + ∑N µi i=1 j β j = . j b1 − 1 + ∑N (1 − µi )(1 − yi ) i=1 j j b1 + b2 − 2 + ∑N (1 − µi ) i=1 . (4) The Newton-Raphson update for optimizing w is given by wt+1 = wt − ηH −1 g, with step length η, gradient vector N g(w) = ∑ µi − σ(w⊤ xi ) xi − Γw, i=1 and Hessian matrix N H(w) = − ∑ σ(w⊤ xi ) 1 − σ(w⊤ xi ) xi x⊤ − Γ. i i=1 10. It may be convenient to specify a prior in terms of the mean µ and variance σ2 . The mean and the variance for a beta prior are given by µ = a/(a + b) and σ2 = ab/((a + b)2 (a + b + 1)). Solving for a and b we get a = (−µ3 + µ2 − µσ2 )/σ2 and b = a(1 − µ)/µ. 1304 L EARNING F ROM C ROWDS 3. Given the sensitivity and speciﬁcity of each annotator and the model parameters, update µi as µi = ai pi , ai pi + bi (1 − pi ) (5) where pi = σ(w⊤ xi ). R ai = ∏[α j ]y [1 − α j ]1−y . j i j i j=1 R bi = ∏[β j ]1−y [1 − β j ]y . j i j i (6) j=1 Iterate (2) and (3) till convergence. 2.7 Discussions 1. Estimate of the gold standard The value of the posterior probability µi is a soft probabilistic estimate of the actual ground truth yi , that is, µi = Pr[yi = 1|y1 , . . . , yR , xi , θ]. The actual i i hidden label yi can be estimated by applying a threshold on µi , that is, yi = 1 if µi ≥ γ and zero otherwise. We can use γ = 0.5 as the threshold. By varying γ we can change the misclassiﬁcation costs and obtain a ground truth with large sensitivity or large speciﬁcity. Because of this in our experimental validation we can actually draw an ROC curve for the estimated ground truth. 2. Log-odds of µ A particularly revealing insight can be obtained in terms of the log-odds or the logit of the posterior probability µi . From (5) the logit of µi can be written as logit(µi ) = ln Pr[yi = 1|y1 , . . . , yR , xi , θ] µi i i = ln 1 − µi Pr[yi = 0|y1 , . . . , yR , xi , θ] i i R = w⊤ xi + c + ∑ yi [logit(α j ) + logit(β j )]. j j=1 j where c = ∑R log 1−α is a constant term which does not depend on i. This indicates that j=1 βj the estimated ground truth (in the logit form of the posterior probability) is a weighted linear combination of the labels from all the experts. The weight of each expert is the sum of the logit of the sensitivity and speciﬁcity. 3. Using any other classiﬁer For ease of exposition we used logistic regression. However, the proposed algorithm can be used with any generalized linear model or in fact with any classiﬁer that can be trained with soft probabilistic labels. In each step of the EM-algorithm, the classiﬁer is trained with instances sampled from µi . This modiﬁcation is easy for most probabilistic classiﬁers. For general black-box classiﬁers where we cannot tweak the training algorithm an alternate approach is to replicate the training examples according to the soft label. For example a probabilistic label µi = 0.8 can be effectively simulated by adding 8 training examples with deterministic label 1 and 2 examples with label 0. 1305 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY 4. Obtaining ground truth with no features In some scenarios we may not have features xi and we wish to obtain an estimate of the actual ground truth based only on the labels from multiple annotators. Here instead of learning a classiﬁer we estimate p which is the prevalence of the positive class, that is, p = Pr[yi = 1]. We further assume a beta prior for the prevalence, that is, Beta(p|p1 , p2 ). The algorithm simpliﬁes as follows. j (a) Initialize µi = (1/R) ∑R yi based on majority voting. j=1 (b) Given µi , estimate the sensitivity and speciﬁcity of each annotator using (4). The prevalence of the positive class is estimated as follows. p = p1 − 1 + ∑N µi i=1 . p1 + p2 − 2 + N (c) Given the sensitivity and speciﬁcity of each annotator and prevalence, reﬁne µi as follows. ai p µi = . ai p + bi (1 − p) Iterate (2) and (3) till convergence. This algorithm is similar to the one proposed by Dawid and Skene (1979) and Smyth et al. (1995). 5. Handling missing labels The proposed approach can easily handle missing labels, that is, when the labels from some experts are missing for some instances. Let Ri be the number of radiologists labeling the ith instance, and let N j be the number of instances labeled by the jth radiologist. Then in the EM algorithm, we just need to replace N by N j for estimating the sensitivity and speciﬁcity in (4), and replace R by Ri for updating µi in (6). 