nips nips2006 nips2006-41 knowledge-graph by maker-knowledge-mining

41 nips-2006-Bayesian Ensemble Learning

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Author: Hugh A. Chipman, Edward I. George, Robert E. Mcculloch

Abstract: We develop a Bayesian “sum-of-trees” model, named BART, where each tree is constrained by a prior to be a weak learner. Fitting and inference are accomplished via an iterative backﬁtting MCMC algorithm. This model is motivated by ensemble methods in general, and boosting algorithms in particular. Like boosting, each weak learner (i.e., each weak tree) contributes a small amount to the overall model. However, our procedure is deﬁned by a statistical model: a prior and a likelihood, while boosting is deﬁned by an algorithm. This model-based approach enables a full and accurate assessment of uncertainty in model predictions, while remaining highly competitive in terms of predictive accuracy. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 McCulloch Graduate School of Business University of Chicago Chicago, IL, 60637 Abstract We develop a Bayesian “sum-of-trees” model, named BART, where each tree is constrained by a prior to be a weak learner. [sent-4, score-0.304]

2 This model is motivated by ensemble methods in general, and boosting algorithms in particular. [sent-6, score-0.212]

3 However, our procedure is deﬁned by a statistical model: a prior and a likelihood, while boosting is deﬁned by an algorithm. [sent-10, score-0.226]

4 This model-based approach enables a full and accurate assessment of uncertainty in model predictions, while remaining highly competitive in terms of predictive accuracy. [sent-11, score-0.123]

5 + gm (x) where each gi denotes a binary regression tree. [sent-16, score-0.112]

6 It is vastly more ﬂexible than a single tree model which does not easily incorporate additive effects. [sent-18, score-0.284]

7 Because multivariate components can easily account for high order interaction effects, a sum-of-trees model is also much more ﬂexible than typical additive models that use low dimensional smoothers as components. [sent-19, score-0.094]

8 We specify a prior, and then obtain a sequence of draws from the posterior using Markov chain Monte Carlo (MCMC). [sent-21, score-0.187]

9 First, with m chosen large, it restrains the ﬁt of each individual gi so that the overall ﬁt is made up of many small contributions in the spirit of boosting (Freund & Schapire (1997), Friedman (2001)). [sent-23, score-0.191]

10 The prior speciﬁcation is kept simple and a default choice is shown to have good out of sample predictive performance. [sent-26, score-0.255]

11 Inferential uncertainty is naturally quantiﬁed in the usual Bayesian way: variation in the MCMC draws of f = gi (evaluated at a set of x of interest) and σ indicates our beliefs about plausible values given the data. [sent-27, score-0.22]

12 Note that the depth of each tree is not ﬁxed so that we infer the level of interaction. [sent-28, score-0.188]

13 Thus, our procedure captures ensemble learning (in which many trees are combined) both in the fundamental sum-of-trees speciﬁcation and in the model-averaging used to obtain the estimate. [sent-30, score-0.197]

14 1 A Sum-of-Trees Model To elaborate the form of a sum-of-trees model, we begin by establishing notation for a single tree model. [sent-33, score-0.188]

15 Let T denote a binary tree consisting of a set of interior node decision rules and a set of terminal nodes, and let M = {µ1 , µ2 , . [sent-34, score-0.435]

16 , µB } denote a set of parameter values associated with each of the B terminal nodes of T . [sent-37, score-0.211]

17 Prediction for a particular value of input vector x is accomplished as follows: If x is associated with terminal node b of T by the sequence of decision rules from top to bottom, it is then assigned the µb value associated with this terminal node. [sent-38, score-0.457]

18 (1) (2) Unlike the single tree model, when m > 1 the terminal node parameter µi given by g(x; Tj , Mj ) is merely part of the conditional mean of Y given x. [sent-41, score-0.435]

19 Such terminal node parameters will represent interaction effects when their assignment depends on more than one component of x (i. [sent-42, score-0.309]

20 Because (1) may be based on trees of varying sizes, the sum-of-trees model can incorporate both direct effects and interaction effects of varying orders. [sent-45, score-0.216]

21 In the special case where every terminal node assignment depends on just a single component of x, the sum-of-trees model reduces to a simple additive function. [sent-46, score-0.31]

22 With a large number of trees, a sum-of-trees model gains increased representation ﬂexibility, which, when coupled with our regularization prior, gives excellent out of sample predictive performance. [sent-47, score-0.086]

23 2 A Regularization Prior The complexity of the prior speciﬁcation is vastly simpliﬁed by letting the Ti be i. [sent-54, score-0.121]

24 Given these independence assumptions we need only choose priors for a single tree T , a single µ, and σ. [sent-59, score-0.188]

25 Motivated by our desire to make each g(x; Ti , Mi ) a small contribution to the overall ﬁt, we put prior weight on small trees and small µi,b . [sent-60, score-0.211]

26 For the tree prior, we use the same speciﬁcation as in Chipman, George & McCulloch (1998). [sent-61, score-0.188]

27 In all examples we use the same prior corresponding to the choice α = . [sent-63, score-0.088]

28 With this choice, trees with 1, 2, 3, 4, and ≥ 5 terminal nodes receive prior probability of 0. [sent-65, score-0.422]

29 Note that even with this prior, trees with many terminal nodes can be grown if the data demands it. [sent-71, score-0.334]

