{"title": "Inter-time segment information sharing for non-homogeneous dynamic Bayesian networks", "book": "Advances in Neural Information Processing Systems", "page_first": 901, "page_last": 909, "abstract": "Conventional dynamic Bayesian networks (DBNs) are based on the homogeneous Markov assumption, which is too restrictive in many practical applications. Various approaches to relax the homogeneity assumption have therefore been proposed in the last few years. The present paper aims to improve the flexibility of two recent versions of non-homogeneous DBNs, which either (i) suffer from the need for data discretization, or (ii) assume a time-invariant network structure. Allowing the network structure to be fully flexible leads to the risk of overfitting and inflated inference uncertainty though, especially in the highly topical field of systems biology, where independent measurements tend to be sparse. In the present paper we investigate three conceptually different regularization schemes based on inter-segment information sharing. We assess the performance in a comparative evaluation study based on simulated data. We compare the predicted segmentation of gene expression time series obtained during embryogenesis in Drosophila melanogaster with other state-of-the-art techniques. We conclude our evaluation with an application to synthetic biology, where the objective is to predict a known regulatory network of five genes in Saccharomyces cerevisiae.", "full_text": "Inter-time segment information sharing for\n\nnon-homogeneous dynamic Bayesian networks\n\nDirk Husmeier & Frank Dondelinger\n\nBiomathematics & Statistics Scotland (BioSS)\n\nJCMB, The King\u2019s Buildings, Edinburgh EH93JZ, United Kingdom\n\ndirk@bioss.ac.uk, frank@bioss.ac.uk\n\nUniversit\u00b4e de Strasbourg, LSIIT - UMR 7005, 67412 Illkirch, France\n\nSophie L`ebre\n\nsophie.lebre@lsiit-cnrs.unistra.fr\n\nAbstract\n\nConventional dynamic Bayesian networks (DBNs) are based on the homogeneous\nMarkov assumption, which is too restrictive in many practical applications. Vari-\nous approaches to relax the homogeneity assumption have recently been proposed,\nallowing the network structure to change with time. However, unless time series\nare very long, this \ufb02exibility leads to the risk of over\ufb01tting and in\ufb02ated infer-\nence uncertainty. In the present paper we investigate three regularization schemes\nbased on inter-segment information sharing, choosing different prior distributions\nand different coupling schemes between nodes. We apply our method to gene ex-\npression time series obtained during the Drosophila life cycle, and compare the\npredicted segmentation with other state-of-the-art techniques. We conclude our\nevaluation with an application to synthetic biology, where the objective is to pre-\ndict a known in vivo regulatory network of \ufb01ve genes in yeast.\n\nIntroduction\n\n1\nThere is currently considerable interest in structure learning of dynamic Bayesian networks (DBNs),\nwith a variety of applications in signal processing and computational biology; see e.g. [1, 2, 3]. The\nstandard assumption underlying DBNs is that time-series have been generated from a homogeneous\nMarkov process. This assumption is too restrictive in many applications and can potentially lead to\nerroneous conclusions. While there have been various efforts to relax the homogeneity assumption\nfor undirected graphical models [4, 5], relaxing this restriction in DBNs is a more recent research\ntopic [1, 2, 3, 6, 7, 8]. At present, none of the proposed methods is without its limitations, leav-\ning room for further methodological innovation. The method proposed in [3, 8] is non-Bayesian.\nThis requires certain regularization parameters to be optimized \u201cexternally\u201d, by applying informa-\ntion criteria (like AIC or BIC), cross-validation or bootstrapping. The \ufb01rst approach is suboptimal,\nthe latter approaches are computationally expensive1. In the present paper we therefore follow the\nBayesian paradigm, like [1, 2, 6, 7]. These approaches also have their limitations. The method\nproposed in [2] assumes a \ufb01xed network structure and only allows the interaction parameters to vary\nwith time. This assumption is too rigid when looking at processes where changes in the overall\nregulatory network structure are expected, e.g. in morphogenesis or embryogenesis. The method\nproposed in [1] requires a discretization of the data, which incurs an inevitable information loss.\nThese limitations are addressed in [6, 7], where the authors propose a method for continuous data\nthat allows network structures associated with different nodes to change with time in different ways.\nHowever, this high \ufb02exibility causes potential problems when applied to time series with a low num-\nber of measurements, as typically available from systems biology, leading to over\ufb01tting or in\ufb02ated\n\n1See [9] for a demonstration of the higher computational costs of bootstrapping over Bayesian approaches\n\nbased on MCMC.