Ghahramani, Z. (1998). Learning dynamic Bayesian networks.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of X 1 = fX[1] X [T ]g can be written as : P (X i [t]j it ) 70) 19 where it denotes the parents of X i [t] 6. 2 Dependency range and structural inductions In the BNs literature, DBNs are de ned using the assumption that X [t] is Markovian and stationary [10] 9] 20] [15] [13] The time dependency properties of X i [t] determines the parents, and hence the network structure at time t. If the process is stationary, then the network structure is repeating for each time instant. However, care must be taken for the boundary regions depending on the dependency range ....

Z. Ghahramani. Learning dynamic bayesian networks, 1997. ../ghahramani /ghahramani97b.ps.

....space mixture distributions are described by the mixture weights c jn , c mean vectors and diagonal covariance matrices . x x t 1 x o o t 1 o t 1 q q t Figure 1: Bayesian network representing a factor analysed hidden Markov model. Dynamic Bayesian networks (DBN) [6] are often presented in conjunction with generative models to illustrate the conditional independence assumptions made in a statistical model. A DBN describing a FAHMM is shown in Figure 1. The square nodes represent discrete random variables 3 such as the HMM state q and state and ....

Z. Ghahramani. Learning dynamic Bayesian networks. In C.L. Giles and M. Gori, editors, Adaptive Processing of Sequences and Data Structures, volume 1387 of Lecture Notes in Computer Science, pages 168--197. Springer, 1998.

....according to dynamic regime. Otherwise, an averaged evolution is learned, which leads to alarm states that are invalid for time series with deviating dynamics. Suggested approaches include the HMM clustering approach in [8] mixtures of regressors clustering [1] and switching state space models [2]. A 1 2 3 4 5 6 1 .98 .02 .00 0 0 0 2 .02 .96 .02 0 0 0 3 .00 .02 .97 .01 0 0 4 .00 .00 .01 .96 .02 .01 5 0 .00 .00 .03 .95 .02 6 0 0 .00 .00 .03 .97 B 1 2 3 4 5 6 1 .04 .16 .79 .01 .00 .00 2 .01 .00 .06 .91 .03 .00 3 .00 .00 .00 .02 .62 .34 4 .00 .00 .00 .02 .01 .15 5 .00 .00 0 0 .00 ....

Z. Ghahramani. Learning dynamic bayesian networks. In C.L. Giles and M. Gori, editors, Adaptive processing of temporal information, 1997.

....according to dynamic regime. Otherwise, an averaged evolution is learned, which leads to alarm states that are invalid for time series with deviating dynamics. Suggested approaches include the HMM clustering approach in [8] mixtures of regressors clustering [1] and switching state space models [2]. II AII 1 I 2 I 3 I 4 I 5 I 6 Illl B II 1 I 2 I 3 I 4 I 5 I 6 I i .98 .02 .00 0 0 0 i .04 .16 .79 .01 .00 .00 2 .02 .96 .02 0 0 0 2 .01 .00 .06 .91 .03 .00 3 .00 .02 .97 .01 0 0 3 .00 .00 .00 .02 .62 .34 4 .00 .00 .01 .96 .02 .01 4 .00 .00 .00 .02 .01 .15 5 0 .00 .00 .03 .95 .02 5 .00 .00 0 ....

Z. Ghahramani. Learning dynamic bayesian networks. In C.L. Giles and M. Gori, editors, Adaptive processing of temporal information, 1997.

....by both observation processes when k = p and v = 0. Factorial hidden Markov models [18] use distributed representation of the discrete state space so that several independent HMMs can be viewed to have produced the observation vectors. 2. 4 Bayesian Networks In this paper, Bayesian networks [14] are used to illustrate the statistical independencies between di erent random variables in the probabilistic models. Bayesian networks are directed acyclic graphs, also known as graphical models. The notation is adopted from [33] where round nodes were used to denote continuous and squared nodes ....