6. Evaluating a classiﬁer We can use the probability scores µi directly to evaluate classiﬁers. If zi are the labels obtained from any other classiﬁer, then sensitivity and speciﬁcity can be estimated as ∑N (1 − µi )(1 − zi ) ∑N µi zi , β = i=1 N . α = i=1 ∑N µi ∑i=1 (1 − µi ) i=1 7. Posterior approximation At the end of each EM iteration a crude approximation to the posterior is obtained as N N α j ∼ Beta α j |a1 + ∑ µi yi , a2 + ∑ µi (1 − yi ) , j j i=1 N j j i=1 N β j ∼ Beta β j |b1 + ∑ (1 − µi )(1 − yi ), b2 + ∑ (1 − µi )yi j j i=1 j j . i=1 3. Multi-class Classiﬁcation In this section we describe how the proposed approach for binary classiﬁcation can be extended to categorical data. Suppose there are K ≥ 2 categories. An example for categorical data from the CAD domain is in LungCAD, where the radiologist needs to label whether a nodule (known to be precursors of cancer) is a solid, a part-solid, or a ground glass opacity—which are three 1306 L EARNING F ROM C ROWDS different kinds on nodules. We can extend the previous model and introduce a vector of multinomial j j j parameters αc = (αc1 , . . . , αcK ) for each annotator, where αck := Pr[y j = k|y = c] j and ∑K αck = 1. Here αck denotes the probability that the annotator j assigns class k to an instance k=1 j j given the true class is c. When K = 2, α11 and α00 are sensitivity and speciﬁcity, respectively. A similar EM algorithm can be derived. In the E-step, we estimate j j R K Pr[yi = c|Y , α] ∝ Pr[yi = c|xi ] ∏ ∏ (αck )δ(yi ,k) , j j j=1 k=1 where δ(u, v) = 1 if u = v and 0 otherwise and in the M-step we learn a multi-class classiﬁer and update the multinomial parameter as j αck = j ∑N Pr[yi = c|Y , α]δ(yi , k) i=1 . ∑N Pr[yi = c|Y , α] i=1 One can also assign a Dirichlet prior for the multinomial parameters, and this results in a smoothing term in the above updates in the MAP estimate. 4. Ordinal Regression We now consider the situation where the outputs are categorical and have an ordering among the labels. In the CAD domain the radiologist often gives a score (for example, 1 to 5 from lowest to highest) to indicate how likely she thinks it is malignant. This is different from a multi-class setting in which we do not have any preference among the multiple class labels. j Let yi ∈ {1, . . . , K} be the label assigned to the ith instance by the jth expert. Note that there is an ordering in the labels 1 < . . . < K. A simple approach is to convert the ordinal data into a series of binary data (Frank and Hall, 2001). Speciﬁcally the K class ordinal labels are transformed into K − 1 binary class labels as follows: jc yi = j 1 if yi > c 0 otherwise c = 1, . . . , K − 1. Applying the same procedure used for binary labels we can estimate Pr[yi > c] for c = 1, . . . , K − 1. The probability of the actual class values can then be obtained as Pr[yi = c] = Pr[yi > c − 1 and yi ≤ c] = Pr[yi > c − 1] − Pr[yi > c]. The class with the maximum probability is assigned to the instance. 5. Regression In this section we develop a similar algorithm to learn a regression function using annotations from multiple experts. In the CAD domain as a part of the annotation process a common task for a radiologist is to measure the diameter of a suspicious lesion. 1307 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY 5.1 Model for Annotators j Let yi ∈ R be the continuous target value assigned to the ith instance by the jth annotator. Our model is that the annotator provides a noisy version of the actual true value yi . For the jth annotator we will assume a Gaussian noise model with mean yi (the true unknown value) and inverse-variance (precision) τ j , that is, j j Pr[yi |yi , τ j ] = N (yi |yi , 1/τ j ), (7) where the Gaussian distribution is deﬁned as N (z|m, σ2 ) = (2πσ2 )−1/2 exp(−(z − m)2 /2σ2 ). The unknown inverse-variance τ j measures the accuracy of each annotator—the larger the value of τ j the more accurate the annotator. We have assumed that τ j does not depend on the instance xi . For example, in the CAD domain, this means that the radiologist’s accuracy does not depend on the nodule she is measuring. While this a practical assumption, it is not entirely true. It is known that some nodules are harder to measure than others. 