\n\n1\n\n\finference uncertainty. The objective of the work described in our paper is to propose a model that\naddresses the principled shortcomings of the three Bayesian methods mentioned above. Unlike [1],\nour model is continuous and therefore avoids the information loss inherent in a discretization of the\ndata. Unlike [2], our model allows the network structure to change among segments, leading to\ngreater model \ufb02exibility. As an improvement on [6, 7], our model introduces information sharing\namong time series segments, which provides an essential regularization effect.\n\n2 Background: non-homogeneous DBNs without information coupling\nThis section summarizes brie\ufb02y the non-homogeneous DBN proposed in [6, 7], which combines\nthe Bayesian regression model of [10] with multiple changepoint processes and pursues Bayesian\ninference with reversible jump Markov chain Monte Carlo (RJMCMC) [11]. In what follows, we\nwill refer to nodes as genes and to the network as a gene regulatory network. The method is not\nrestricted to molecular systems biology, though.\n\ni , where \u03bei = (\u03be0\n\n2.1 Model\nMultiple changepoints: Let p be the number of observed genes, whose expression values y =\n{yi(t)}1\u2264i\u2264p,1\u2264t\u2264N are measured at N time points. M represents a directed graph, i.e. the network\nde\ufb01ned by a set of directed edges among the p genes. Mi is the subnetwork associated with target\ngene i, determined by the set of its parents (nodes with a directed edge feeding into gene i). The\nregulatory relationships among the genes, de\ufb01ned by M, may vary across time, which we model\nwith a multiple changepoint process. For each target gene i, an unknown number ki of changepoints\nde\ufb01ne ki + 1 non-overlapping segments. Segment h = 1, .., ki + 1 starts at changepoint \u03beh\u22121\nand\ni . To delimit the bounds, \u03be0\nstops before \u03beh\ni = 2\n= N + 1. Thus vector \u03bei has length |\u03bei| = ki + 2. The set of changepoints is denoted by \u03be =\nand \u03beki+1\n{\u03bei}1\u2264i\u2264p. This changepoint process induces a partition of the time series, yh\n, with\ndifferent structures Mh\ni associated with the different segments h \u2208 {1, . . . , ki + 1}. Identi\ufb01ability is\nsatis\ufb01ed by ordering the changepoints based on their position in the time series.\nRegression model: For all genes i, the random variable Yi(t) refers to the expression of gene i at\ntime t. Within any segment h, the expression of gene i depends on the p gene expression values\ni parents\nmeasured at the previous time point through a regression model de\ufb01ned by (a) a set of sh\ndenoted by Mh\ni );\ni | = sh\nij)j\u22080..p, \u03c3h\nij \u2208 R, \u03c3h\ni . For all genes i, for all time points t in segment h\nah\n(\u03beh\u22121\ni ), the random variable Yi(t) depends on the p variables {Yj(t \u2212 1)}1\u2264j\u2264p according to\n\n} \u2286 {1, . . . , p}, |Mh\nij = 0 if j /\u2208 Mh\n\ni , and (b) a set of parameters ((ah\n\ni > 0. For all j 6= 0, ah\n\ni = (yi(t))\u03beh\u22121\n\ni = {j1, ..., jsh\n\ni , ..., \u03beki+1\n\n) with \u03beh\u22121\n\ni , ..., \u03beh\u22121\n\n\u2264 t < \u03beh\n\ni \u2264t<\u03beh\ni\n\n< \u03beh\n\n, \u03beh\n\ni\n\ni\n\ni\n\ni\n\ni\n\ni\n\ni\n\nYi(t) = ah\n\ni0 +Xj\u2208Mh\n\ni\n\nah\nij Yj(t \u2212 1) + \u03b5i(t)\n\n(1)\n\nwhere the noise \u03b5i(t) is assumed to be Gaussian with mean 0 and variance (\u03c3h\nWe de\ufb01ne ah\n\nij)j\u22080..p.\n\ni = (ah\n\ni )2, \u03b5i(t) \u223c N (0, (\u03c3h\n\ni )2).\n\n2.2 Prior\nThe ki + 1 segments are delimited by ki changepoints, where ki is distributed a priori as a truncated\nPoisson random variable with mean \u03bb and maximum k = N \u22122: P (ki|\u03bb) \u221d \u03bbki\nki! 1l{ki\u2264k} . Conditional\non ki changepoints, the changepoint positions vector \u03bei = (\u03be0\n) takes non-overlapping\ninteger values, which we take to be uniformly distributed a priori. There are (N \u2212 2) possible posi-\ntions for the ki changepoints, thus vector \u03bei has prior density P (\u03bei|ki) = 1/\u201cN \u22122\nki\u201d. For all genes i\ni of parents for node i follows a truncated Poisson distribution2 with\nand all segments h, the number sh\ni , the prior for the parent set\nmean \u039b and maximum s = 5: P (sh\n).\nMh\nThe overall prior on the network structures is given by marginalization:\n\ni is a uniform distribution over all parent sets with cardinality sh\n\ni \u2264s}. Conditional on sh\n\ni |\u039b) \u221d \u039bsh\ni\nsh\ni !\n\ni ) = 1/( p\nsh\ni\n\ni , ..., \u03beki+1\n\ni : P (Mh\n\ni | = sh\n\ni , \u03be1\n\n1l{sh\n\ni\n\ni \u02db\u02db|Mh\n\nP (Mh\n\ni |\u039b) = Xs\n\nsh\ni =1\n\nP (Mh\n\ni |sh\n\ni )P (sh\n\ni |\u039b)\n\n(2)\n\n2A restrictive Poisson prior encourages sparsity of the network, and is therefore comparable to a sparse\n\nexponential prior, or an approach based on the LASSO.