jmlr jmlr2010 jmlr2010-64 knowledge-graph by maker-knowledge-mining

64 jmlr-2010-Learning Non-Stationary Dynamic Bayesian Networks

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Author: Joshua W. Robinson, Alexander J. Hartemink

Abstract: Learning dynamic Bayesian network structures provides a principled mechanism for identifying conditional dependencies in time-series data. An important assumption of traditional DBN structure learning is that the data are generated by a stationary process, an assumption that is not true in many important settings. In this paper, we introduce a new class of graphical model called a nonstationary dynamic Bayesian network, in which the conditional dependence structure of the underlying data-generation process is permitted to change over time. Non-stationary dynamic Bayesian networks represent a new framework for studying problems in which the structure of a network is evolving over time. Some examples of evolving networks are transcriptional regulatory networks during an organism’s development, neural pathways during learning, and trafﬁc patterns during the day. We deﬁne the non-stationary DBN model, present an MCMC sampling algorithm for learning the structure of the model from time-series data under different assumptions, and demonstrate the effectiveness of the algorithm on both simulated and biological data. Keywords: Bayesian networks, graphical models, model selection, structure learning, Monte Carlo methods

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Non-stationary dynamic Bayesian networks represent a new framework for studying problems in which the structure of a network is evolving over time. [sent-10, score-0.39]

2 Some examples of evolving networks are transcriptional regulatory networks during an organism’s development, neural pathways during learning, and trafﬁc patterns during the day. [sent-11, score-0.34]

3 But during development, these regulatory networks are evolving over time, with certain conditional dependencies between gene products being created as the organism develops, and others being destroyed. [sent-32, score-0.367]

4 While the transition model of an htERGM allows for nearly unlimited generality in characterizing how the network structure changes with time, it is restricted to functions of temporally adjacent network structures. [sent-56, score-0.611]

5 Additionally, the total number of segments or epochs in the piecewise stationary process is assumed known a priori, thereby limiting application of the method in situations where the number of epochs is not known. [sent-68, score-0.594]

6 , the set of conditional dependencies) of a Bayesian network is typically expressed using Bayes’ rule, where the posterior probability of a given network structure G (i. [sent-131, score-0.403]

7 The prior over networks P(G) can be used to incorporate prior knowledge about the existence of speciﬁc edges (Bernard and Hartemink, 2005) or the overall topology of the network (e. [sent-135, score-0.398]

8 While multiple moves may result in a transition from state x to state x′ (and vice versa), typically there is only a single move for each transition. [sent-187, score-0.466]

9 In such a case, the sums over M and M ′ each only include one term, and the proposal ratio can be split into two terms: one is the ratio of the proposal probabilities for move types and the other is the ratio of selecting a particular state given the current state and the move type. [sent-188, score-0.375]

10 3653 ROBINSON AND H ARTEMINK Figure 1: An example nsDBN with labeled components (namely, transition times t1 and t2 and edge change sets ∆g1 and ∆g2 ). [sent-203, score-0.482]

11 Note that the network at each epoch is actually a DBN drawn in compact form, where each edge represents a statistical dependence between a node at time t and its parent at the previous time t − 1. [sent-205, score-0.398]

12 We deﬁne the transition time ti to be the time at which Gi is replaced by Gi+1 in the data-generation process. [sent-220, score-0.326]

13 We call the period of time between consecutive transition times—during which a single network of conditional dependencies is operative—an epoch. [sent-221, score-0.482]

14 Consider the simplest case where m = 2 and the transition time t1 at which the structure of the nsDBN evolves from network G1 to network G2 is known. [sent-227, score-0.519]

15 If prior knowledge about the problem dictates that the networks will not vary dramatically across adjacent epochs, information about the networks learned in adjacent epochs can be leveraged to increase the accuracy of the network learned in the current epoch. [sent-234, score-0.666]

16 Expanding the simple formulation to multiple epochs, assume there exist m different epochs with transition times T = {t1 , . [sent-235, score-0.65]

17 The network Gi+1 prevailing in epoch i + 1 differs from network Gi prevailing in epoch i by a set of edge changes we call ∆gi . [sent-239, score-0.655]

18 We assume the prior over networks can be further split into components describing the initial network and subsequent edge changes, shown below: P(G1 , ∆g1 , . [sent-264, score-0.389]

19 To encode this prior knowledge, we place a truncated geometric prior with parameter p = 1 − e−λs on the number of changes (si ) in each edge change set (∆gi ), with the distribution truncated for si > smax . [sent-285, score-0.487]

