nips nips2013 nips2013-161 knowledge-graph by maker-knowledge-mining

161 nips-2013-Learning Stochastic Inverses

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Author: Andreas Stuhlmüller, Jacob Taylor, Noah Goodman

Abstract: We describe a class of algorithms for amortized inference in Bayesian networks. In this setting, we invest computation upfront to support rapid online inference for a wide range of queries. Our approach is based on learning an inverse factorization of a model’s joint distribution: a factorization that turns observations into root nodes. Our algorithms accumulate information to estimate the local conditional distributions that constitute such a factorization. These stochastic inverses can be used to invert each of the computation steps leading to an observation, sampling backwards in order to quickly ﬁnd a likely explanation. We show that estimated inverses converge asymptotically in number of (prior or posterior) training samples. To make use of inverses before convergence, we describe the Inverse MCMC algorithm, which uses stochastic inverses to make block proposals for a Metropolis-Hastings sampler. We explore the efﬁciency of this sampler for a variety of parameter regimes and Bayes nets. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Goodman Department of Psychology Stanford University Abstract We describe a class of algorithms for amortized inference in Bayesian networks. [sent-2, score-0.187]

2 Our approach is based on learning an inverse factorization of a model’s joint distribution: a factorization that turns observations into root nodes. [sent-4, score-0.342]

3 These stochastic inverses can be used to invert each of the computation steps leading to an observation, sampling backwards in order to quickly ﬁnd a likely explanation. [sent-6, score-0.508]

4 We show that estimated inverses converge asymptotically in number of (prior or posterior) training samples. [sent-7, score-0.474]

5 To make use of inverses before convergence, we describe the Inverse MCMC algorithm, which uses stochastic inverses to make block proposals for a Metropolis-Hastings sampler. [sent-8, score-1.036]

6 Humans and robots are in the setting of amortized inference: they have to solve many similar inference problems, and can thus ofﬂoad part of the computational work to shared precomputation and adaptation over time. [sent-17, score-0.253]

7 While much of this work is focused on adaptation for a single posterior inference, amortized inference calls for adaptation across many different inferences. [sent-21, score-0.379]

8 In this setting, we will often have considerable training data available in the form of posterior samples from previous inferences; how should we use this data to adapt our inference procedure? [sent-22, score-0.395]

9 We consider using training samples to learn the inverse structure of a directed model. [sent-23, score-0.438]

10 Posterior inference is the task of inverting a probabilistic model: Bayes’ theorem turns p(d|h) into p(h|d); vision is commonly understood as inverse graphics (Horn, 1977) and, more recently, as inverse physics (Sanborn et al. [sent-24, score-0.527]

11 However, while this is a good description of the problem that inference solves, conditional sampling usually does not proceed backwards step-by-step. [sent-28, score-0.226]

12 ally and actually learning the inverse conditionals needed to invert the model. [sent-30, score-0.334]

13 Knowing the conditionals for this inverse factorization would allow us to rapidly sample the latent variables given an observation. [sent-33, score-0.375]

14 In this paper, we will explore what these factorizations look like for Bayesian networks, how to learn them, and how to use them to construct block proposals for MCMC. [sent-34, score-0.323]

15 We say that a Bayesian network H expresses an inverse factorization of p if the observations y do not have parents (but may themselves be parents of some xi ): m p(xi |paH (xi )) p(x, y) = p(y) i=1 As an example, consider the forward and inverse networks shown in Figure 1. [sent-44, score-0.892]

16 We call the conditional distributions p(xi |paH (xi )) stochastic inverses, with inputs paH (xi ) and output xi . [sent-45, score-0.213]

17 If we could sample from these distributions, we could produce samples from p(x|y) for arbitrary y, which solves the problem of inference for all queries with the same set of observation nodes. [sent-46, score-0.222]

18 This fact will be important in Section 4 when we resample subsets of inverse graphs. [sent-49, score-0.239]

19 Algorithm 1 gives a heuristic method for computing an inverse factorization given Bayes net G, observation nodes y, and desired leaf node xi . [sent-50, score-0.622]

20 We then add the nodes in order to the inverse graph, with dependencies determined by the graph structure of the original network. [sent-52, score-0.392]

21 In the setting of amortized inference, past tasks provide approximate posterior samples for the corresponding observations. [sent-53, score-0.342]