\n\n2\n\n\fConditional on the parent set Mh\n(ah\n\ni0, (ah\n\nij)j\u2208Mh\n\ni\n\n), are assumed zero-mean multivariate Gaussian with covariance matrix (\u03c3h\n\ni + 1 regression coef\ufb01cients, denoted by aMh\ni )2\u03a3Mh\n\ni\n\n=\n,\n\ni of size sh\n\ni , the sh\n\ni\n\n(3)\n\nP (ah\n\ni |Mh\n\ni , \u03c3h\n\ni )=|2\u03c0(\u03c3h\n\ni )2\u03a3Mh\n\ni\n\n2exp0\n|\u2212 1\n@\u2212\n\na\u2020\nMh\ni\n\n\u03a3\u22121\nMh\ni\n2(\u03c3h\n\ni\n\naMh\n\ni )2 1\nA\n(y) and DMh\n\n(y)DMh\n\ni\n\ni\n\nMh\ni\n\n) \u00d7 (sh\n\n= \u03b4\u22122D\u2020\n\nwhere the symbol \u2020 denotes matrix transposition, \u03a3Mh\n(y) is the\ni + 1) matrix whose \ufb01rst column is a vector of 1 (for the constant in model (1))\ni \u2212 \u03beh\u22121\n(\u03beh\nand each (j + 1)th column contains the observed values (yj(t))\u03beh\u22121\ni \u22121 for all factor gene j\ni . This prior was also used in [10] and is motivated in [12]. Finally, the conjugate prior for\nin Mh\ni )2) = IG(\u03c50, \u03b30). Following [6, 7], we\nthe variance (\u03c3h\nset the hyper-hyperparameters for shape, \u03c50 = 0.5, and scale, \u03b30 = 0.05, to \ufb01xed values that give\na vague distribution. The terms \u03bb and \u039b can be interpreted as the expected number of changepoints\nand parents, respectively, and \u03b42 is the expected signal-to-noise ratio. These hyperparameters are\ndrawn from vague conjugate hyperpriors, which are in the (inverse) gamma distribution family:\nP (\u039b) = P (\u03bb) = Ga(0.5, 1) and P (\u03b42) = IG(2, 0.2).\n\ni )2 is the inverse gamma distribution, P ((\u03c3h\n\ni \u22121\u2264t<\u03beh\n\ni\n\ni\n\n2.3 Posterior\nEquation (1) implies that\n\nP (yh\n\ni |\u03beh\u22121\n\ni\n\n, \u03beh\n\ni , Mh\n\ni , ah\n\ni , \u03c3h\n\ni ) = \u201c\u221a2\u03c0\u03c3h\n\ni \u201d\u2212(\u03beh\n\nh\u22121\ni \u2212\u03be\ni\n\n(yh\n\ni \u2212 D\n\n)\n\nexp 0\n@\u2212\n\n(y)a\n\nMh\ni\n\n)\u2020 (yh\n\nMh\ni\n2(\u03c3h\n\ni )2\n\ni \u2212 D\n\nMh\ni\n\n(y)a\n\nMh\ni\n\n)\n\n1\nA\n\n(4)\n\nFrom Bayes theorem, the posterior is given by the following equation, where all prior distributions\nhave been de\ufb01ned above:\n\np\n\nki\n\nP (k, \u03be, M, a, \u03c3, \u03bb, \u039b, \u03b42|y) \u221d P (\u03b42)P (\u03bb)P (\u039b)\n\nP (ki|\u03bb)P (\u03bei|ki)\n\nP (Mh\n\ni |\u039b)\n\n(5)\n\nYi=1\ni , [\u03c3h\ni |Mh\n\nYh=1\ni |\u03beh\u22121\n\ni\n\nP ([\u03c3h\n\ni ]2)P (ah\n\ni ]2, \u03b42)P (yh\n\n, \u03beh\n\ni , Mh\n\ni , ah\n\ni , [\u03c3h\n\ni ]2)\n\nInference\n\n2.4\nAn attractive feature of the chosen model is that the marginalization over the parameters a and \u03c3 in\nthe posterior distribution of (5) is analytically tractable:\n\nP (k,\u03be,M,\u03bb,\u039b,\u03b42|y) = Z P (k,\u03be,M,a,\u03c3,\u03bb,\u039b,\u03b42|y)dad\u03c3\n\n(6)\n\nSee [6, 10] for details and an explicit expression. The number of changepoints and their location,\nk, \u03be, the network structure M and the hyperparameters \u03bb, \u039b, \u03b42 can be sampled from the posterior\nP (k, \u03be, M, \u03bb, \u039b, \u03b42|y) with RJMCMC [11]. A detailed description can be found in [6, 10].\n\n3 Model improvement: information coupling between segments\nAllowing the network structure to change between segments leads to a highly \ufb02exible model. How-\never, this approach faces a conceptual and a practical problem. The practical problem is potential\nmodel over-\ufb02exibility. If subsequent changepoints are close together, network structures have to be\ninferred from short time series segments. This will almost inevitably lead to over\ufb01tting (in a maxi-\nmum likelihood context) or in\ufb02ated inference uncertainty (in a Bayesian context). The conceptual\nproblem is the underlying assumption that structures associated with different segments are a priori\nindependent. This is not realistic. For instance, for the evolution of a gene regulatory network during\nembryogenesis, we would assume that the network evolves gradually and that networks associated\nwith adjacent time intervals are a priori similar.\n\nTo address these problems, we propose three methods of information sharing among time series\nsegments, as illustrated in Figure 1. The \ufb01rst method is based on hard information coupling between\nthe nodes, using the exponential distribution proposed in [13]. The second scheme is also based\non hard information coupling, but uses a binomial distribution with conjugate Beta prior. The third\nscheme is based on the same distributional assumptions as the second scheme, but replaces the hard\nby a soft information coupling scheme.\n\n3\n\n\f(a) Hard Node Coupling\n\n(b) Soft Node Coupling\n\nFigure 1: Hierarchical Bayesian models for inter-segment and inter-node information coupling. 