20 Therefore, a (truncated) geometric prior on each si is essentially equivalent to a (truncated) geometric prior on the sufﬁcient statistic s, the total number of edge changes. [sent-300, score-0.344]

21 When the transition times are a priori unknown, we would like to estimate them. [sent-301, score-0.388]

22 We assume that the joint prior over the evolutionary behavior of the network and the locations of transition times can be decomposed into two independent components. [sent-324, score-0.504]

23 Any choice for P(T ) can be made, but for the purposes of this paper, we have no prior knowledge about the transition times; therefore, we assume a uniform prior on T so that all settings of transition times are equally likely for a given value of m. [sent-326, score-0.76]

24 Finally, when neither the transition times nor the number of epochs are known a priori, both the number and times of transitions must be estimated a posteriori. [sent-327, score-0.774]

25 , a transition does not occur at every observation), we can include this knowledge by placing a non-uniform prior on m, for example a (truncated) geometric prior with parameter p = 1 − eλm on the number of epochs m, with the distribution truncated for m > N. [sent-330, score-0.704]

26 A geometric prior on the number of epochs is equivalent to a geometric prior on epoch lengths (see Appendix A). [sent-331, score-0.527]

27 , few epochs exist) and large values of λs encode the strong prior belief that the structure changes smoothly (i. [sent-335, score-0.452]

28 Fortunately, the likelihood component of the posterior does not change whether we know the transition times a priori or not. [sent-338, score-0.517]

29 Therefore, any uncertainty about the transition times can be incorporated into the evaluation of the prior, leaving the evaluation of the likelihood unchanged. [sent-339, score-0.353]

30 An epoch is deﬁned between adjacent transition times while an interval is deﬁned as the union of consecutive epochs during which a particular parent set is operative (which may include all epochs). [sent-348, score-0.869]

31 In particular, when the transition times are not known a priori, the posterior is highly multimodal because structures with slightly different transition times likely have similar posterior probabilities. [sent-374, score-0.964]

32 To provide quick convergence, we want to ensure that every move in the move set efﬁciently jumps between posterior modes. [sent-377, score-0.381]

33 The different settings will be abbreviated according to the type of uncertainty: whether the number of transitions is known (KN) or unknown (UN) and whether the transition times themselves are known (KT) or unknown (UT). [sent-381, score-0.413]

34 1 Known Number and Known Times of Transitions (KNKT) In the KNKT setting, we know all of the transition times a priori; therefore, we only need to identify the most likely initial network G1 and sets of edge changes ∆g1 , . [sent-384, score-0.624]

35 In the KNKT setting, the transition times t j are known a priori, but they must be estimated in the two other settings. [sent-399, score-0.353]

36 In the KNUT setting, the number of epochs m is still known, but the transition times themselves are not. [sent-400, score-0.65]

37 Finally, in the UNUT setting, even the number of epochs is unknown; instead a truncated geometric prior is placed on m. [sent-401, score-0.356]

38 Structures with the same edge sets but slightly different transition times will probably have similar likelihoods. [sent-409, score-0.482]

39 Therefore, we can add a new move that proposes a local shift to one of the transition times: let d be some small positive integer and let the new time ti′ be drawn from a discrete uniform distribution ti′ ∼ DU(ti − d,ti + d) with the constraint that ti−1 < ti′ < ti+1 . [sent-410, score-0.415]

40 Initially, we set the m − 1 transition times so that the epochs are roughly equal in length. [sent-411, score-0.65]

41 This placement allows the transition times ample room to locally shift without “bumping” into each other too early in the sampling 3659 ROBINSON AND H ARTEMINK procedure. [sent-412, score-0.39]

42 As with the last setting, the number of epochs does not change; therefore, only the prior on the number of edge changes s is used. [sent-414, score-0.535]

43 3 Unknown Number and Unknown Times of Transitions (UNUT) In the most general UNUT setting, both the transition times T and the number of transitions are unknown and must be estimated. [sent-416, score-0.413]

44 Since both the number of edge changes s and the number of epochs m are allowed to vary, we need to incorporate both priors mentioned in Section 3. [sent-421, score-0.512]

45 To allow the number of epochs m to change during sampling, we introduce merge and split operations to the move set. [sent-423, score-0.423]

46 The transition time of the new edge set is selected to be the mean of the previous locations weighted by the size of each edge set: ti′ = (siti + si+1ti+1 )/(si + si+1 ). [sent-425, score-0.547]

47 For the split operation, an edge set ∆gi is randomly chosen and randomly partitioned into two new edge sets ∆g′ and ∆g′ with all subsequent edge sets re-indexed appropriately. [sent-426, score-0.387]