30 At any non-terminal node, the prior on the associated decision rule puts equal probability on each available variable and then equal probability on each available rule given the variable. [sent-72, score-0.088]

31 For the prior on a µ, we start by simply shifting and rescaling Y so that we believe the prior proba2 bility that E(Y | x) ∈ (−. [sent-73, score-0.176]

32 For example if k = 2 there is a 95% (conditional) prior probability that the mean of Y is in (−. [sent-99, score-0.088]

33 k = 2 is our default choice and in practice we typically rescale the response y so that its observed values range from -5. [sent-102, score-0.142]

34 Note that this prior increases the shrinkage of µi,b (toward zero) as m increases. [sent-105, score-0.13]

35 For the prior on σ we start from the usual inverted-chi-squared prior: σ 2 ∼ ν λ/χ2 . [sent-106, score-0.088]

36 99, and set λ so that the qth quantile of the prior on σ is located at σ , that is P (σ < σ ) = q. [sent-112, score-0.175]

37 For automatic use, we recommend the default setting (ν, q) = (3, 0. [sent-118, score-0.111]

38 Strong prior beliefs that σ is very small could lead to over-ﬁtting. [sent-122, score-0.088]

39 3 A Backﬁtting MCMC Algorithm Given the observed data y, our Bayesian setup induces a posterior distribution p((T1 , M1 ), . [sent-123, score-0.087]

40 For notational convenience, let T(i) be the set of all trees in the sum except Ti , and similarly deﬁne M(i) . [sent-129, score-0.123]

41 The Gibbs sampler here entails m successive draws of (Ti , Mi ) conditionally on (T(i) , M(i) , σ): (T1 , M1 )|T(1) , M(1) , σ, y (T2 , M2 )|T(2) , M(2) , σ, y . [sent-130, score-0.1]

42 (Tm , Mm )|T(m) , M(m) , σ, y, (3) followed by a draw of σ from the full conditional: σ|T1 , . [sent-133, score-0.075]

43 (4) Hastie & Tibshirani (2000) considered a similar application of the Gibbs sampler for posterior sampling for additive and generalized additive models with σ ﬁxed, and showed how it was a stochastic generalization of the backﬁtting algorithm for such models. [sent-140, score-0.245]

44 In contrast with the stagewise nature of most boosting algorithms (Freund & Schapire (1997), Friedman (2001), Meek, Thiesson & Heckerman (2002)), the backﬁtting MCMC algorithm repeatedly resamples the parameters of each learner in the ensemble. [sent-142, score-0.233]

45 The idea is that given (T(i) , M(i) ) and σ we may subtract the ﬁt from (T(i) , M(i) ) from both sides of (1) leaving us with a single tree model with known error variance. [sent-143, score-0.188]

46 This draw may be made following the approach of Chipman et al. [sent-144, score-0.105]

47 These methods draw (Ti , Mi ) | T(i) , M(i) , σ, y as Ti | T(i) , M(i) , σ, y followed by Mi | Ti , T(i) , M(i) , σ, y. [sent-146, score-0.075]

48 The ﬁrst draw is done by the Metropolis-Hastings algorithm after integrating out Mi and the second is a set of normal draws. [sent-147, score-0.075]

49 The draw of σ is easily accomplished by subtracting all the ﬁt from both sides of (1) so the the are considered to be observed. [sent-148, score-0.171]

50 The Metropolis-Hastings draw of Ti | T(i) , M(i) , σ, y is complex and lies at the heart of our method. [sent-150, score-0.075]

51 (1998) proposes a new tree based on the current tree using one of four moves. [sent-152, score-0.376]

52 The moves and their associated proposal probabilities are: growing a terminal node (0. [sent-153, score-0.247]

53 Although the grow and prune moves change the implicit dimensionality of the proposed tree in terms of the number of terminal nodes, by integrating out Mi from the posterior, we avoid the complexities associated with reversible jumps between continuous spaces of varying dimensions (Green 1995). [sent-158, score-0.424]

54 At each iteration, each tree may increase or decrease the number of terminal nodes by one, or change one or two decision rules. [sent-160, score-0.399]

55 It is not uncommon for a tree to grow large and then subsequently collapse back down to a single node as the algorithm iterates. [sent-162, score-0.368]

56 The sum-of-trees model, with its abundance of unidentiﬁed parameters, allows for “ﬁt” to be freely reallocated from one tree to another. [sent-163, score-0.188]

57 Compared to the single tree model MCMC approach of Chipman et al. [sent-165, score-0.218]

58 When only single tree models are considered, the MCMC algorithm tends to quickly gravitate toward a single large tree and then gets stuck in a local neighborhood of that tree. [sent-167, score-0.376]

59 In some ways backﬁtting MCMC is a stochastic alternative to boosting algorithms for ﬁtting linear combinations of trees. [sent-170, score-0.138]

60 It is distinguished by the ability to sample from a posterior distribution. [sent-171, score-0.087]

61 At each iteration, we get a new draw f ∗ = g(x; T1 , M1 ) + g(x; T2 , M2 ) + . [sent-172, score-0.075]

62 + g(x; Tm , Mm ) (5) corresponding to the draw of Tj and Mj . [sent-175, score-0.075]