1(a): Hard\ncoupling between nodes with common hyperparameter \u0398 regulating the strength of the coupling between struc-\ntures associated with adjacent segments, Mh\n. This corresponds to the models in Section 3.1, with\n\u0398 = \u03b2, \u03a8 = [0, 10], and no \u2126, and Section 3.2, with \u0398 = {a, b}, \u03a8 = {\u03b1, \u03b1, \u03b3, \u03b3}, and \u2126 = [0, 20]. 1(b):\nSoft coupling between nodes, with node-speci\ufb01c hyperparameters \u0398i coupled via level2-hyperparameters \u03a8.\nThis corresponds to the model in Section 3.3, with \u0398i = {ai, bi}, \u03a8 = {\u03b1, \u03b1, \u03b3, \u03b3}, and \u2126 = [0, 20].\n\ni and Mh+1\n\ni\n\n3.1 Hard information coupling based on an exponential prior\nDenote by Ki := ki + 1 the total number of partitions in the time series associated with node i,\nand recall that each time series segment yh\ni , 1 \u2264 h \u2264\n, \u03b2) on the structures, and the joint probability\nKi. We impose a prior distribution P (Mh\ndistribution factorizes according to a Markovian dependence:\n\ni is associated with a separate subnetwork Mh\ni |Mh\u22121\n\ni\n\nP (y1\n\ni , . . . , yKi\n\ni\n\n, M1\n\ni , . . . , MKi\n\ni\n\n, \u03b2) =\n\nKi\n\nYh=1\n\nP (yh\n\ni |Mh\n\ni )P (Mh\n\ni |Mh\u22121\n\ni\n\nSimilar to [13] we de\ufb01ne\n\nP (Mh\n\ni |Mh\u22121\n\ni\n\n, \u03b2) =\n\nexp(\u2212\u03b2|Mh\n\ni \u2212 Mh\u22121\n\n|)\n\nZi(\u03b2, Mh\u22121\n\ni\n\ni\n)\n\n, \u03b2)P (\u03b2)\n\n(7)\n\n(8)\n\ni\n\ni\n\n, and |.| denotes the Hamming distance. For h = 1, P (Mh\n\nfor h \u2265 2, where \u03b2 is a hyperparameter that de\ufb01nes the strength of the coupling between Mh\nand Mh\u22121\ni ) is given by (2). The\ndenominator Z(\u03b2, Mh\u22121\n) in (8) is a normalizing constant, also known as the partition function:\ni \u2208M e\u2212\u03b2|Mh\n| where M is the set of all valid subnetwork structures. If we ignore\nZ(\u03b2) = PMh\nany fan-in restriction that might have been imposed a priori (via s), then the expression for the parti-\ntion function can be simpli\ufb01ed: Z(\u03b2) \u2248 Qp\n| = 1 + e\u2212\u03b2\nj=1 Zj(\u03b2), where Zj(\u03b2) = P1\nand hence Z(\u03b2) = `1 + e\u2212\u03b2\u00b4p. Inserting this expression into (8) gives:\n\nj =0 e\u2212\u03b2|eh\n\ni \u2212Mh\u22121\n\nj \u2212eh\u22121\n\neh\n\ni\n\nj\n\ni\n\n(9)\n\nP (Mh\n\ni |Mh\u22121\n\ni\n\n, \u03b2) =\n\n(1 + e\u2212\u03b2)p\n\nexp(\u2212\u03b2|Mh\n\ni \u2212 Mh\u22121\n\n|)\n\ni\n\nIt is straightforward to integrate the proposed model into the RJMCMC scheme of [6, 7] as described\nin Section 2.4. When proposing a new network structure Mh\ni for segment h, the prior\nprobability ratio has to be replaced by: P (Mh+1\ni |Mh\u22121\n. An additional MCMC step is\nP (Mh+1\ni |Mh\u22121\nintroduced for sampling the hyperparameter \u03b2 from the posterior distribution. For a proposal move\n\u03b2 \u2192 \u02dc\u03b2 with symmetric proposal probability Q( \u02dc\u03b2|\u03b2) = Q(\u03b2| \u02dc\u03b2) we get the following acceptance\n(1+e\u2212 \u02dc\u03b2)p , 1\ufb00 where in our study\nprobability: A( \u02dc\u03b2|\u03b2) = min\uf6be P ( \u02dc\u03b2)\nthe hyperprior P (\u03b2) was chosen as the uniform distribution on the interval [0, 10].\n\ni ,\u03b2)P ( \u02dcMh\ni ,\u03b2)P (Mh\n\ni=1QKi\n\nP (\u03b2) Qp\n\nexp(\u2212 \u02dc\u03b2|Mh\nexp(\u2212\u03b2|Mh\n\ni \u2212Mh\u22121\ni \u2212Mh\u22121\n\ni \u2192 \u02dcMh\n\n(1+e\u2212\u03b2)p\n\n| \u02dcMh\n|Mh\n\nh=2\n\n,\u03b2)\n\n,\u03b2)\n\n|)\n\n|)\n\ni\n\ni\n\ni\n\ni\n\ni\n\ni\n\n3.2 Hard information coupling based on a binomial prior\nAn alternative way of information sharing among segments and nodes is by using a binomial prior:\n\nP (Mh\n\ni |Mh\u22121\n\ni\n\n, a, b) = aN 1\n\n1 [h,i](1 \u2212 a)N 0\n\n1 [h,i]bN 0\n\n0 [h,i](1 \u2212 b)N 1\n\n0 [h,i]\n\n(10)\n\n4\n\n\fwhere we have de\ufb01ned the following suf\ufb01cient statistics: N 1\nthat are matched by an edge in Mh\nedge in Mh\nis the number of coinciding non-edges in Mh\u22121\njoint distribution can be expressed as:\n\n0 [h, i] is the number of edges in Mh\n\n1 [h, i] is the number of edges in Mh\u22121\nfor which there is no\n0 [h, i]\ni . Since the hyperparameters are shared, the\n\ni for which there is no edge in Mh\u22121\nand Mh\n\n1 [h, i] is the number of edges in Mh\u22121\n\n, and N 0\n\ni , N 1\n\ni , N 0\n\ni\n\ni\n\ni\n\ni\n\nP ({Mh\n\ni }|a, b) =\n\np\n\nYi=1\n\nP (M1\ni )\n\nP (Mh\n\ni |Mh\u22121\n\ni\n\n, a, b) = aN 1\n\n1 (1 \u2212 a)N 0\n\n1 bN 0\n\n0 (1 \u2212 b)N 1\n\n0\n\np\n\nYi=1\n\nP (M1\n\ni ) (11)\n\nk[h, i], and the right-hand side follows from Eq. (10).