22 We therefore investigate learning inverses from such samples, and ways of using approximate stochastic inverses for improving the efﬁciency of solving future inference tasks. [sent-54, score-0.893]

23 (Learning from prior samples) Let H be an inverse factorization. [sent-57, score-0.246]

24 We now show that valid inverse factorizations allow us to learn from posterior samples as well. [sent-60, score-0.49]

25 (Learning from posterior samples) Let H be an inverse factorization. [sent-62, score-0.338]

26 The nodes y are root nodes in H and hence do not descend from xi . [sent-68, score-0.328]

27 In addition, we can combine samples from distributions conditioned on several different observation sets to produce more accurate estimates of the inverse conditionals. [sent-71, score-0.363]

28 , (z (k) , xi ) in S based on distance to z (1) (k) 2: construct density estimate q on xi , . [sent-80, score-0.321]

29 Then, use a consistent density estimator to construct a density estimate on the nearby previous outputs and sample an output xi from the estimated distribution. [sent-84, score-0.26]

30 4 Inverse MCMC We have described how to compute the structure of inverse Bayes nets, and how to learn the associated conditional distributions and densities from prior and posterior samples. [sent-89, score-0.461]

31 We propose the following Inverse MCMC procedure (Algorithm 3): Ofﬂine, use Algorithm 1 to compute an inverse graph for each latent node and train each local inverse in this graph from (posterior or prior) samples. [sent-92, score-0.591]

32 With little training data, we will want to make small proposals (small k) in order to achieve a reasonable acceptance rate; with more training data, we can make larger proposals and expect to succeed. [sent-94, score-0.668]

33 Let G be a Bayesian network, let θ be a consistent estimator (for inverse conditionals), let {Hi }i∈1. [sent-96, score-0.276]

34 m be a collection of inverse graphs produced using Algorithm 1, and assume a source of training samples (prior or posterior) with full support. [sent-98, score-0.386]

35 Then, as training set size |S| → ∞, Inverse MCMC with proposal size k converges to block-Gibbs sampling where blocks are the last k nodes in each Hi . [sent-99, score-0.288]

36 In particular, it converges to Gibbs sampling for proposal size k = 1 and to exact posterior sampling for k = |G|. [sent-100, score-0.288]

37 We must show that proposals are made from the conditional posterior in the limit of large training data. [sent-102, score-0.45]

38 Fix an inverse H, and let x be the last k variables in H. [sent-103, score-0.214]

39 Now the conditional distribution over x factorizes along the inverse graph: |H| p(x|paH (x)) = i=k p(xi |paH (xi )). [sent-106, score-0.253]

40 Hence, using the estimated inverses to sequentially sample the x variables results, in the limit, in samples from the conditional distribution given remaining variables. [sent-108, score-0.531]

41 (Note that, in the limit, these proposals will always be accepted. [sent-109, score-0.21]

42 4 Algorithm 3: Inverse MCMC Algorithm 4: Inverse MCMC proposer Input: Prior or posterior samples S Output: Samples x(1) , . [sent-115, score-0.219]

43 m do 4: train inverse θS (xj |paHi (xj )) 5: end for 6: end for Online (MH with inverse proposals): 1: for t in 1 . [sent-124, score-0.472]

44 T do 2: x , pfw , pbw from Algorithm 4 3: x ← x with MH acceptance rule 4: end for Input: State x, observations y, ordered inverse graphs {Hi }i∈1. [sent-127, score-0.488]

45 m , proposal size kmax , inverses θ Output: Proposed state x , forward and backward probabilities pfw and pbw 1: H ∼ Uniform({Hi }i∈1. [sent-129, score-0.715]

46 , kmax − 1}) 3: x ← x 4: pfw , pbw ← 0 5: for j in n − k, . [sent-134, score-0.209]

47 , n do 6: let xl be jth variable in H 7: xl ∼ θ(xl |paH (xl )) 8: pfw ← pfw · pθ (xl |paH (xl )) 9: pbw ← pbw · pθ (xl |paH (xl )) 10: end for Instead of learning the k=1 “Gibbs” conditionals for each inverse graph, we can often precompute these distributions to “seed” our sampler. [sent-137, score-0.901]