48 The move set is completed with the inclusion of the add transition time and delete transition time operations. [sent-428, score-0.737]

49 For example, if either move M1 or move M2 is randomly selected, the sampling procedure is as follows: random variables xi and x j are selected, if the edge xi → x j exists in G1 , it is deleted, otherwise it is added (subject to the maximum of pmax parents constraint). [sent-433, score-0.468]

50 E1 is the total number of edges in G1 , pmax is the maximum parent set size, smax is the maximum number of edge changes allowed in a single transition time, and si is the number of edge changes in the set ∆gi . [sent-445, score-1.031]

51 The epochs in which the various networks are active are shown in the horizontal bars, roughly to scale. [sent-462, score-0.406]

52 The horizontal bars represent the segmentation of the 1020 observations, with the transition times labeled below. [sent-463, score-0.353]

53 The learned networks and most likely transition times are highly accurate (only missing two edges in G1 and all predicted transition times close to the truth). [sent-466, score-0.947]

54 To obtain a consensus (model averaged) structure prediction, an edge is considered present at a particular time if the posterior probability of the edge is greater than 0. [sent-473, score-0.433]

55 2 KNUT S ETTING In this setting, transition times are unknown and must be estimated a posteriori. [sent-481, score-0.353]

56 3662 L EARNING N ON -S TATIONARY DYNAMIC BAYESIAN N ETWORKS KNUT Setting Figure 4: Posterior probability of transition times when learning an nsDBN in the KNUT setting. [sent-484, score-0.353]

57 The blue triangles on the baseline represent the true transition times and the red dots represent one standard deviation from the mean probability, which is drawn as a black line. [sent-485, score-0.353]

58 The highly probable transition times correspond closely with the true transition times. [sent-487, score-0.642]

59 The estimated structure and transition times are very close to the truth. [sent-491, score-0.399]

60 All edges are correct, with the exception of two missing edges in G1 , and the predicted transition times are all within 10 of the true transition times. [sent-492, score-0.853]

61 We can also examine the posterior probabilities of transition times over all sampled structures. [sent-493, score-0.519]

62 The blue triangles on the baseline represent the true transition times, and spikes represent transition times that frequently occurred in the sampled structures. [sent-495, score-0.642]

63 While the highest probability regions do occur near the true transition times, some uncertainty exists about the exact locations of t3 and t4 since the fourth epoch is exceedingly short. [sent-496, score-0.401]

64 To obtain more accurate measures of the posterior quantities of interest (such as the locations of transition times), we generate samples from multiple chains; we use 25 in the KNUT setting. [sent-497, score-0.418]

65 Combining the samples from several chains allows us to estimate both the probability of a transition occurring at a certain time and the variance of that estimate. [sent-498, score-0.327]

66 This unexpected result implies that the posterior over transition times is rather smooth; therefore, the mixing rate is not greatly affected when sampling transition times. [sent-501, score-0.808]

67 Figure 5 shows the posterior probabilities of transition times for various settings of λs and λm . [sent-509, score-0.519]

68 Essentially, when λm is large, only the few transition times that best characterize the non-stationary behavior of the data will be identiﬁed. [sent-511, score-0.353]

69 On the other hand, when λm is very small, noises within the data begin to be identiﬁed as transition times, leading to poor estimates of transition times. [sent-512, score-0.578]

70 A smaller λm results in more predicted epochs and less conﬁdence about the most probable value of m. [sent-515, score-0.35]

71 2 Larger Simulated Data Set To evaluate the scalability of our technique in the most difﬁcult UNUT setting, we also simulate data from a 100 variable network with an average of 50 edges over ﬁve epochs spanning 4800 observations, with one to three edges changing between each epoch. [sent-524, score-0.547]

72 The posterior probabilities of transition times and the number of epochs (corresponding to Figures 5 and 6) for one of the simulated data sets are shown in Figure 8. [sent-527, score-0.875]

73 The number of epochs with the highest posterior probability is ﬁve for all choices of priors, which is exactly the true number of epochs for this data set. [sent-529, score-0.723]

74 3664 L EARNING N ON -S TATIONARY DYNAMIC BAYESIAN N ETWORKS UNUT Setting Figure 5: The posterior probabilities of transition times from the sampled structures in the UNUT setting for various values of λs and λm . [sent-531, score-0.519]

75 As in Figure 4, the blue triangles on the baseline represent the true transition times and the red dots represent one standard deviation from the mean probability obtained from several runs, which is drawn as a black line. [sent-532, score-0.353]