63 These draws are a (dependent) sample from the posterior distribution on the “true” f . [sent-176, score-0.187]

64 Rather than pick the “best” f ∗ from these draws, the set of multiple draws can be used to further enhance inference. [sent-177, score-0.142]

65 We estimate f by the posterior mean of f which is approximated by averaging the f ∗ over the draws. [sent-178, score-0.087]

66 For example, we can use the 5% and 95% quantiles of f ∗ (x) to obtain 90% posterior intervals for f (x). [sent-180, score-0.191]

67 All datasets correspond to regression problems with between 3 and 28 numeric predictors and 0 to 6 categorical predictors. [sent-202, score-0.201]

68 As competitors we considered linear regression with L1 regularization (the Lasso) (Efron, Hastie, Johnstone & Tibshirani 2004) and four black-box models: Friedman’s (2001) gradient boosting, random forests (Breiman 2001), and neural networks with one layer of hidden units. [sent-205, score-0.269]

69 We considered two versions of our Bayesian ensemble procedure BART. [sent-208, score-0.106]

70 In BART-cv, the prior hyperparameters (ν, q, k, m) were treated as operational parameters to be tuned via cross-validation. [sent-209, score-0.153]

71 For both BART-cv and BART-default, all speciﬁcations of the quantile q were made relative to the least squares linear regression estimate σ , and the number of burn-in steps and MCMC iterations used were determined by inspection of a ˆ single long run. [sent-212, score-0.146]

72 For example, with 20 predictors we used 3, 8 and 21 as candidate values for the number of hidden units. [sent-222, score-0.103]

73 Finally, in order to enable performance comparisons across all datasets, after possible nonlinear transformation, the resultant response was scaled to have sample mean 0 and standard deviation 1 prior to any train/test splitting. [sent-229, score-0.119]

74 Each learner has the following percentage of ratios larger than 2. [sent-249, score-0.095]

75 0, which are not plotted above: Neural net: 5%, BART-cv: 6%, BART-default and Boosting: 7%, Random forests 10% and Lasso 21%. [sent-250, score-0.115]

76 each of the 840 experiments, the learner with smallest RMSE was identiﬁed. [sent-251, score-0.095]

77 The relative ratio for each learner is the raw RMSE divided by the smallest RMSE. [sent-252, score-0.095]

78 Thus a relative RMSE of 1 means that the learner had the best performance in a particular experiment. [sent-253, score-0.095]

79 The strong performance of our “default” ensemble is especially noteworthy, since it requires no selection of operational parameters. [sent-261, score-0.139]

80 This results in a huge computational savings, since under cross-validation, the number of times a learner must be trained is equal to the number of settings times the number of folds. [sent-263, score-0.095]

81 Not only is the default version of BART faster, but it also provides valid statistical inference, a beneﬁt not available to any of the other learners considered. [sent-270, score-0.142]

82 We applied BART to all 506 observations of the Boston Housing data using the default setting (ν, q, k, m) = (3, 0. [sent-272, score-0.111]

83 At each of the 506 predictor values x, we used 5% and ˆ 95% quantiles of the MCMC draws to obtain 90% posterior intervals for f (x). [sent-274, score-0.291]

84 An appealing feature of these posterior intervals is that they widen when there is less information about f (x). [sent-275, score-0.197]

85 (b) Partial dependence plot for the effect of crime on the response (log median property value), with 90% uncertainty bounds. [sent-295, score-0.227]

86 To see how the width of the 90% posterior intervals corresponded to Dx , we plotted them together in Figure 3(a). [sent-297, score-0.153]

87 Since BART provides posterior draws for f (x), calculation of a posterior distribution for the partial dependence function is straightforward. [sent-300, score-0.274]

88 For the Boston Housing data, Figure 3(b) shows the partial dependence plot for crime, with 90% posterior intervals. [sent-303, score-0.087]

89 The vast majority of data values occur for crime < 5, causing the intervals to widen as crime increases and the data become more sparse. [sent-304, score-0.302]

90 5 Discussion Our approach is a fully Bayesian approach to learning with ensembles of tree models. [sent-305, score-0.188]

91 Because of the nature of the underlying tree model, we are able to specify simple, effective priors and fully exploit the beneﬁts of Bayesian methodology. [sent-306, score-0.188]

92 Our prior provides the regularization needed to obtain good predictive performance. [sent-307, score-0.174]

93 In particular, our default prior, which is minimially dependent on the data, performs well compared to other methods which rely on cross-validation to pick model parameters. [sent-308, score-0.153]

94 We obtain inference in the natural Bayesian way from the variation in the posterior draws. [sent-309, score-0.087]

95 In this case, gauging the inferential uncertainty is essential. [sent-311, score-0.116]

96 A common concern with Bayesian approaches is sensitivity to prior parameters. [sent-321, score-0.088]

97 (2006) found that results were robust to a reasonably wide range of prior parameters, including ν, q, σµ , as well as the number of trees, m. [sent-323, score-0.088]

98 (2006), BART: Bayesian additive regression trees, Technical report, University of Chicago. [sent-351, score-0.122]

99 (2001), ‘Greedy function approximation: A gradient boosting machine’, The Annals of Statistics 29, 1189–1232. [sent-366, score-0.138]

100 (2007), ‘Bayesian CART: Prior speciﬁcation and posterior simulation’, Journal of Computational and Graphical Statistics. [sent-393, score-0.087]