\nwhere we have de\ufb01ned N l\nThe conjugate prior for the hyperparameters a, b is a beta distribution, P (a, b|\u03b1, \u03b1, \u03b3, \u03b3) \u221d a(\u03b1\u22121)(1 \u2212\na)(\u03b1\u22121)b(\u03b3\u22121)(1 \u2212 b)(\u03b3\u22121) , which allows the hyperparameters to be integrated out in closed form:\n\ni=1PKi\n\nh=2 N l\n\nKi\n\nYh=1\nk = Pp\n\nP ({Mh\n\ni }|\u03b1, \u03b1, \u03b3, \u03b3) = Z Z P ({Mh\n\u0393(N 1\n\u0393(N 1\n\n\u0393(\u03b1 + \u03b1)\n\u0393(\u03b1)\u0393(\u03b1)\n\n\u221d\n\ni }|a, b)P (a, b|\u03b1, \u03b1, \u03b3, \u03b3)dadb\n\n(12)\n\n1 + \u03b1)\u0393(N 0\n1 + \u03b1 + N 0\n\n1 + \u03b1)\n1 + \u03b1)\n\n\u0393(\u03b3 + \u03b3)\n\u0393(\u03b3)\u0393(\u03b3)\n\n\u0393(N 0\n\u0393(N 0\n\n0 + \u03b3)\u0393(N 1\n0 + \u03b3 + N 1\n\n0 + \u03b3)\n0 + \u03b3)\n\ni , the structures Mh\n\ni \u2192 \u02dcMh\ni }|\u03b1, \u03b1, \u03b3, \u03b3), as P ({M1\n\nThe level-2 hyperparameters \u03b1, \u03b1, \u03b3, \u03b3 are given a uniform hyperprior over [0, 20]. The MCMC\nscheme of Section 2.4 has to be modi\ufb01ed as follows. When proposing a new network structure\nfor node i and segment h, Mh\ni enter the prior probability\ni=1|\u03b1,\u03b1,\u03b3,\u03b3)\n. Note that as\nratio via the expression P ({Mh\ni=1|\u03b1,\u03b1,\u03b3,\u03b3)\na consequence of integrating out the hyperparameters, all network structures become interdepen-\n0 .\ndent, and information about the structures is contained in the suf\ufb01cient statistics N 1\n0 , N 0\nA new proposal move for the level-2 hyperparameters is added to the existing RJMCMC scheme\nof Section 2.4. New values for the level-2 hyperparameters x \u2208 {\u03b1, \u03b1, \u03b3, \u03b3} are proposed from\na uniform distribution over a \ufb01xed interval. For a move x \u2192 \u02dcx, the acceptance probability is:\n, 1\ufb00 where {\u03b1, \u03b1, \u03b3, \u03b3} \\ x corresponds to {\u03b1, \u03b3, \u03b3} if\nA(\u02dcx|x) = min\uf6be P ({M1\nx designates hyperparameter \u03b1, and similarly for \u03b1, \u03b3, \u03b3.\n\ni and \u02dcMh\nKi\ni }p\ni ,...,M\nKi\ni }p\ni ,...,M\n\ni=1|\u02dcx,{\u03b1,\u03b1,\u03b3,\u03b3}\\\u02dcx)\ni=1|x,{\u03b1,\u03b1,\u03b3,\u03b3}\\x)\n\ni ,..., \u02dcMh\ni ,...,Mh\n\ni ,...,M\ni ,...,M\n\nKi\ni }p\nKi\ni }p\n\n1 , N 1\n\n1 , N 0\n\nP ({M1\n\nP ({M1\n\ni\n\n(1 \u2212 ai)(\u03b1\u22121)b(\u03b3\u22121)\n, ai, bi) = (ai)N 1\n\ni\n\n3.3 Soft information coupling based on a binomial prior\nWe can relax the information sharing scheme from a hard to a soft coupling by introducing\nnode-speci\ufb01c hyperparameters ai, bi that are softly coupled via a common level-2 hyperprior,\nP (ai, bi|\u03b1, \u03b1, \u03b3, \u03b3) \u221d a(\u03b1\u22121)\n\n(1 \u2212 bi)(\u03b3\u22121) , as illustrated in Figure 1(b):\n\nk[i] = PKi\n\ni |Mh\u22121\n\nP (Mh\n\n(13)\nThis leads to a straightforward modi\ufb01cation of eq. (11) \u2013 replacing a, b by ai, bi \u2013 from which we\nget as an equivalent to (13), using the de\ufb01nition N l\n\n1 [h,i](1 \u2212 ai)N 0\n\n0 [h,i](1 \u2212 bi)N 1\n\n1 [h,i](bi)N 0\n\n0 [h,i]\n\ni\n\nh=2 N l\n\nk[h, i]:\n\nP (M1\n\ni , . . . , M\n\nKi\ni\n\n|\u03b1, \u03b1, \u03b3, \u03b3) \u221d\n\n\u0393(\u03b1 + \u03b1)\n\u0393(\u03b1)\u0393(\u03b1)\n\n\u0393(N 1\n\u0393(N 1\n\n1 [i] + \u03b1)\u0393(N 0\n1 [i] + \u03b1 + N 0\n\n1 [i] + \u03b1)\n1 [i] + \u03b1)\n\n\u0393(\u03b3 + \u03b3)\n\u0393(\u03b3)\u0393(\u03b3)\n\n\u0393(N 0\n\u0393(N 0\n\n0 [i] + \u03b3)\u0393(N 1\n0 [i] + \u03b3 + N 1\n\n0 [i] + \u03b3)\n0 [i] + \u03b3)\n\n(14)\n\nAs in Section 3.2, we extend the RJMCMC scheme from Section 2.4 so that when proposing a new\nnetwork structure, Mh\ni , the acceptance probability has to be updated with the prior ratio:\nP (M1\n. In addition, we have to add a new level-2 hyperparameter update move\nP (M1\nx \u2192 \u02dcx, where the prior and proposal probabilities are the same as in Section 3.2, and the acceptance\n\ni ,..., \u02dcMh\ni ,...,Mh\n\ni \u2192 \u02dcMh\n\ni ,...,M\ni ,...,M\n\n|\u03b1,\u03b1,\u03b3,\u03b3)\n\n|\u03b1,\u03b1,\u03b3,\u03b3)\n\nKi\ni\nKi\ni\n\ni=1\n\nKi\ni\nKi\ni\n\n|\u02dcx,{\u03b1,\u03b1,\u03b3,\u03b3}\\\u02dcx)\n\n|x,{\u03b1,\u03b1,\u03b3,\u03b3}\\x)\n\nP (M1\nP (M1\n\n, 1\ufb00 .\n\ni ,...,M\ni ,...,M\n\nprobability becomes: A(\u02dcx|x) = min\uf6beQp\n4 Results\nThe methods described in this paper have been implemented in R, based on code from [6, 7]. Our\nprogram sets up an RJMCMC simulation to sample the network structure, the changepoints and\nthe hyperparameters from the posterior distribution. As a convergence diagnostic we monitor the\npotential scale reduction factor (PSRF) [14], computed from the within-chain and between-chain\nvariances of marginal edge posterior probabilities. Values of PSRF\u22641.1 are usually taken as indi-\ncation of suf\ufb01cient convergence. In our simulations, we extended the burn-in phase until a value of\n\n5\n\n\f0\n0\n0\n0\n\n.\n.\n.\n.\n\n1\n1\n1\n1\n\n8\n8\n8\n8\n\n.\n.\n.\n.\n\n0\n0\n0\n0\n\n0\n0\n0\n0\n\n.\n.\n.\n.\n\n1\n1\n1\n1\n\n8\n8\n8\n8\n\n.\n.\n.\n.\n\n0\n0\n0\n0\n\n \n\ne\nr\no\nc\nS\nC\nO\nR\nU\nA\n\n6\n6\n6\n6\n\n.\n.\n.\n.\n\n0\n0\n0\n0\n\n4\n4\n4\n4\n\n.\n.\n.\n.\n\n0\n0\n0\n0\n\n \n\ne\nr\no\nc\nS\nC\nR\nP\nU\nA\n\n6\n6\n6\n6\n\n.\n.\n.\n.\n\n0\n0\n0\n0\n\n4\n4\n4\n4\n\n.\n.\n.\n.\n\n0\n0\n0\n0\n\n2\n2\n2\n2\n\n.\n.\n.\n.\n\n0\n0\n0\n0\n\n0\n0\n0\n0\n\n.\n.\n.\n.