5.2 Linear Regression Model for Features As before we consider the family of linear regression functions: F = { fw}, where for any x, w ∈ Rd , fw(x) = w⊤ x. We assume that the actual target response yi is given by the deterministic regression function fw with additive Gaussian noise, that is, yi = w⊤ xi + ε, where ε is a zero-mean Gaussian random variable with inverse-variance (precision) γ. Hence Pr[yi |xi , w, γ] = N (yi |w⊤ xi , 1/γ). (8) 5.3 Combined Model Combining both the annotator (7) and the regressor (8) model we have Pr[yi |xi , w, τ j , γ] = j Z Pr[yi |yi , τ j ]Pr[yi |xi , w, γ]dyi = N (yi |w⊤ xi , 1/γ + 1/τ j ). j j Since the two precision terms (γ and τ j ) are grouped together they are not uniquely identiﬁable. Hence we will deﬁne a new precision term λ j as 1 1 1 = + j. j λ γ τ So we have the following model Pr[yi |xi , w, λ j ] = N (yi |w⊤ xi , 1/λ j ). j j (9) 5.4 Estimation/Learning Problem Given the training data D consisting of N instances with annotations from R experts, that is, D = {xi , y1 , . . . , yR }N , the task is to estimate the weight vector w and the precision λ = [λ1 , . . . , λR ] of i i=1 i all the annotators. 1308 L EARNING F ROM C ROWDS 5.5 Maximum-likelihood Estimator Assuming the instances are independent the likelihood of the parameters θ = {w, λ} given the observations D can be factored as N Pr[D |θ] = ∏ Pr[y1 , . . . , yR |xi , θ]. i i i=1 Conditional on the instance xi we assume that y1 , . . . , yR are independent, that is, the annotators i i provide their responses independently. Hence from (9) the likelihood can be written as N R Pr[D |θ] = ∏ ∏ N (yi |w⊤ xi , 1/λ j ). j i=1 j=1 The maximum-likelihood estimator is found by maximizing the log-likelihood θML = {λ, w} = arg max{ln Pr[D |θ]}. θ By equating the gradient of the log-likelihood to zero we obtain the following update equations for the precision and the weight vector. 1 λj = 1 N ∑ yij − w⊤ xi N i=1 N w = ∑ −1 2 . (10) N ∑R λ j yi j=1 i=1 ∑R λ j j=1 ∑ xi xi x⊤ i i=1 j . (11) As the parameters w and λ are coupled together we iterate these two steps till convergence. 5.6 Discussions 1. Is this standard least-squares? Deﬁne the design matrix X = [x1 , . . . , xN ]⊤ and the rej j sponse vector for each annotator as y j = [y1 , . . . , yN ]⊤ . Using matrix notation Equation 11 can be written as w = (X ⊤ X)−1 X ⊤ y where y = ∑R λ j y j j=1 ∑R λ j j=1 . (12) Equation 12 is essentially the solution to a standard linear regression model, except that we are training a linear regression model with y as the ground truth, which is a precision weighted mean of the response vectors from all the annotators. The variance of each annotator is estimated using (10). The ﬁnal algorithm iteratively establishes a particular gold standard (y), measures the performance of the annotators and learns a regressor given that gold standard, and reﬁnes the gold standard based on the performance measures. 2. Are we better than the best annotator? If we assume λ is ﬁxed (i.e., we ignore the variability and assume that it is well estimated) then w is an unbiased estimator of w and the ˆ covariance matrix is given by Cov(w) = Cov(y) X ⊤ X ˆ 1309 −1 = 1 ∑R λ j j=1 X ⊤X −1 . R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY Since ∑R λ j > max j (λ j ) the proposed method has a lower variance than the regressor learnt j=1 with the best annotator (i.e., the one with the minimum variance). 3. Are we better than the average? For a ﬁxed X the error in w depends only on the variance of y j . If we know the true λ j then yi is the best linear unbiased estimator for yi which minij mizes the variance. To see this consider any linear estimator of the form yi = ∑ j a j (yi − b j ). The variance is given by Var[yi ] = ∑ j (a j )2 /λ j . Since E[yi ] = yi ∑ j a j , for the bias of this estimator to be zero we require that ∑ j a j = 1. Solving the constrained minimization problem we see that a j = λ j / ∑ j λ j minimizes the variance. 4. Obtaining a consensus without features When no features are available the same algorithm can be simpliﬁed to get a consensus estimate of the actual ground truth and also evaluate the annotators. Essentially we have to iterate the following two updates till convergence ∑R λ j yi j=1 1 ∑R λ j j=1 λj j yi = = 1 N ∑ y j − yi N i=1 i 2 . 