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Hidden Markov models (HMMs), on the other hand, are such that the number of underlying random variables changes depending on the desired length (which can be a random variable), and HMMs are applicable even without knowing this length as they can be extended indeﬁnitely using online inference. HMMs have been generalized to dynamic Bayesian networks (DBNs) and temporal conditional random ﬁelds (CRFs), where an underlying set of variables gets repeated as needed to ﬁll any ﬁnite but unbounded length. Probabilistic relational models (PRMs) [5] allow for a more complex template that can be expanded in multiple dimensions simultaneously. An attribute common to all of the above cases is that the speciﬁcation of rules for expanding any particular instance of a model is ﬁnite. In other words, these forms of GM allow the speciﬁcation of models with an unlimited number of random variables (RVs) using a ﬁnite description. This is achieved using parameter tying, so while the number of RVs increases without bound, the number of parameters does not. In this paper, we introduce a new class of model we call multi-dynamic Bayesian networks. MDBNs are motivated by our research into the application of graphical models to the domain of statistical machine translation (MT) and they have two key attributes from the graphical modeling perspective. First, an MDBN generalizes a DBN in that there are multiple “streams” of variables that can get unrolled, but where each stream may be unrolled by a differing amount. In the most general case, connecting these different streams together would require the speciﬁcation of conditional probabil- ity tables with a varying and potentially unlimited number of parents. To avoid this problem and retain the template’s ﬁnite description length, we utilize a switching parent functionality (also called value-speciﬁc independence). Second, in order to capture the notion of fertility in MT-systems (deﬁned later in the text), we employ a form of existence uncertainty [7] (that we call switching existence), whereby the existence of a given random variable might depend on the value of other random variables in the network. Being fully propositional, MDBNs lie between DBNs and PRMs in terms of expressiveness. While PRMs are capable of describing any MDBN, there are, in general, advantages to restricting ourselves to a more speciﬁc class of model. For example, in the DBN case, it is possible to provide a bound on inference costs just by looking at attributes of the DBN template only (e.g., the left or right interfaces [12, 2]). Restricting the model can also make it simpler to use in practice. MDBNs are still relatively simple, while at the same time making possible the easy expression of MT systems, and opening doors to novel forms of probabilistic inference as we show below. In section 2, we introduce MDBNs, and describe their application to machine translation showing how it is possible to represent even complex MT systems. In section 3, we describe MDBN learning and decoding algorithms. In section 4, we present experimental results in the area of statistical machine translation, and future work is discussed in section 5. 2 MDBNs A standard DBN [4] template consists of a directed acyclic graph G = (V, E) = (V1 ∪ V2 , E1 ∪ → E2 ∪ E2 ) with node set V and edge set E. For t ∈ {1, 2}, the sets Vt are the nodes at slice t, Et → are the intra-slice edges between nodes in Vt , and Et are the inter-slice edges between nodes in V1 and V2 . To unroll a DBN to length T , the nodes V2 along with the edges adjacent to any node in V2 are cloned T − 1 times (where parameters of cloned variables are constrained to be the same as the template) and re-connected at the corresponding places. An MDBN with K streams consists of the union of K DBN templates along with a template structure specifying rules to connect the various streams together. An MDBN template is a directed graph (k) G = (V, E) = ( V (k) , E (k) ∪ E ) k (k) (k) th k (k) where (V , E ) is the k DBN, and the edges E are rules specifying how to connect stream k to the other streams. These rules are general in that they specify the set of edges for all values of Tk . There can be arbitrary nesting of the streams such as, for example, it is possible to specify a model that can grow along several dimensions simultaneously. An MDBN also utilizes “switching existence”, meaning some subset of the variables in V bestow existence onto other variables in the network. We call these variables existence bestowing (or ebnodes). The idea of bestowing existence is well deﬁned over a discrete space, and is not dissimilar to a variable length DBN. For example, we may have a joint distribution over lengths as follows: p(X1 , . . . , XN , N ) = p(X1 , . . . , Xn |N = n)p(N = n) where here N is an eb-node that determines the number of other random variables in the DGM. Our notion of eb-nodes allows us to model certain characteristics found within machine translation systems, such as “fertility” [3], where a given English word is cloned a random number of times in the generative process that explains a translation from French into English. This random cloning might happen simultaneously at all points along a given MDBN stream. This means that even for a given ﬁxed stream length Ti = ti , each stream could have a randomly varying number of random variables. Our graphical notation for eb-nodes consists of the eb-node as a square box containing variables whose existence is determined by the eb-node. We start by providing a simple example of an expanded MDBN for three well known MT