\n\n0\n0\n0\n0\n\nSame Segs\n\nDifferent Segs\n\n2\n2\n2\n2\n\n.\n.\n.\n.\n\n0\n0\n0\n0\n\n0\n0\n0\n0\n\n.\n.\n.\n.\n\n0\n0\n0\n0\n\nSame Segs\n\nDifferent Segs\n\n(a) AUROC Score Comparison\n\n(b) AUPRC Score Comparison\n\nFigure 2: Network reconstruction performance comparison of AUROC and AUPRC reconstruction\nscores for the four methods, HetDBN-0 (white), HetDBN-Exp (light grey), HetDBN-Bino1 (dark\ngrey, left), HetDBN-Bino2 (dark grey, right). The boxplots show the distributions of the scores for 10\ndatasets with 4 network segments each, where the horizontal bar shows the median, the box margins\nshow the 25th and 75th percentiles, the whiskers indicate data within 2 times the interquartile range,\nand circles are outliers. \u201cSame Segs\u201d means that all segments in a dataset have the same structure,\nwhile \u201cDifferent Segs\u201d indicates that structure changes are applied to the segments sequentially.\n\nPSRF\u2264 1.05 was reached, and then sampled 1000 network and changepoint con\ufb01gurations in inter-\nvals of 200 RJMCMC steps. From these samples we compute the marginal posterior probabilities\nof all potential interactions, which de\ufb01nes a ranking of the edges in the recovered network. When\nthe true network is known, this allows us to construct the Receiver Operating Characteristic (ROC)\ncurve (plotting the sensitivity or recall against the complementary speci\ufb01city) and the precision-\nrecall (PR) curve (plotting the precision against the recall), and to assess the network reconstruction\naccuracy in terms of the areas under these graphs (AUROC and AUPRC, respectively); see [15].\n\n4.1 Comparative evaluation on simulated data\nWe randomly generated 10 networks with 10 nodes each, with the number of parents per node drawn\nfrom a Poisson distribution with mean \u03bb = 3. To simulate changes in the network structure, we cre-\nated 4 different network segments by drawing the number of changes from a Poisson distribution\nand applying the changes uniformly at random to edges and non-edges in the previous segment. For\neach segment, we generated a time series of length 15 using a linear regression model. The regres-\nsion weights were drawn from a Gaussian N (0, 1), and Gaussian observation noise N (0, 1) was\nadded. We compared the network reconstruction accuracy of the non-homogeneous DBN without\ninformation sharing proposed in [6, 7] (HetDBN-0) with the three information sharing approaches,\nbased on the exponential prior from Section 3.1 (HetDBN-Exp), the binomial prior with hard node\ncoupling from Section 3.2 (HetDBN-Bino1), and the binomial prior with soft node coupling from\nSection 3.3 (HetDBN-Bino2). Figures 2(a) and 2(b) shows the network reconstruction performance\nof the different information sharing methods in terms of AUROC and AUPRC scores. All infor-\nmation sharing methods show a clear improvement in network reconstruction over HetDBN-0, as\ncon\ufb01rmed by paired t-tests (p < 0.01). We investigated two different situations, the case where all\nsegment structures are the same (although edge weights are allowed to vary) and the case where\nchanges are applied sequentially to the segments3. Information sharing is most bene\ufb01cial for the\n\ufb01rst case, but even when we introduce changes we still see an increase in the network reconstruction\nscores compared to HetDBN-0. When all segments are the same, HetDBN-Bino1 and HetDBN-\nBino2 outperform HetDBN-Exp (p < 0.05), but there is no signi\ufb01cant difference between the two\nbinomial methods. Paired t-tests showed that all other differences in mean are signi\ufb01cant. When the\nsegments are different, all information sharing methods outperform HetDBN-0 (p < 0.05), but the\ndifference between the information sharing methods is not signi\ufb01cant.\n\n4.2 Morphogenesis in Drosophila melanogaster\nWe applied our methods to a gene expression time series for eleven genes involved in the muscle\ndevelopment of Drosophila melanogaster [16]. The microarray data measured gene expression lev-\nels during all four major stages of morphogenesis: embryo, larva, pupa and adult. We investigated\nwhether our methods were able to infer the correct changepoints corresponding to the known transi-\ntions between stages. Figure 3(a) shows the posterior probabilities of inferred changepoints for any\ngene using HetDBN-0, while Figure 3(c) shows the posterior probabilities for the information shar-\n\n3We chose to draw the number of changes from a Poisson with mean 1 for each node.\n\n6\n\n\fy\nt\ni\nl\ni\n\nb\na\nb\no\nr\nP\n\n \nr\no\ni\nr\ne\n\nt\ns\no\nP\n\ny\nt\ni\nl\ni\n\nb\na\nb\no\nr\nP\n\n \nr\no\ni\nr\ne\n\nt\ns\no\nP\n\n0\n1\n\n.\n\n8\n0\n\n.\n\n6\n\n.\n\n0\n\n4\n\n.\n\n0\n\n2\n0\n\n.\n\n0\n0\n\n.\n\n0\n\n.\n\n1\n\n8\n0\n\n.\n\n6\n\n.\n\n0\n\n4\n\n.\n\n0\n\n2\n0\n\n.\n\n0\n0\n\n.