6. Experimental Validation We now experimentally validate the proposed algorithms on both simulated and real data. 6.1 Classiﬁcation Experiments We use two CAD and one text data set in our experiments. The CAD data sets include a digital mammography data set and a breast MRI data set, both of which are biopsy proven, that is, the gold standard is available. For the digital mammography data set we simulate the radiologists in order to validate our methods. The breast MRI data has annotations from four radiologists. We also report results on a Recognizing Textual Entailment data collected by Snow et al. (2008) using the Amazon’s Mechanical Turk which has annotations from 164 annotators. 6.1.1 D IGITAL M AMMOGRAPHY WITH S IMULATED R ADIOLOGISTS Mammograms are used as a screening tool to detect early breast cancer. CAD systems search for abnormal areas (lesions) in a digitized mammographic image. These lesions generally indicate the presence of malignant cancer. The CAD system then highlights these areas on the images, alerting the radiologist to the need for a further diagnostic mammogram or a biopsy. In classiﬁcation terms, given a set of descriptive morphological features for a region on a image, the task is to predict whether it is potentially malignant (1) or not (0). In order to train such a classiﬁer, a set of mammograms is collected from hospitals. The ground truth (whether it is cancer or not) is obtained from biopsy. Since biopsy is an expensive, tedious, and an invasive process, very often CAD systems are built from labels collected from multiple expert radiologists who visually examine the mammograms and mark the lesion locations—this constitutes our ground truth (multiple labels) for learning. In this experiment we use a proprietary biopsy-proven data set (Krishnapuram et al., 2008) containing 497 positive and 1618 negative examples. Each instance is described by a set of 27 morphological features. In order to validate our proposed algorithm, we simulate multiple radiologists according to the two-coin model described in § 2.1. Based on the labels from multiple radiologists, 1310 L EARNING F ROM C ROWDS we can simultaneously (1) learn a logistic-regression classiﬁer, (2) estimate the sensitivity and speciﬁcity of each radiologist, and (3) estimate the golden ground truth. We compare the results with the classiﬁer trained using the biopsy proved ground truth as well as the majority-voting baseline. For the ﬁrst set of experiments we use 5 radiologists with sensitivity α = [0.90 0.80 0.57 0.60 0.55] and speciﬁcity β = [0.95 0.85 0.62 0.65 0.58]. This corresponds to a scenario where the ﬁrst two radiologists are experts and the last three are novices. Figure 1 summarizes the results. We compare on three different aspects: (1) How good is the learnt classiﬁer? (2) How well can we estimate the sensitivity and speciﬁcity of each radiologist? (3) How good is the estimated ground truth? The following observations can be made. 1. Classiﬁer performance Figure 1(a) plots the ROC curve of the learnt classiﬁer on the training set. The dotted (black) line is the ROC curve for the classiﬁer learnt using the actual ground truth. The solid (red) line is the ROC curve for the proposed algorithm and the dashed (blue) line is for the classiﬁer learnt using the majority-voting scheme. The classiﬁer learnt using the proposed method is as good as the one learnt using the golden ground truth. The area under the ROC curve (AUC) for the proposed algorithm is around 3.5% greater than that learnt using the majority-voting scheme. 2. Radiologist performance The actual sensitivity and speciﬁcity of each radiologist is marked as a black × in Figure 1(b). The end of the solid red line shows the estimates of the sensitivity and speciﬁcity from the proposed method. We used a uniform prior on all the parameters. The ellipse plots the contour of one standard deviation as obtained from the beta posterior estimates. The end of the dashed blue line shows the estimate obtained from the majorityvoting algorithm. We see that the proposed method is much closer to the actual values of sensitivity and speciﬁcity. 