systems, namely the IBM models 1 and 2 [3], and the “HMM” model [15].1 We adopt the convention in [3] that our goal is to translate from a string of French words F = f of length M = m into a string of English words E = e of length L = l — of course these can be any two languages. The basic generative (noisy channel) approach when translating from French to English is to represent the joint 1 We will refer to it as M-HMM to avoid confusion with regular HMMs. distribution P (f , e) = P (f |e)P (e). P (e) is a language model specifying the prior over the word string e. The key goal is to produce a ﬁnite-description length representation for P (f |e) where f and e are of arbitrary length. A hidden alignment string, a, speciﬁes how the English words align to the French word, leading to P (f |e) = a P (f , a|e). Figure 1(a) is a 2-stream MDBN expanded representation of the three models, in this case ℓ = 4 and m = 3. As shown, it appears that the fan-in to node fi will be ℓ and thus will grow without bound. However, a switching mechanism whereby P (fi |e, ai ) = P (fi |eai ) limits the number of parameters regardless of L. This means that the alignment variable ai indicates the English word eai that should be aligned to French word fi . The variable e0 is a null word that connects to French words not explained by any of e1 , . . . , eℓ . The graph expresses all three models — the difference is that, in Models 1 and 2, there are no edges between aj and aj+1 . In Model 1, p(aj = ℓ) is uniform on the set {1, . . . , L}; in Model 2, the distribution over aj is a function only of its position j, and on the English and French lengths ℓ and m respectively. In the M-HMM model, the ai variables form a ﬁrst order Markov chain. l e0 ℓ e1 e3 e2 e1 e4 e2 e3 φ1 φ2 φ3 m’ φ0 τ01 a1 f2 a2 f3 a3 m (a) Models 1,2 and M-HMM τ12 τ13 τ21 π02 π11 π12 π13 π21 f2 f3 f4 f5 f6 a1 u v τ11 f1 f1 τ02 a2 a3 a4 a5 a6 π01 w y x m (b) Expanded M3 graph Figure 1: Expanded 2-stream MDBN description of IBM Models 1 and 2, and the M-HMM model for MT; and the expanded MDBN description of IBM Model 3 with fertility assignment φ0 = 2, φ1 = 3, φ2 = 1, φ3 = 0. From the above, we see that it would be difﬁcult to express this model graphically using a standard DBN since L and M are unequal random variables. Indeed, there are two DBNs in operation, one consisting of the English string, and the other consisting of the French string and its alignment. Moreover, the fully connected structure of the graph in the ﬁgure can represent the appropriate family of model, but it also represents models whose parameter space grows without bound — the switching function allows the model template to stay ﬁnite regardless of L and M . With our MDBN descriptive abilities complete, it is now possible to describe the more complex IBM models 3, and 4[3] (an MDBN for Model3 is depicted in ﬁg. 1(b)). The top most random variable, ℓ, is a hidden switching existence variable corresponding to the length of the English string. The box abutting ℓ includes all the nodes whose existence depends on the value of ℓ. In the ﬁgure, ℓ = 3, thus resulting in three English words e1 , e2 , and e3 connected using a second-order Markov chain. To each English word ei corresponds a conditionally dependent fertility eb-node φi , which indicates how many times ei is used by words in the French string. Each φi in turn controls the existence of a set of variables under it. Given the fertilities (the ﬁgure depicts the case φ1 = 3, φ2 = 1, φ3 = 0), for each word ei , φi French word variables are granted existence and are denoted by τi1 , τi2 , . . . , τiφi , what is called the tablet [3] of ei . The values taken by the τ variables need to match the actual observed French sequence f1 , . . . , fm . This is represented as a shared constraint between all the f , π, and τ variables which have incoming edges into the observed variable v. v’s conditional probability table is such that it is one only when the associated constraint is satisﬁed2 . The variable 2 This type of encoding of constraints corresponds to the standard mechanism used by Pearl [14]. A naive implementation, however, would enumerate a number of conﬁgurations exponential in the number of constrained variables, while typically only a small fraction of the conﬁgurations would have positive probability. πi,k ∈ {1, . . . , m} is a switching dependency parent with respect to the constraint variable v and determines which fj participates in an equality constraint with τi,k . The bottom variable m is a switching existence node (observed to be 6 in the ﬁgure) with corresponding French word sequence and alignment variables. The French sequence participates in the v constraint described above, while the alignment variables aj ∈ {1, . . . , ℓ}, j ∈ 1, . . . , m constrain the fertilities to take their unique allowable values (for the given alignment). Alignments also restrict the domain of permutation variables, π, using the constraint variable x. Finally, the domain size of each aj has to lie in the interval [0, ℓ] and that is enforced by the variable u. The dashed edges connecting the alignment a variables represent an extension to implement an M3/M-HMM hybrid. ℓ The null submodel involving the deterministic node m′ (= i=1 φi ) and eb-node φ0 accounts for French words that are not explained by any of the English words e1 , . . . , eℓ . In this submodel, successive permutation variables are ordered and this constraint is implemented using the observed child w of π0i and π0(i+1) . Model 4 [3] is similar to Model 3 except that the former is based on a more elaborate distortion model that uses relative instead of absolute positions both within and between tablets. 