\n\n0\n5\n\n0\n4\n\n0\n3\n\n0\n2\n\n0\n1\n\n0\n\ne\nc\nn\ne\nr\ne\n\nf\nf\ni\n\nD\n\n \nr\ne\n\nt\n\n \n\ne\nm\na\nr\na\nP\nn\no\ns\ns\ne\nr\ng\ne\nR\n\ni\n\n0\n\n10\n\n20\n\n30\nTimepoints\n\n40\n\n50\n\n60\n\n(a) Drosophila CPs with HetDBN-0\n\n0\n\n10\n\n20\n\n30\n40\nTimepoints\n\n50\n\n60\n\n(b) Drosophila CPs with TESLA\n\nHetDBN\u2212Exp\nHetDBN\u2212Bino1\nHetDBN\u2212Bino2\n\nHetDBN\u22120\nHetDBN\u2212Exp\nHetDBN\u2212Bino1\nHetDBN\u2212Bino2\n\ny\nt\ni\nl\ni\n\nb\na\nb\no\nr\nP\n\n \nr\no\ni\nr\ne\n\nt\ns\no\nP\n\n0\n\n.\n\n1\n\n8\n0\n\n.\n\n6\n\n.\n\n0\n\n4\n\n.\n\n0\n\n2\n0\n\n.\n\n0\n0\n\n.\n\n0\n\n10\n\n20\n\n30\nTimepoints\n\n40\n\n50\n\n60\n\n0\n\n5\n\n10\n\n15\n\n20\nTimepoints\n\n25\n\n30\n\n35\n\n(c) Drosophila CPs with HetDBN-Exp and HetDBN-Bino\n\n(d) Synthetic Network CPs with HetDBN\n\nFigure 3: Changepoints inferred on gene expression data related to morphogenesis in Drosophila\nmelanogaster, and synthetic biology in Saccharomyces cerevisiae (yeast). All \ufb01gures using HetDBN\nplot the posterior probability of a changepoint occurring for any node at a given time plotted against\ntime. 3(a): HetDBN-0 changepoints for Drosophila (no information sharing) 3(b): TESLA, L1-\nnorm of the difference of the regression parameter vectors associated with two adjacent time points\nplotted against time. 3(c): HetDBN changepoints for Drosophila with information sharing; the\nmethod is indicated by the legend. 3(d) HetDBN changepoints for the synthetic gene regulatory\nnetwork in yeast. In 3(a)-3(c), the vertical dotted lines indicate the three morphogenic transitions,\nwhile in 3(d) the line indicates the boundary between \u201cswitch on\u201d and \u201cswitch off\u201d data.\n\ning methods. For comparison, we applied the method proposed in [3], using the authors\u2019 software\npackage TESLA (Figure 3(b)). Robinson and Hartemink applied the discrete non-homogeneous\nDBN in [1] to the same data set, and a plot corresponding to Figure 3(b) can be found in their paper.\n\nOur non-homogeneous DBN methods are generally more successful than TESLA, in that they re-\ncover changepoints for all three transitions (embryo \u2192 larva, larva \u2192 pupa, and pupa \u2192 adult).\nFigure 3(b) indicates that the last transition, pupa \u2192 adult, is less clearly detected with TESLA,\nand it is completely missing in [1]. Both our method as well as TESLA detect additional transitions\nduring the embryo stage, which are missing in [1]. We would argue that a complex gene regulatory\nnetwork is unlikely to transition into a new morphogenic phase all at once, and some pathways might\nhave to undergo activational changes earlier in preparation for the morphogenic transition. As such,\nit is not implausible that additional transitions at the gene regulatory network level occur. However,\na failure to detect known morphogenic transitions can clearly be seen as a shortcoming of a method,\nand on these grounds our model appears to outperform the two alternative ones. We note that the\nmain effect of information sharing is to reduce the size of the smaller peaks, while keeping the three\nmost salient peaks (corresponding to larva \u2192 pupa, and pupa \u2192 adult, and an extra transition in the\nembryo phase). This re\ufb02ects the fact that these changepoints are associated with signi\ufb01cant changes\nin network structure, and adds to the interpretability of the results. The drawback is that the third\nmorphological transition (embryo \u2192 larva) is less pronounced.\n\n4.3 Reconstruction of a synthetic gene regulatory network in Saccharomyces cerevisiae\nThe highly topical \ufb01eld of synthetic biology enables biologists to design known gene regulatory\nnetworks in living cells. In the work described in [17], a synthetic regulatory network of 5 genes was\nconstructed in Saccharomyces cerevisiae (yeast), and gene expression time series were measured\nwith RT-PCR for 16 and 21 time points under two experimental conditions, related to the carbon\nsource: galactose (\u201cswitch on\u201d) and glucose (\u201cswitch off\u201d). The authors tried to reconstruct the\nknown gold-standard network from these time series with two established state-of-the-art methods\nfrom computational systems biology, one based on ordinary differential equations (ODEs), called\n\n7\n\n\fPrecision\u2212Recall for Switch On\n\nPrecision\u2212Recall for Switch Off\n\ni\n\ni\n\nn\no\ns\nc\ne\nr\nP\n\n0\n\n.\n\n1\n\n8\n\n.\n\n0\n\n6\n\n.\n\n0\n\n4\n\n.\n\n0\n\n2\n\n.\n\n0\n\n0\n\n.\n\n0\n\nTSNI\n\nBanjo\n\nHetDBN\u22120\nHetDBN\u2212Exp\nHetDBN\u2212Bino1\nHetDBN\u2212Bino2\n\ni\n\ni\n\nn\no\ns\nc\ne\nr\nP\n\n0\n\n.\n\n1\n\n8\n\n.\n\n0\n\n6\n\n.\n\n0\n\n4\n\n.\n\n0\n\n2\n\n.\n\n0\n\n0\n\n.\n\n0\n\nBanjo\n and \n TSNI\n\nHetDBN\u22120\nHetDBN\u2212Exp\nHetDBN\u2212Bino1\nHetDBN\u2212Bino2\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\n0.0\n\n0.2\n\n0.4\n\n0.6\n\n0.8\n\n1.0\n\nRecall\n\nRecall\n\nFigure 4: Reconstruction of a known gene regulatory network from synthetic biology in yeast.