3. Actual ground truth Since the estimates of the actual ground truth are probabilistic scores, we can also plot the ROC curves of the estimated ground truth. From Figure 1(b) we can see that the ROC curve for the proposed method dominates the majority voting ROC curve. Furthermore, the area under the ROC curve (AUC) is around 3% higher. The estimate obtained by majority voting is closer to the novices since they form a majority (3/5). It does not have an idea of who is an expert and who is a novice. The proposed algorithm appropriately weights each radiologist based on their estimated sensitivity and speciﬁcity. The improvement obtained is quite large in Figure 2 which corresponds a situation where we have only one expert and 7 novices. 4. Joint Estimation To learn a classiﬁer, Smyth et al. (1995) proposed to ﬁrst estimate the golden ground truth and then use the probabilistic ground truth to learn a classiﬁer. In contrast, our proposed algorithm learns the classiﬁer and the ground truth jointly as a part of the EM algorithm. Figure 3 shows that the classiﬁer and the ground truth learnt obtained by the proposed algorithm is superior than that obtained by other procedures which ﬁrst estimates the ground truth and then learns the classiﬁer. 6.1.2 B REAST MRI In this example, each radiologist reviews the breast MRI data and assesses the malignancy of each lesion on a BIRADS scale of 1 to 5. The BIRADS scale is deﬁned as follows: 1 Negative, 2 Benign, 1311 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY Majority Voting Estimated 1 Estimated 2 Estimated 3 Estimated 4 Estimated 5 EM algorithm Estimated 1 Estimated 2 Estimated 3 Estimated 4 Estimated 5 True 1 True 2 True 3 True 4 True 5 x x x x x 0.0217 0.5869 0.2391 0.1521 0.0000 0 0 0 1 0 x x x x x 0.0000 0.1785 0.1071 0.2500 0.4642 True 1 True 2 True 3 True 4 True 5 x x x x x 0.0000 0.6957 0.1304 0.1739 0.0000 0 0 0 1 0 x x x x x 0.0000 0.1428 0.0000 0.3214 0.5357 Table 1: The confusion matrix for the estimate obtained using majority voting and the proposed EM algorithm. The x indicates that there was no such category in the true labels (the gold standard). The gold-standard is obtained by the biopsy which can conﬁrm whether it is benign (BIRADS=2) or malignant (BIRADS=5). 3 Probably Benign, 4 Suspicious abnormality, and 5 Highly suggestive of malignancy. Our data set comprises of 75 lesions with annotations from four radiologists, and the true labels from biopsy. Based on eight morphological features, we have to predict whether a lesion is malignant or not. For the ﬁrst experiment we reduce the BIRADS scale to a binary one: any lesion with a BIRADS > 3 is considered malignant and benign otherwise. The set included 28 malignant and 47 benign lesions. Figure 4 summarizes the results. We show the leave-one-out cross validated ROC for the classiﬁer. The cross-validated AUC of the proposed method is approximately 6% better than the majority voting baseline. We also consider the BIRADS labels as a set of ordinal measurements since there is an ordering among the BIRADS label. The confusion matrix in Table 1 shows that the EM algorithm is signiﬁcantly superior than the majority voting in estimating the true BIRADS. 6.1.3 R ECOGNIZING T EXTUAL E NTAILMENT Finally we report results on Recognizing Textual Entailment data collected by Snow et al. (2008) using the Amazon’s Mechanical Turk. In this task, the annotator is presented with two sentences and given a choice of whether the second sentence can be inferred from the ﬁrst. The data has 800 tasks and 164 distinct readers, with 10 annotations per task along with the golden ground truth. The majority of the entries (94 %) in the 800x164 matrix are missing. There is one annotator who has labeled all the tasks. We use this data set to obtain an estimate of the actual ground truth. Figure 5 plots the accuracy of the estimated ground truth as a function of the number of annotators. The proposed EM algorithm achieves a higher accuracy than majority voting. In other words to achieve a desired accuracy the proposed algorithm needs fewer annotators than the majority voting scheme. 