Z. Ghahramani. Learning dynamic Bayesian networks. In C.L. Giles and M. Gori, editors, Adaptive Processing of Sequences and Data Structures, volume 1387 of Lecture Notes in Computer Science, pages 168-197. Springer, 1998.

....until we have modeled the baseline interactions among individuals, and thus in this study we focus on the latter. 3 The Influence Model In seeking a model appropriate to our goals, we turned to the work on dynamic Bayes nets (DBN s) in the graphical models community (for a review of DBNs see [3]) The most straightforward approach would be to model each participant with an HMM and then take the outer product of all the state spaces. Unfortunately, the number of states would be exponential in the number of chains N where Q is the number of states per chain, i.e. Q , and the number of ....

Ghahramani Z, "Learning Dynamic Bayesian Networks." Adaptive Processing of Sequences and Data Structures. International Summer School on Neural Networks `E.R. Caianiello'. Tutorial Lectures. Springer Verlag, 1998.

....models or belief networks, is a combination of probability theory and graph theory in which dependencies between random variables is expressed graphically. Hence a Bayesian network can be defined as a graphical model for representing conditional independencies between a set of random variables [Ghahramani97]. Let us consider an example from a tutorial by Ghahramani. Figure 1 shows a graphical representation of the joint probability P(W,X,Y,Z) that can be factorized as a set of conditional independence relations, as follows: P(W,X,Y,Z) P(W) P(X) P(Y W) P(Z X,Y) Given the values of X and Y, we can ....

....and observation probabilities P(Y X ) are both decomposed into deterministic and stochastic components. If both the transition and output functions are linear and time invariant, and the distribution of states and observation noise variables is Gaussian, we get a Linear Gaussian state space model [Ghahramani97]: X t = A X t 1 w where A is the state transition matrix and w t is a noise vector Y t = C X t v t where C is the observation matrix and v t is a noise vector Such models, also known as Kalman filters [Kalman61] are used extensively in control and signal processing applications. In ....

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Ghahramani, Zoubin. 1997. Learning Dynamic Bayesian Networks. Adaptive Processing of Temporal Information. Lecture Notes in Artificial Intelligence. Springer-Verlag. See related tutorial paper here - http://www.cs.utoronto.ca/~zoubin/

....by both observation processes when k = p and v = 0. Factorial hidden Markov models [18] use distributed representation of the discrete state space so that several independent HMMs can be viewed to have produced the observation vectors. 2. 4 Bayesian Networks In this paper, Bayesian networks [14] are used to illustrate the statistical independencies between di erent random variables in the probabilistic models. Bayesian networks are directed acyclic graphs, also known as graphical models. The notation is adopted from [33] where round nodes were used to denote continuous and squared nodes ....

Z. Ghahramani. Learning dynamic Bayesian networks. In C.L. Giles and M. Gori, editors, Adaptive Processing of Sequences and Data Structures, volume 1387 of Lecture Notes in Computer Science, pages 168-197. Springer, 1998.

....to be special cases of belief propagation and typically of Pearl s algorithm. Such reductions usually have simplified and illuminated the original derivation. Examples of this phenomena include Fourier and Hadamard transforms, a number of decoding algorithms (Gallager Tanner Wiberg, Turbo, etc. [2, 30, 18], Kalman filters (the RauchTung Streibel smoother) the EM algorithm for hiddden Markov models [39] and the inside outside algorithm for stochastic context free grammars [7] Both emprirical and theoretical evidence suggest that at least in some situations the simple message passing algorithm ....

.... instance of belief propagation in belief networks with loops (see [30] for details) While belief propagation in general remains NP complete, approximate algorithms can often be derived using Monte Carlo methods, such as Gibbs sampling [20, 46] and variational methods, such as mean field theory [36, 25, 18], sometimes leveraging the particular structure of a network (see also [23, 15, 45] for an interesting learning algorithm for a particular class of Bayesian networks) Gibbs sampling is particularly attractive for Bayesian networks, both because of its simplicity and generality. 8.3.7 Gibbs ....