3 Inference, Parameter Estimation and MPE Multi-dynamic Bayesian Networks are amenable to any type of inference that is applicable to regular Bayesian networks as long as switching existence relationships are respected and all the constraints (aggregation for example) are satisﬁed. Unfortunately DBN inference procedures that take advantage of the repeatable template and can preprocess it ofﬂine, are not easy to apply to MDBNs. A case in point is the Junction Tree algorithm [11]. Triangulation algorithms exist that create an ofﬂine triangulated version of the input graph and do not re-triangulate it for each different instance of the input data [12, 2]. In MDBNs, due to the ﬂexibility to unroll templates in several dimensions and to specify dependencies and constraints spanning the entire unrolled graph, it is not obvious how we can exploit any repetitive patterns in a Junction Tree-style ofﬂine triangulation of the graph template. In section 4, we discuss sampling inference methods we have used. Here we discuss our extension to a backtracking search algorithm with the same performance guarantees as the JT algorithm, but with the advantage of easily handling determinism, existence uncertainty, and constraints, both learned and explicitly stated. Value Elimination (VE) ([1]), is a backtracking Bayesian network inference technique that caches factors associated with portions of the search tree and uses them to avoid iterating again over the same subtrees. We follow the notation introduced in [1] and refer the reader to that paper for details about VE inference. We have extended the VE inference approach to handle explicitly encoded constraints, existence uncertainty, and to perform approximate local domain pruning (see section 4). We omit these details as well as others in the original paper and brieﬂy describe the main data structure required by VE and sketch the algorithm we refer to as FirstPass (ﬁg. 1) since it constitutes the ﬁrst step of the learning procedure, our main contribution in this section. A VE factor, F , is such that we can write the following marginal of the joint distribution P (X = x, Y = y, Z) = F.val × f (Z) X=x such that (X∪Y)∩Z = ∅, F.val is a constant, and f (Z) a function of Z only. Y is a set of variables previously instantiated in the current branch of search tree to the value vector y. The pair (Y, y) is referred to as a dependency set (F.Dset). X is referred to as a subsumed set (F.Sset). By caching the tuple (F.Dset, F.Sset, F.val), we avoid recomputing the marginal again whenever (1) F.Dset is active, meaning all nodes stored in F.Dset are assigned their cached values in the current branch of the search tree; and (2) none of the variables in F.Sset are assigned yet. FirstPass (alg. 1) visits nodes in the graph in Depth First fashion. In line 7, we get the values of all Newly Single-valued (NSV) CPTs i.e., CPTs that involve the current node, V , and in which all We use a general directed domain pruning constraint. Deterministic relationships then become a special case of our constraint whereby the domain of the child variable is constrained to a single value with probability one. Variable traversal order: A, B, C, and D. Factors are numbered by order of creation. *Fi denotes the activation of factor i. Tau values propagated recursively F7: Dset={} Sset={A,B,C,D} val=P(E=e) F7.tau = 1.0 = P(Evidence)/F7.val A F5: Dset={A=0} Sset={B,C,D} F2 D *F1 *F2 Factor values needed for c(A=0) and c(C=0,B=0) computation: F5.val=P(B=0|A=0)*F3.val+P(B=1|A=0)*F4.val F3.val=P(C=0|B=0)*F1.val+P(C=1|B=0)*F2.val F4.val=P(C=0|B=1)*F1.val+P(C=1|B=1)*F2.val F1.val=P(D=0|C=0)P(E=e|D=0)+P(D=1|C=0)P(E=e|D=1) F2.val=P(D=0|C=1)P(E=e|D=0)+P(D=1|C=1)P(E=e|D=1) First pass C *F3 *F4 Second pass D B F4 C F6.tau = F7.tau * P(A=1) 1 B F3: Dset={B=0} Sset={C,D} F1 F5.tau = F7.tau * P(A=0) F6 0 F3.tau = F5.tau * P(B=0|A=0) + F6.tau * P(B=0|A=1) = P(B=0) F4.tau = F5.tau * P(B=1|A=0) + F6.tau * P(B=1|A=1) = P(B=1) F1.tau = F3.tau * P(C=0|B=0) + F4.tau * P(C=0|B=1) = P(C=0) F2.tau = F3.tau * P(C=1|B=0) + F4.tau * P(C=1|B=1) = P(C=1) c(A=0)=(1/P(e))*(F7.tau*P(A=0)*F5.val)=(1/P(e))(P(A=0)*P(E=e|A=0))=P(A=0|E=e) c(C=0,B=0)=(1/P(e))*F3.tau*P(C=0|B=0)*F1.val =(1/P(e) * (P(A=0,B=0)+P(A=1,B=0)) * P(C=0|B=0) * F1.val =(1/P(e)) * P(B=0) * P(C=0|B=0) * F1.val =(1/P(e)) * P(B=0) * P(C=0|B=0) * F1.val =(1/P(e)) * P(C=0,B=0) * F1.val =P(C=0,B=0,E=e)/P(e)=P(C=0,B=0|E=e) Figure 2: Learning example using the Markov chain A → B → C → D → E, where E is observed. In the ﬁrst pass, factors (Dset, Sset and val) are learned in a bottom up fashion. Also, the normalization constant P (E = e) (probability of evidence) is obtained. In the second pass, tau values are updated in a top-down fashion and used to calculate expected counts c(F.head, pa(F.head)) corresponding to each F.head (the ﬁgure shows the derivations for (A=0) and (C=0,B=0), but all counts are updated in the same pass). other variables are already assigned (these variables and their values are accumulated into Dset). We also check for factors that are active, multiply their values in, and accumulate subsumed vars in Sset (to avoid branching on them). In line 10, we add V to the Sset. In line 11, we cache a new factor F with value F.val = sum. We store V into F.head, a pointer to the last variable to be inserted into F.Sset, and needed for parameter estimation described below. F.Dset consists of all the variables, except V , that appeared in any NSV CPT or the Dset of an activated factor at line 6. Regular Value Elimination is query-based, similar to variable elimination and