1312 L EARNING F ROM C ROWDS ROC Curve for the classifier 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Golden ground truth AUC=0.915 Proposed EM algorithm AUC=0.913 Majority voting baseline AUC=0.882 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (a) ROC Curve for the estimated true labels 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Proposed EM algorithm AUC=0.991 Majority voting baseline AUC=0.962 0.2 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (b) Figure 1: Results for the digital mammography data set with annotations from 5 simulated radiologists. (a) The ROC curve of the learnt classiﬁer using the golden ground truth (dotted black line), the majority voting scheme (dashed blue line), and the proposed EM algorithm (solid red line). (b) The ROC curve for the estimated ground truth. The actual sensitivity and speciﬁcity of each of the radiologists is marked as a ×. The end of the dashed blue line shows the estimates of the sensitivity and speciﬁcity obtained from the majority voting algorithm. The end of the solid red line shows the estimates from the proposed method. The ellipse plots the contour of one standard deviation. 1313 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY ROC Curve for the classifier 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Golden ground truth AUC=0.915 Proposed EM algorithm AUC=0.906 Majority voting baseline AUC=0.884 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (a) ROC Curve for the estimated true labels 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 Proposed EM algorithm AUC=0.967 Majority voting baseline AUC=0.872 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (b) Figure 2: Same as Figure 1 except with 8 different radiologist annotations. 1314 L EARNING F ROM C ROWDS ROC Curve for the classifier 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 Proposed EM algorithm [Joint Estimation] AUC=0.905 Decoupled Estimation AUC=0.884 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (a) ROC Curve for the estimated true labels 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 Proposed EM algorithm [Joint Estimation] AUC=0.972 Decoupled Estimation AUC=0.921 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (b) Figure 3: ROC curves comparing the proposed algorithm (solid red line) with the Decoupled Estimation procedure (dotted blue line), which refers to the algorithm where the ground truth is ﬁrst estimated using just the labels from the ﬁve radiologists and then a logistic regression classiﬁer is trained using the soft probabilistic labels. In contrast the proposed EM algorithm estimates the ground truth and learns the classiﬁer simultaneously during the EM algorithm. 1315 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY Leave−One−Out ROC Curve for the classifier 1 True Positive Rate (sensitivity) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Golden ground truth AUC=0.909 Majority voting baseline AUC=0.828 Proposed EM algorithm AUC=0.879 0.2 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (a) ROC Curve for the estimated true labels 1 0.9 True Positive Rate (sensitivity) 0.8 0.7 0.6 0.5 0.4 Proposed EM algorithm AUC=0.944 Majority voting baseline AUC=0.937 0.3 0.2 0.1 0 0 0.2 0.4 0.6 False Positive Rate (1−specifcity) 0.8 1 (b) Figure 4: Breast MRI results. (a) The leave-one-out cross validated ROC. (b) ROC for the estimated ground truth. 1316 L EARNING F ROM C ROWDS 0.95 0.9 0.85 Accuracy 0.8 0.75 0.7 0.65 Majority Voting EM Algorithm 0.6 0.55 0.5 20 40 60 80 100 120 Number of Annotators 140 160 Figure 5: The mean and the one standard deviation error bars for the accuracy of the estimated ground truth for the Recognizing Textual Entailment task as a function of the number of annotators. The plot was generated by randomly sampling the annotators 100 times. 6.2 Regression Experiments We ﬁrst illustrate the algorithm on a toy dataset and then present a case study for automated polyp measurements. 6.2.1 I LLUSTRATION Figure 6 illustrates the the proposed algorithm for regression on a one-dimensional toy data set with three annotators. The actual regression model (shown as a blue dotted line) is given by y = 5x − 2. We simulate 20 samples from three annotators with precisions 0.01, 0.1, and 1.0. The data are shown by the annotators’s number. While we can ﬁt a regression model using each annotators’s response, we see that only the model for annotator three (with highest precision) is close to the true regression model. The green dashed line shows the model learnt using the average response from all the three annotators. The red line shows the model learnt by the proposed algorithm. 