Z. Ghahramani. Learning dynamic Bayesian networks. In M. Gori and C. L. Giles, editors, Adaptive Processing of Temporal Information. Lecture Notes in Artifical Intelligence. Springer Verlag, Heidelberg, 1998. In press.

.... HMMs (see Figure 4) Unfortunately, since the state space is exponential in the number of variables, exact inference in these models is computationally intractable (requiring O(TS 2n ) time and space, which is exponential in n) We therefore have to use approximate inference (see e.g. [Ghahramani, 1998]) 9 Y1 A1 B1 C1 A2 B2 C2 A3 B3 C3 . Y2 Y3 Figure 4: A factorial hidden Markov model. The hidden state is represented in a distributed fashion, in terms of three a priori independent discrete random variables, X t = A t ; B t ; C t ) However, after observing the output Y t , the hidden ....

....is a way to compute P (X t jy 1:T ; u 1:T ) in an LDS, and is entirely analogous to the forwards backwards algorithm for HMMs. 1 To do parameter estimation in an LDS, we can use the EM algorithm, using the RTS algorithm as a subroutine to compute the required expected sucient statistics (see [Ghahramani, 1998] for details) 1 In the general case, the RTS algorithm takes O(Tm 3 ) time and O(Tn 2 ) space, where Y t 2 IR m and X t 2 IR n , because at each step, the algorithm must invert an m m matrix and must store an n n covariance matrix. 10 3.3.2 Non linear systems Inference, and ....

Ghahramani, Z. (1998). Learning Dynamic Bayesian Networks. In Giles, C. and Gori, M., editors, Adaptive Processing of Sequences and Data Structures. Lecture Notes in Articial Intelligence, pages 168-197. Springer-Verlag.

....until we have modeled the baseline interactions among individuals, and thus in this study we focus on the latter. 3 The Influence Model In seeking a model appropriate to our goals, we turned to the work on dynamic Bayes nets (DBN s) in the graphical models community (for a review of DBNs see [3]) The most straightforward approach would be to model each participant with an HMM and then take the outer product of all the state spaces. Unfortunately, the number of states would be exponential in the number of chains N where Q is the number of states per chain, i.e. Q N , and the number of ....

Ghahramani Z, "Learning Dynamic Bayesian Networks." Adaptive Processing of Sequences and Data Structures. International Summer School on Neural Networks `E.R. Caianiello'. Tutorial Lectures. Springer Verlag, 1998.

....propagation in the computation of the hidden state probabilities (a.k. a E Step [6] The maximisation equations for the observation models are identical to those for HMMs [4, 6] Thus we only describe here the derivation of the forward backward recursions for the CHMM and refer the reader to [12] for background literature. For a standard HMMs we have: ff t 1 j P (O t 1 t ; S t 1) P (O t 1 jS t 1) Z P (O t 1 ; S t )P (S t 1 jS t )dS t = P (O t 1 jS t 1) Z P (S t 1 ; S t jO t 1 )dS t = P (O t 1 jS t 1)P (S t 1 jO t 1 ) 1) and by conditional independence = P (O t 1 1 ....

Z. Ghahramani. Learning dynamic bayesian networks. In C.L. Giles and M. Gori, editors, Adaptive Processing of Sequences and Data Structures, pages 168--197. Berlin: Springer-Verlag, 1998.

....(x , t) may be used to re ne the estimate of A t . The Markov structure of the generative model permits the pdf p(a t j X T ) to be found from a forward pass through the data, followed by a backward sweep in which the in uence of future observations on a t is evaluated. See, for example, [8] for a detailed exposition of forward backward recursions. In the forward pass the joint probability p(a t ; x 1 ; x t ) t = Z t 1 p(a t j a t 1 ) p(x t j a t ) da t 1 (24) is recursively evaluated. In the backward sweep the condtional probability p(x t 1 ; x T j a t ) t ....