48 This suggests a bootstrapping procedure for amortized inference on observations y (1) , . [sent-138, score-0.212]

49 , y (t) : ﬁrst, precompute the “Gibbs” distributions so that k=1 proposals will be reasonably effective; then iterate between training on previously generated approximate posterior samples and doing inference on the next observation. [sent-141, score-0.675]

50 For networks with near-deterministic dependencies, Gibbs may be unable to generate training samples of sufﬁcient quality. [sent-143, score-0.221]

51 This poses a chicken-and-egg problem: we need a sufﬁciently good posterior sampler to generate the data required to train our sampler. [sent-144, score-0.239]

52 To address this problem, we propose a simple annealing scheme: We introduce a temperature parameter t that controls the extent to which (almost-)deterministic dependencies in a network are relaxed. [sent-145, score-0.241]

53 We produce a sequence of trained samplers, one for each temperature, by generating samples for a network with temperature ti+1 using a sampler trained on approximate samples for the network with next-higher temperature ti . [sent-146, score-0.585]

54 Finally, we discard all samplers except for the sampler trained on the network with t = 0, the network of interest. [sent-147, score-0.242]

55 We start by studying the behavior of the Inverse MCMC algorithm with empirical frequency estimator on a 225-node rectangular grid network from the UAI 2008 inference competition. [sent-150, score-0.299]

56 This network has binary nodes and approximately 50% deterministic dependencies, which we relax to dependencies with strength . [sent-151, score-0.214]

57 We select the 15 nodes on the diagonal as observations and remove any nodes below, leaving a triangular network with 120 nodes and treewidth 15 (Figure 2). [sent-153, score-0.342]

58 , 2010), and calculate the error of our estimates P s as error = 1 N N i=1 1 |Xi | |P ∗ (Xi = xi ) − P s (Xi = xi )|. [sent-155, score-0.344]

59 xi ∈Xi We generate 20 inference tasks as sources of training samples by sampling values for the 15 observation nodes uniformly at random. [sent-156, score-0.615]

60 We precompute the “ﬁnal” inverse conditionals as outlined above, producing a Gibbs sampler when k=1. [sent-157, score-0.449]

61 For each inference task, we use this sampler to generate 105 approximate posterior samples. [sent-158, score-0.294]

62 00 Error in marginals Gibbs Inverses (10x10) Inverses (10x100) Inverses (10x1000) Figure 3: The effect of training on approximate posterior samples for 10 inference tasks. [sent-169, score-0.446]

63 As the number of training samples per task increases, Inverse MCMC with proposals of size 20 performs new inference tasks more quickly. [sent-170, score-0.516]

64 1e+01 1e+02 1e+03 1e+04 1e+05 Number of training samples Figure 4: Learning an inverse distribution for the brightness constancy model (Figure 1) from prior samples using the KNN density predictor. [sent-171, score-0.596]

65 More training samples result in better estimates after the same number of MCMC steps. [sent-172, score-0.172]

66 Figures 3 and 5 show the effect of training the frequency estimator on 10 inference tasks and testing on a different task (averaged over 20 runs). [sent-173, score-0.311]

67 Inverse proposals of (up to) size k=20 do worse than pure Gibbs sampling with little training (due to higher rejection rate), but they speed convergence as the number of training samples increases. [sent-174, score-0.502]

68 More generally, large proposals are likely to be rejected without training, but improve convergence after training. [sent-175, score-0.21]

69 Figure 6 illustrates how the number of inference tasks inﬂuences error and MH acceptance ratio in a setting where the total number of training samples is kept constant. [sent-176, score-0.424]

70 Surprisingly, increasing the number of training tasks from 5 to 15 has little effect on error and acceptance ratio for this network. [sent-177, score-0.23]

71 That is, it seems relatively unimportant which posterior the training samples are drawn from; we may expect different results when posteriors are more sparse. [sent-178, score-0.296]

72 Figure 7 shows how different sources of training data affect the quality of the trained sampler (averaged over 20 runs). [sent-179, score-0.175]

73 As the strength of near-deterministic dependencies increases, direct training on Gibbs samples becomes infeasible. [sent-180, score-0.224]

74 In this regime, we can still train on prior samples and on Gibbs samples for networks with relaxed dependencies. [sent-181, score-0.348]

75 01, 0]—that is, we start by learning inverses for the relaxed network where all CPT probabilities are constrained to lie within [. [sent-188, score-0.502]