recursive conditioning—what this means is that to answer a query of the type P (Q|E = e), where Q is query variable and E a set of evidence nodes, we force Q to be at the top of the search tree, run the backtracking algorithm and then read the answers to the queries P (Q = q|E = e), q ∈ Dom[Q], along each of the outgoing edges of Q. Parameter estimation would require running a number of queries on the order of the number of parameters to estimate. We extend VE into an algorithm that allows us to obtain Expectation Maximization sufﬁcient statistics in a single run of Value Elimination plus a second pass, which can never take longer than the ﬁrst one (and in practice is much faster). This two-pass procedure is analogous to the collect-distribute evidence procedure in the Junction Tree algorithm, but here we do this via a search tree. Let θX=x|pa(X)=y be a parameter associated with variable X with value x and parents Y = pa(X) when they have value y. Assuming a maximum likelihood learning scenario3 , to estimate θX=x|pa(X)=y , we need to compute f (X = x, pa(X) = y, E = e) = P (W, X = x, pa(X) = y, E = e) W\{X,pa(X)} which is a sum of joint probabilities of all conﬁgurations that are consistent with the assignment {X = x, pa(X) = y}. If we were to turn off factor caching, we would enumerate all such variable conﬁgurations and could compute the sum. When standard VE factors are used, however, this is no longer possible whenever X or any of its parents becomes subsumed. Fig. 2 illustrates an example of a VE tree and the factors that are learned in the case of a Markov chain with an evidence node at the end. We can readily estimate the parameters associated with variables A and B as they are not subsumed along any branch. C and D become subsumed, however, and we cannot obtain the correct counts along all the branches that would lead to C and D in the full enumeration case. To address this issue, we store a special value, F.tau, in each factor. F.tau holds the sum over all path probabilities from the ﬁrst level of the search tree to the level at which the factor F was 3 For Bayesian networks the likelihood function decomposes such that maximizing the expectation of the complete likelihood is equivalent to maximizing the “local likelihood” of each variable in the network. either created or activated. For example, F 6.tau in ﬁg. 2 is simply P (A = 1). Although we can compute F 3.tau directly, we can also compute it recursively using F 5.tau and F 6.tau as shown in the ﬁgure. This is because both F 5 and F 6 subsume F 3: in the context {F 5.Dset}, there exists a (unique) value dsub of F 5.head4 s.t. F 3 becomes activable. Likewise for F 6. We cannot compute F 1.tau directly, but we can, recursively, from F 3.tau and F 4.tau by taking advantage of a similar subsumption relationship. In general, we can show that the following recursive relationship holds: F pa .tau × N SVF pa .head=dsub × F.tau ← F pa ∈F pa Fact .val F.val Fact ∈Fact (1) where F pa is the set of factors that subsume F , Fact is the set of all factors (including F ) that become active in the context of {F pa .Dset, F pa .head = dsub } and N SVF pa .head=dsub is the product of all newly single valued CPTs under the same context. For top-level factors (not subsumed by any factor), F.tau = Pevidence /F.val, which is 1.0 when there is a unique top-level factor. Alg. 2 is a simple recursive computation of eq. 1 for each factor. We visit learned factors in the reverse order in which they were learned to ensure that, for any factor F ′ , F ′ .tau is incremented (line 13) by any F that might have activated F ′ (line 12). For example, in ﬁg. 2, F 4 uses F 1 and F 2, so F 4.tau needs to be updated before F 1.tau and F 2.tau. In line 11, we can increment the counts for any NSV CPT entries since F.tau will account for the possible ways of reaching the conﬁguration {F.Dset, F.head = d} in an equivalent full enumeration tree. Algorithm 1: FirstPass(level) 1 2 3 4 5 6 7 8 9 10 Input: Graph G Output: A list of learned factors and Pevidence Select var V to branch on if V ==NONE then return Sset={}, Dset={} for d ∈ Dom[V ] do V ←d prod = productOfAllNSVsAndActiveFactors(Dset, Sset) if prod != 0 then FirstPass(level+1) sum += prod Sset = Sset ∪ {V } cacheNewFactor(F.head ← V ,F.val ← sum, F.Sset ← Sset, F.Dset ← Dset); Algorithm 2: SecondPass() 1 2 3 4 5 6 7 8 9 10 11 12 13 Input: F : List of factors in the reverse order learned in the ﬁrst pass and Pevidence . Result: Updated counts foreach F ∈ F do if F.Dset = {} then F.tau ← Pevidence /F.val else F.tau ← 0.0 Assign vars in F.Dset to their values V ← F.head (last node to have been subsumed in this factor) foreach d ∈ Dom[V ] do prod = productOfAllNSVsAndActiveFactors() prod∗ = F.tau foreach newly single-valued CPT C do count(C.child,C.parents)+=prod/Pevidence F ′ =getListOfActiveFactors() for F ′ ∈ F ′ do F ′ .tau+ = prod/F ′ .val Most Probable Explanation We compute MPE using a very similar two-pass algorithm. In the ﬁrst pass, factors are used to store a maximum instead of a summation over variables in the Sset. We also keep track of the value of F.head at which the maximum is achieved. In the second pass, we recursively ﬁnd the optimal variable conﬁguration by following the trail of factors that are activated when we assign each F.head variable to its maximum value starting from the last learned factor. 