76 3665 ROBINSON AND H ARTEMINK UNUT Setting Figure 6: The posterior probabilities of the number of epochs for various values of λs and λm . [sent-536, score-0.463]

77 The posterior estimates on the number of epochs m are closest to the true value of 7 when λm is 1 or 2. [sent-539, score-0.426]

78 3667 ROBINSON AND H ARTEMINK UNUT Setting Figure 8: The posterior probabilities of transition times and number of epochs from one of the larger (100 variables, 5 epochs, and 4800 observations) simulated data sets under the UNUT setting for various values of λs and λm . [sent-576, score-0.875]

79 Top: The blue triangles on the baseline represent the true transition times and the red dots represent one standard deviation from the mean probability obtained from several runs, which is drawn as a black line. [sent-578, score-0.353]

80 The nsDBN in Figure 10C was learned using the KNKT setting with transition times deﬁned at the borders between the embryonic, larval, pupal, and adult stages. [sent-604, score-0.398]

81 If we transition to the UNUT setting, we can also examine the posterior probabilities of transition times and epochs. [sent-628, score-0.808]

82 We generate data from an nsDBN with 66 observations and transition times at 30, 40, and 58 to mirror the number of observations in embryonic, larval, pupal, and adult stages of the experimental ﬂy data. [sent-634, score-0.398]

83 0 0 65 Time 1 2 3 4 5 6 7 8 9 10 Number of epochs Figure 11: Learning nsDBN structure in the UNUT setting using the Drosophila muscle development data. [sent-648, score-0.484]

84 A: Posterior probabilities of transition times using λm = λs = 2. [sent-649, score-0.39]

85 3671 ROBINSON AND H ARTEMINK Figure 12: An nsDBN was learned on simulated data that mimicked the number of nodes, connectivity, and transition behavior of the experimental ﬂy data. [sent-655, score-0.348]

86 As discussed earlier, to obtain posterior estimates of quantities of interest, such as the number of epochs or transition times, we generate many samples from several chains; averaging over chains provides a more efﬁcient exploration of the sample space. [sent-660, score-0.753]

87 2 0 0 0 1 2 3 4 5 6 0 0 1 Time (s) 2 3 Time (s) 4 5 6 Number of epochs Figure 13: Posterior results of learning nsDBNs under the UNUT setting for two birds presented with two different stimuli (white noise and song). [sent-712, score-0.389]

88 The estimated number of epochs is three or four, with strong support for the value four when the bird is presented with a song. [sent-716, score-0.343]

89 The posterior transition time probabilities and the posterior number of epochs for two birds presented with two different stimuli under the UNUT setting (λs = λm = 2) can be seen in Figure 13. [sent-717, score-0.973]

90 When listening to a song, an additional transition is predicted 300–400 ms after the onset of a song but not after the onset of white noise. [sent-720, score-0.455]

91 For example, recording just the transition times and number of epochs adds only a few seconds to the runtime on the larger simulated data set. [sent-746, score-0.745]

92 We have demonstrated the feasibility of learning an nsDBN in all three settings using simulated data sets of various numbers of transition times, observations, variables, epochs, and connection densities. [sent-755, score-0.348]

93 The predicted transition times and number of epochs also correspond to the known times of large developmental changes. [sent-758, score-0.767]

94 Although each connection on the predicted Drosophila muscle development network is difﬁcult to verify, simulated experiments of a similar scale demonstrate highly accurate predictions, even with moderately noisy data and one replicate. [sent-759, score-0.345]

95 While we decided to place a (truncated) geometric prior on the number of epochs m, other priors may be considered. [sent-789, score-0.392]

96 Here, we prove that a geometrically distributed prior on epoch lengths is equivalent to a geometrically distributed prior on the number of epochs. [sent-794, score-0.328]

97 Letting li be the length of epoch i and N be the total number of observations, we can write a geometrically distributed prior on epoch lengths as: m m ∏(1 − p)li −1 p = pm ∏(1 − p)li −1 i=1 i=1 = p (1 − p)∑i=1 li −1 m m = pm (1 − p)N−m m p = (1 − p)N . [sent-795, score-0.4]

98 1− p Since N is the same for every nsDBN, we see that the geometrically distributed prior is simply a function of the number of epochs m. [sent-796, score-0.405]

99 Therefore, a geometrically distributed prior on epoch lengths with success probability p = 1−q < 1/2 is equivalent to a geometrically distributed 2−q prior on the number of epochs with success probability q. [sent-799, score-0.625]

100 Informative structure priors: Joint learning of dynamic regulatory networks from multiple types of data. [sent-823, score-0.33]

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