76 8]; we then use these inverses as proposers for MCMC inference on a network constrained to CPT probabilities in [. [sent-190, score-0.568]

77 While the empirical frequency estimator used in the above experiments provides an attractive asymptotic convergence guarantee (Theorem 3), it is likely to generalize slowly from small amounts of training data. [sent-193, score-0.203]

78 We compare inference using a logistic regression estimator with L2 regularization (with and without interaction terms) to inference using the empirical frequency estimator. [sent-198, score-0.328]

79 The regression estimator with interaction terms results in signiﬁcantly better results when training on few posterior samples, but is ultimately overtaken by the consistent empirical estimator. [sent-200, score-0.263]

80 Next, we use the KNN density predictor to learn inverse distributions for the continuous Bayesian network shown in Figure 1. [sent-201, score-0.403]

81 16 Maximum proposal size Maximum proposal size 30 4 1 Log10(training samples per task) 2 3 4 Log10(training samples per task) Figure 5: Without training, big inverse proposals result in high error, as they are unlikely to be accepted. [sent-219, score-0.77]

82 As we increase the number of approximate posterior samples used to train the MCMC sampler, the acceptance probability for big proposals goes up, which decreases overall error. [sent-220, score-0.567]

83 , samples for other observations) we train on has little effect on acceptance ratio (and error) if we keep the total number of training samples constant. [sent-239, score-0.405]

84 For others, we can use prior samples, Gibbs samples for relaxed networks, and samples from a sequence of annealed Inverse samplers. [sent-241, score-0.255]

85 As we reﬁne the inverses using forward samples, the error in the estimated marginals decreases towards 0, providing evidence for convergence towards a posterior sampler (Figure 4). [sent-244, score-0.698]

86 To evaluate Inverse MCMC in more breadth, we run the algorithm on all binary Bayes nets with up to 500 nodes that have been submitted to the UAI 08 inference competition (216 networks). [sent-245, score-0.23]

87 Since many of these networks exhibit strong determinism, we train on prior samples and apply the annealing scheme outlined above to generate approximate posterior samples. [sent-246, score-0.421]

88 While performance varies across network classes—with extremely deterministic networks making the acquisition of training data challenging—the comparison with Gibbs suggests that learned block proposals frequently help. [sent-253, score-0.44]

89 Overall, these results indicate that Inverse MCMC is of practical beneﬁt for learning block proposals in reasonably large Bayes nets and using a realistic amount of training data (an amount that might result from amortizing over ﬁve or ten inferences). [sent-254, score-0.36]

90 2 10 Gibbs error integral 20 50 100 200 500 Number of training samples Figure 8: Each mark represents a single run of a model from the UAI 08 inference competition. [sent-263, score-0.295]

91 1 q qq q qqqq qqqqq q q q q q qq q qq q q q q qq q q q q q qq q q q qq q q q qq qq qq qqq q q qq q q q q q qq q q q qq q Frequency estimator Logistic regression (L2) Logistic regression (L2 + ^2) 0. [sent-266, score-1.874]

92 0 Inverse MCMC error integral q Figure 9: Integrated error (over 1s of inference) as a function of the number of samples used to train inverses, comparing logistic regression with and without interaction terms to an empirical frequency estimator. [sent-272, score-0.255]

93 Related work A recognition network (Morris, 2001) is a multilayer perceptron used to predict posterior marginals. [sent-273, score-0.196]

94 By learning local inverses our technique generalizes in a more ﬁne-grained way, and can be combined with MCMC to provide unbiased samples. [sent-275, score-0.397]

95 These techniques typically learn Bayes nets which are directed “forward”, which means that the conditional distributions must be learned from posterior samples, creating a chicken-and-egg problem. [sent-281, score-0.282]

96 Because our trained model is directed “backwards”, we can learn from both prior and posterior samples. [sent-282, score-0.235]

97 It is well-known that this can be addressed using block proposals, but such proposals typically need to be built manually for each model. [sent-284, score-0.242]

98 In our framework, block proposals are learned from past samples, with a natural parameter for adjusting the block size. [sent-285, score-0.274]

99 We have given simple methods for estimating and using these inverses as part of an MCMC algorithm. [sent-287, score-0.397]

100 Jags: A program for analysis of bayesian graphical models using gibbs sampling. [sent-346, score-0.175]

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