\nThe network was reconstructed from two gene expression time series obtained with RT-PCR in\ntwo experimental conditions, re\ufb02ecting the switch in the carbon source from galactose (\u201cswitch\non\u201d) to glucose (\u201cswitch off\u201d). The reconstruction accuracy of the methods proposed in Section 3,\nwhere the legend is explained, is shown in terms of precision (vertical axis) - recall (horizontal\naxis) curves. Results were averaged over 10 independent MCMC simulations. For comparison,\n\ufb01xed precision/recall scores are shown for two state-of-the-art methods reported in [17]: Banjo, a\nconventional DBN, and TSNI, a method based on ODEs.\n\nTSNI, the other based on conventional DBNs, called Banjo; see [17] for details. Both methods\nare optimization-based and output a single network. By comparison with the known gold standard,\nthe authors obtained the precision (proportion of predicted interactions that are correct) and recall\n(proportion of predicted true interactions) scores. In our study, we merged the time series from\nthe two experimental conditions under exclusion of the boundary point4, and applied the four non-\nhomogeneous DBNs described before. Figure 3(d) shows the inferred marginal posterior probability\nof potential changepoints. The most signi\ufb01cant changepoint is at the boundary between \u201cswitch\non\u201d and \u201cswitch off\u201d data, con\ufb01rming that the known true changepoint is consistently identi\ufb01ed.\nThe biological mechanism behind the other peaks is not known, and they are potentially spurious.\nInterestingly, the application of the proposed information-coupling schemes reduces the height of\nthese peaks, with the binomial models having a stronger effect than the exponential one.\n\nAs we pursue a Bayesian inference scheme, we also obtain a ranking of the potential gene interac-\ntions in terms of their marginal posterior probabilities. From this we computed the precision-recall\ncurves [15] shown in Figure 4. Our non-homogeneous DBNs with information sharing outperform\nBanjo and TSNI both in the \u201cswitch on\u201d and the \u201cswitch off\u201d phase. They also perform better than\nHetDBN-0 on the \u201cswitch off\u201d data, but are slightly worse on the \u201cswitch on\u201d data. Note that the\nreconstruction accuracy on the \u201cswitch off\u201d data is generally poorer than on the \u201cswitch on\u201d data\n[17]. Our results are thus plausible, suggesting that information sharing boosts the reconstruction\naccuracy on the poorer time series segment at the cost of a degraded performance on the stronger\none. This effect is more pronounced for the exponential prior than for the binomial one, indicating\na tighter coupling. The average areas under the PR curves, averaged over both phases (\u201cswitch on\nand off\u201d), are as follows. HetDBN-0= 0.70, HetDBN-Exp= 0.77, HetDBN-Bino1= 0.75, HetDBN-\nBino2= 0.75. Hence, the overall effect of information sharing is a performance improvement.\n\n5 Conclusions\nWe have described a non-homogeneous DBN, which has various advantages over existing schemes:\nit does not require the data to be discretized (as opposed to [1]); it allows the network structure\nto change with time (as opposed to [2]); it includes three different regularization schemes based on\ninter-time segment information sharing (as opposed to [6, 7]); and it allows all hyperparameters to be\ninferred from the data via a consistent Bayesian inference scheme (as opposed to [3]). An evaluation\non simulated data has demonstrated an improved performance over [6, 7] when information sharing\nis introduced. The application of our method to gene expression time series taken during the life cy-\ncle of Drosophila melanogaster has revealed better agreement with known morphogenic transitions\nthan the methods of [1] and [3]. We have carried out a comparative evaluation of different informa-\ntion coupling schemes: a binomial versus an exponential prior, and hard versus soft coupling. In an\napplication to data from a topical study in synthetic biology, our methods have outperformed two\nestablished network reconstruction methods from computational systems biology.\n\n4When merging two time series (x1, . . . , xm) and (y1, . . . , yn), only the pairs xi \u2192 xj and yi \u2192 yj are\n\npresented to the DBN, while the pair xm \u2192 y1 is excluded due to the obvious discontinuity.\n\n8\n\n\fReferences\n[1] J. W. Robinson and A. J. Hartemink. Non-stationary dynamic Bayesian networks. In D. Koller,\nD. Schuurmans, Y. Bengio, and L. Bottou, editors, Advances in Neural Information Processing\nSystems (NIPS), volume 21, pages 1369\u20131376. Morgan Kaufmann Publishers, 2009.\n\n[2] M. Grzegorczyk and D. Husmeier. Non-stationary continuous dynamic Bayesian networks. In\nY. Bengio, D. Schuurmans, J. Lafferty, C. 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