....the hidden state is now comprised of a t and the states of the sources s t , and predictions and corrections for the full state should be made. Since the sources are independent, predictions for the each source and a t may be made independently and the system is a factorial hidden Markov model [8]. A number of source predictors have been implemented, including the Kalman lter, AR models and Gaussian mixture models. However, the fundamental indeterminacy of the source scales ren 8 ders the combined tracker unstable. The instability arises because the change in observation from x t 1 ....

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Z. Ghahramani. Learning Dynamic Bayesian Networks. In C.L. Giles and M. Gori, editors, Adaptive Processing of Temporal Information, Lecture Notes in Articial Intelligence. Springer-Verlag, 1999.

....(x , t) may be used to refine the estimate of A t . The Markov structure of the generative model permits the pdf p(a t jX T ) to be found from a forward pass through the data, followed by a backward sweep in which the influence of future observations on a t is evaluated. See, for example, [7] for a detailed exposition of forward backward recursions. Figure 6 illustrates tracking both by smoothing and causal filtering. As before the elements of the mixing matrix vary sinusoidally with time except for discontinous jumps at t = 600 and 1200. Both the filtering and forward backward ....

....In common with most tracking methods, the state noise covariance Q and the observational noise covariance R are parameters which must be set. Although we have not addressed the issue here, it is straight forward, though laborious, to obtain maximum likelihood estimates for them using the EM method [7]. Although we have modelled the source densities here with generalised exponentials, which permits the separation of a wide range of sources, it is possible to both generalise or restrict the source model. More complicated (possibly multi modal) densities may be represented by a mixture of ....

Z. Ghahramani. Learning Dynamic Bayesian Networks. In C.L. Giles and M. Gori, editors, Adaptive Processing of Temporal Information, Lecture Notes in Artificial Intelligence. Springer-Verlag, 1999.

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Ghahramani, Z. (1998). Learning dynamic Bayesian networks.

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Zoubin Ghahramani. Learning dynamic Bayesian networks. Lecture Notes in Computer Science, 1387, 1998.

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Z. Ghahramani. Learning dynamic Bayesian networks. In C. L. Giles and M. Gori, editors, Adaptive Processing of Temporal Information, pages 168--197. Springer-Verlag, 1998.

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Z. Ghahramani. Learning dynamic Bayesian networks. In C. L. Giles and M. Gori, editors, Adaptive Processing of Temporal Information. Lecture Notes in Artificial Intelligence. Springer-Verlag, 1997.

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Z. Ghahramani. Learning dynamic Bayesian networks. In C.L. Giles and M. Gori, editors, Adaptive Processing of Sequences and Data Structures, volume 1387 of Lecture Notes in Computer Science, pages 168--197. Springer, 1998.

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Z. Ghahramani. Learning dynamic bayesian networks. In C. L. Giles and M. Gori, editors, Adaptive Processing of Temporal Information. LNAI, SpringerVerlag, Berlin, 1997.

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Z. Ghahramani. Learning dynamic Bayesian networks. In C.L. Giles and M. Gori, editors, Adaptive Processing of Sequences and Data Structures, volume 1387 of Lecture Notes in Computer Science, pages 168--197. Springer, 1998.

No context found.

Z. Ghahramani. Learning dynamic Bayesian networks. In C.L. Giles and M. Gori, editors, Adaptive Processing of Sequences and Data Structures, volume 1387 of Lecture Notes in Computer Science, pages 168--197. Springer, 1998.

No context found.

Z. Ghahramani. Learning dynamic Bayesian networks. In Adaptive Processing of Sequences and Data Structures, Lecture Notes in Artificial Intelligence, pages 168--197. Springer-Verlag, 1998.

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Z. Ghahramani. Learning Dynamic Bayesian Networks. In C. L. Giles and M. Gori (eds), Adaptive Processing of Sequences and Data Structures. Berlin:SpringerVerlag.

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Ghahramani, Z.: Learning Dynamic Bayesian Network. In: Giles, C.L. and Gori, M. (Eds.): Adaptive Processing of Temporal Information. Springer, New York, MA (1998)

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