4 Recall, F.head is the last variable to be added to a newly created factor in line 10 of alg. 1 4 MACHINE TRANSLATION WORD ALIGNMENT EXPERIMENTS A major motivation for pursuing the type of representation and inference described above is to make it possible to solve computationally-intensive real-world problems using large amounts of data, while retaining the full generality and expressiveness afforded by the MDBN modeling language. In the experiments below we compare running times of MDBNs to GIZA++ on IBM Models 1 through 4 and the M-HMM model. GIZA++ is a special-purpose optimized MT word alignment C++ tool that is widely used in current state-of-the-art phrase-based MT systems [10] and at the time of this writing is the only publicly available software that implements all of the IBM Models. We test on French-English 107 hand-aligned sentences5 from a corpus of the European parliament proceedings (Europarl [9]) and train on 10000 sentence pairs from the same corpus and of maximum number of words 40. The Alignment Error Rate (AER) [13] evaluation metric quantiﬁes how well the MPE assignment to the hidden alignment variables matches human-generated alignments. Several pruning and smoothing techniques are used by GIZA and MDBNs. GIZA prunes low lexical (P (f |e)) probability values and uses a default small value for unseen (or pruned) probability table entries. For models 3 and 4, for which there is no known polynomial time algorithm to perform the full E-step or compute MPE, GIZA generates a set of high probability alignments using an MHMM and hill-climbing and collects EM counts over these alignments using M3 or M4. For MDBN models we use the following pruning strategy: at each level of the search tree we prune values which, together, account for the lowest speciﬁed percentage of the total probability mass of the product of all newly active CPTs in line 6 of alg. 1. This is a more effective pruning than simply removing low-probability values of each CPD because it factors in the joint contribution of multiple active variables. Table 1 shows a comparison of timing numbers obtained GIZA++ and MDBNs. The runtime numbers shown are for the combined tasks of training and decoding; however, training time dominates given the difference in size between train and test sets. For models 1 and 2 neither GIZA nor MDBNs perform any pruning. For the M-HMM, we prune 60% of probability mass at each level and use a Dirichlet prior over the alignment variables such that long-range transitions are exponentially less likely than shorter ones.6 This model achieves similar times and AER to GIZA’s. Interestingly, without any pruning, the MDBN M-HMM takes 160 minutes to complete while only marginally improving upon the pruned model. Experimenting with several pruning thresholds, we found that AER would worsen much more slowly than runtime decreases. Models 3 and 4 have treewidth equal to the number of alignment variables (because of the global constraints tying them) and therefore require approximate inference. Using Model 3, and a drastic pruning threshold that only keeps the value with the top probability at each level, we were able to achieve an AER not much higher than GIZA’s. For M4, it achieves a best AER of 31.7% while we do not improve upon Model3, most likely because a too restrictive pruning. Nevertheless, a simple variation on Model3 in the MDBN framework achieves a lower AER than our regular M3 (with pruning still the same). The M3-HMM hybrid model combines the Markov alignment dependencies from the M-HMM model with the fertility model of M3. MCMC Inference Sampling is widely used for inference in high-treewidth models. Although MDBNs support Likelihood Weighing, it is very inefﬁcient when the probability of evidence is very small, as is the case in our MT models. Besides being slow, Markov chain Monte Carlo can be problematic when the joint distribution is not positive everywhere, in particular in the presence of determinism and hard constraints. Techniques such as blocking Gibbs sampling [8] try to address the problem. Often, however, one has to carefully choose a problem-dependent proposal distribution. We used MCMC to improve training of the M3-HMM model. We were able to achieve an AER of 32.8% (down from 39.1%) but using 400 minutes of uniprocessor time. 5 CONCLUSION The existing classes of graphical models are not ideally suited for representing SMT models because “natural” semantics for specifying the latter combine ﬂavors of different GM types on top of standard directed Bayesian network semantics: switching parents found in Bayesian Multinets [6], aggregation relationships such as in Probabilistic Relational Models [5], and existence uncertainty [7]. We 5 Available at http://www.cs.washington.edu/homes/karim French and English have similar word orders. On a different language pair, a different prior might be more appropriate. With a uniform prior, the MDBN M-HMM has 36.0% AER. 6 Model Init M1 M2 M-HMM M3 M4 M3-HMM GIZA++ M1 M-HMM 1m45s (47.7%) N/A 2m02s (41.3%) N/A 4m05s (35.0%) N/A 2m50 (45%) 5m20s (38.5%) 5m20s (34.8%) 7m45s (31.7%) N/A MDBN M1 3m20s (48.0%) 5m30s (41.0%) 4m15s (33.0%) 12m (43.6%) 25m (43.6%) 9m30 (41.0%) M-HMM N/A N/A N/A 9m (42.5%) 23m (42.6%) 9m15s (39.1%) MCMC 400m (32.8%) Table 1: MDBN VE-based learning versus GIZA++ timings and %AER using 5 EM iterations. The columns M1 and M-HMM correspond to the model that is used to initialize the model in the corresponding row. The last row is a hybrid Model3-HMM model that we implemented using MDBNs and is not expressible using GIZA. have introduced a generalization of dynamic Bayesian networks to easily and concisely build models consisting of varying-length parallel asynchronous and interacting data streams. We have shown that our framework is useful for expressing various statistical machine translation models. 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