6.2.2 AUTOMATED P OLYP M EASUREMENTS Colorectal polyps are small colonic ﬁndings that may develop into cancer at a later stage. The diameter of the polyp is one of the key factors which decides the malignancy of a suspicious polyp. Hence accurate size estimation is crucial to decide the action to be taken on a polyp. We have developed various algorithms to segment a polyp. Multiple segmentation algorithms give rise to a set of features which are correlated with the diameter of the polyp. We want to learn a regression function which can predict the diameter of a polyp as a function of these features. In order to learn a regression function we collect our ground truth by asking many radiologists to manually measure the the diameter of the polyps from the three-dimensional images. In practice there is a lot of disagreement among the radiologists as to the actual size of the polyp. 1317 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY 2 1 N=50 examples R=3 annotators 2 2 5 Actual regression model Model with annotator 1 (Precision=0.01) 2 1 Model with annotator 2 (Precision=0.10) 2 Model with annotator 3 (Precision=1.00) 3 2 3 2 Model with average response 2 Proposed algorithm 2 3 1 1 4 3 2 2 y (response) 2 2 1 3 2 3 3 2 3 1 −1 3 3 2 2 3 3 1 2 2 2 3 33 3 1 1 3 3 3 3 2 3 3 2 33 33 2 2 2 3 3 3 32 1 2 3 2 1 3 3 2 1 2 1 2 2 1 1 2 3 2 3 2 3 1 3 3 −3 2 3 3 2 3 3 3 2 3 3 2 1 3 3 2 0 −2 2 2 3 2 3 2 3 1 2 2 1 2 3 −4 1 2 2 1 −5 0 2 0.1 0.2 1 0.3 0.4 0.5 0.6 10.8 2 0.7 x (feature) 0.9 1 1 Figure 6: Illustration of the proposed algorithm on a one-dimensional toy data set. The actual regression model (shown as a blue dotted line) is given by y = 5x − 2. We simulate 50 samples from three annotators with precisions 0.01, 0.1, and 1.0. The data are shown by the annotators’s number. While we can ﬁt a regression model using each annotators’s response, we see that only the model for annotator three (with highest precision) is close to the true regression model. The green dashed line shows the model learnt using the average response from all the three annotators. The red line shows the model learnt by the proposed algorithm. 5 5 10 Actual Polyp Diameter (mm) (a) Gold standard model 15 10 5 0 0 Pearson Correlation Coefficient=0.554966 RMSE=2.887740 15 Estimated Polyp Diameter (mm) 10 0 0 Pearson Correlation Coefficient=0.706558 RMSE=1.991576 15 Estimated Polyp Diameter (mm) Estimated Polyp Diameter (mm) Pearson Correlation Coefficient=0.714720 RMSE=1.970815 15 5 10 Actual Polyp Diameter (mm) (b) Proposed model 15 10 5 0 0 5 10 Actual Polyp Diameter (mm) 15 (c) Average model Figure 7: Scatter plot of the actual polyp diameter vs the diameter predicted by the models learnt using (a) the actual gold standard, (b) the proposed algorithm with annotations from ﬁve radiologists, and (c) the average of the radiologist’s annotations. (See § 6.2.2 for a description of the experimental setup.) 1318 L EARNING F ROM C ROWDS We use a proprietary data set containing 393 examples (which point to 285 distinct polyps— the segmentation algorithms generally return multiple marks on the same polyp.) along with the measured diameter (ranging from 2mm to 15mm) as our training set. Each example is described by a set of 60 morphological features which are correlated to the diameter of the polyp. In order to validate the feasibility of our proposed algorithm, we simulate ﬁve radiologists according to the noisy model described in § 5.1 with τ = [0.001 0.01 0.1 1 10]. This corresponds to a situation where the ﬁrst three radiologists are extremely noisy and the last two are quite accurate. Based on the measurements from multiple radiologists, we can simultaneously (1) learn a linear regressor and (2) estimate the precision of each radiologist. We compare the results with the classiﬁer trained using the actual golden ground truth as well as the regressor learnt using the average of the radiologists measurements. The results are validated on an independent test set containing 397 examples (which point to 298 distinct polyps). Figure 7 shows the scatter plot of the actual polyp diameter vs the diameter predicted by the three different models. We compare the performance based on the root mean squared error (RMSE) and also the Pearson’s correlation coefﬁcient. The regressor learnt using the proposed iterative algorithm (Figure 7(b)) is almost as good as the one learnt using the golden ground truth (Figure 7(a)). The correlation coefﬁcient for the proposed algorithm is signiﬁcantly larger than that learnt using the average of the radiologists response. The estimate obtained by averaging is closer to the novices since they form a majority (3/5). The proposed algorithm appropriately weights each radiologist based on their estimated precisions. 7. Conclusions and Future Work In this paper we proposed a probabilistic framework for supervised learning with multiple annotators providing labels but no absolute gold standard. The proposed algorithm iteratively establishes a particular gold standard, measures the performance of the annotators given that gold standard, and then reﬁnes the gold standard based on the performance measures. We speciﬁcally discussed binary/categorical/ordinal classiﬁcation and regression problems. We made two key assumptions: (1) the performance of each annotator does not depend on the feature vector for a given instance and (2) conditional on the truth the experts are independent, that is, they make their errors independently. As we pointed out earlier these assumptions are not true in practice. The annotator performance depends on the instance he is labeling and there is some degree of correlation among the annotators. We brieﬂy discuss some strategies to relax these two assumptions. 7.1 Instance Difﬁculty One drawback of the current model is that it doesn’t estimate difﬁculty of items. It is often observed that for the easy instances all the annotators agree on the labels—thus violating our conditional independence assumption. The difﬁculty of annotating an item can be captured by another latent variable γi for each instance—which modulates the annotators performance. Models for this have been developed in the area of item-response theory (Baker and Kim, 2004) and also in epidemiology (Uebersax and Grove, 1993)—see also Whitehill et al. (2009) for a recent paper in the machine learning community. While these models do not take into account the available features our pro1319 R AYKAR , Y U , Z HAO , VALADEZ , F LORIN , B OGONI AND M OY posed model for sensitivity and speciﬁcity can be extended as follows (in place of (1) and (2)): α j (γi ) := Pr[yi = 1|yi = 1, γi ] = σ(a j1 + b j1 γi ). j β j (γi ) := Pr[yi = 0|yi = 0, γi ] = σ(a j0 + b j0 γi ). j Here the parameters a j1 and a j0 are related to the sensitivity and speciﬁcity of the jth annotator, while the latent term γi captures the difﬁculty of the instance. The key assumption here is that the annotators are independent conditional on both yi and γi . Various assumptions can be made on two parameters b j1 and b j0 to simplify these models further—for example we could set b j1 = b1 and b j0 = b0 for all the annotators. 7.2 Annotators Actually Look at the Data In our model we made the assumption that the sensitivity α j and the speciﬁcity β j of the jth annotator does not depend on the feature vector xi . For example, in the CAD domain, this meant that the radiologist’s performance is consistent across different sub-groups of data—which is not entirely true. It is known that some radiologists are good at detecting certain kinds of malignant lesions based on their training and experience. We can extend the previous model such that the sensitivity and the speciﬁcity depends on the feature vector xi explicitly as follows j⊤ α j (γi , xi ) := Pr[yi = 1|yi = 1, γi , xi ] = σ(a j1 + b j1 γi + wα xi ). j j⊤ α j (γi , xi ) := Pr[yi = 0|yi = 0, γi , xi ] = σ(a j0 + b j0 γi + wβ xi ). j However this change increases the number of parameters to be learned. References P. S. Albert and L. E. Dodd. A cautionary note on the robustness of latent class models for estimating diagnostic error without a gold standard. Biometrics, 60:427–435, 2004. F. B. Baker and S. Kim. Item Response Theory: Parameter Estimation Techniques. CRC Press, 2 edition, 2004. B. Carpenter. 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