F. V. Jensen, S. L. Lauritzen, and K. G. Olesen. 1990. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quaterly, 4:269--282.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Probabilistic reasoning using Belief networks, computing the probability of one or more events given some evidence, is known to be NP hard [Cooper1990] However most commonly used exact algorithms for probabilistic inference such as join tree clustering [Lauritzen and Spiegelhalter1988, Jensen et al..1990] or variable elimination [Dechter1996] exploit the networks structure. These algorithms are time and space exponential in a graph parameter capturing the density of the network called tree width. Yet, for large belief networks, the tree width is often large, making exact inference impractical and ....

....the theory and practice of belief net works, and thus may be more accessible. The idea is as follows. Pearl s BP algorithm on trees was extended to a general propagation algorithm on trees of clusters called join tree clustering or junction tree clustering [Lauritzen and Spiegelhalter1988, Jensen et al..1990] Since this join tree clustering is a message passing algorithm between clusters of functions, it can also be applied to a join graph rather than a join tree. Namely, rather than decomposing the network into a join tree whose clusters are often too big and thus too costly to process, we can ....

F.V. Jensen, S.L Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computation. Computational Statistics Quarterly, 4:269-282, 1990.

.... Probabilistic reasoning using Belief networks, computing the probability of one or more events given some evidence, is known to be NP hard [Cooper1990] However most commonly used exact algorithms for probabilistic inference such as join tree clustering [Lauritzen and Spiegelhalter1988, Jensen et al..1990] or variable elimination [Dechter1996] exploit the networks structure. These algorithms are time and space exponential in a graph parameter capturing the density of the network called tree width. Yet, for large belief networks, the tree width is often large, making exact inference impractical and ....

....the theory and practice of belief net works, and thus may be more accessible. The idea is as follows. Pearl s BP algorithm on trees was extended to a general propagation algorithm on trees of clusters called join tree clustering or junction tree clustering [Lauritzen and Spiegelhalter1988, Jensen et al..1990] Since this join tree clustering is a message passing algorithm between clusters of functions, it can also be applied to a join graph rather than a join tree. Namely, rather than decomposing the network into a join tree whose clusters are often too big and thus too costly to process, we can ....

F.V. Jensen, S.L Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computation. Computational Statistics Quarterly, 4:269--282, 1990.

....Intelligence. Unfortunately, basic computations on belief networks, such as calculating the probability of a query variable given evidence, or finding the probability of the most probable explanation consistent with a certain variable value given evidence, are NP hard. Clique tree propagation [ Jensen et al. 1990 ] is the most popular exact algorithm, which requires time and space exponential in the treewidth of the network s interaction graph. Variable elimination [ Zhang and Poole, 1994; Dechter, 1999 ] is a simplified formulation of this method, which only computes the answer to one query, but which is ....

F. V. Jensen, S. L. Lauritzen, and K. G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, 5(4):269--282, 1990.

....IV. ON HYPERTREE ORGANIZATION A. Classification of loops The difficulty of coherent inference in multiply connected graphical models (those with loops) of probabilistic knowledge is well known and many inference algorithms have been proposed. Those based on message passing, e.g. 20] 14] [10], 23] 4] all convert a multiply connected network into a tree. However, no formal arguments can be found, e.g. in [20] 9] 18] 3] which demonstrate convincingly that message passing cannot be made coherent in multiply connected networks. This leaves the question whether it is ....

F.V. Jensen, S.L. Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, (4):269--282, 1990.

....and conclude with a statement of further research. 1998 Elsevier Science B.V. All rights reserved. Keywords: Shenoy Shafer architecture; Hugin architecture; Computing marginals 1. Introduction In this paper we introduce some modifications to the Shenoy Shafer [10] and Hugin architectures [2,3]. Although both architectures are valid more generally, we will describe our modifications for the case of Bayesian networks. Our main modification to the Shenoy Shafer architecture is the introduction of a new phase called transfer of valuations. The transfer of valuations phase reduces the ....

....4. The Hugin architecture In this section, we briefly describe the computational aspects of the Hugin architecture. The Hugin architecture has its roots in the method proposed by Lauritzen and Spiegelhalter [6] for computing marginals of probability distributions. It was proposed by Jensen et al. [2,3] and is incorporated in the software product Hugin. Recently, Lauritzen and Jensen [5] have described some axioms underlying this architecture so it applies not only for probabilities but also to any domain that satisfies the axioms. The Hugin architecture also has three phases. The first phase ....

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F.V. Jensen, S.L. Lauritzen, K.G. Olesen, Bayesian updating in causal probabilistic networks by local computation, Comput. Statist. Quarterly 4 (1990) 269--282.

....by exact inference algorithms whose complexity is tied to the network s structure and is improved by conditioning. We use JTC(X; e) as a generic name for a class of variable elimination or join tree clustering algorithms that compute the exact posterior beliefs for a variable X given evidence e [15, 3, 11]. It is known that the complexity of JTC(X; e) is time and space exponential in the inducedwidth of the network s moral graph whose evidence variables E are removed. Cutset Sampling Input: A belief network (B) cutset C = fC1 ; Cmg, evidence e. Output: A set of samples c , t = 1: Tc . ....

F.V. Jensen, S.L Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computation. Computational Statistics Quarterly, 4:269{ 282, 1990.

....of symptoms in our theory, are not so succinctly described. Finally, TNT allows for roles to act as evidence for or against other roles. This would result in a multiply connected network topology, which requires more complex methods for inference, such as clustering [47] cutset conditioning [24,26,36], or stochastic simulation [23] 9. Conclusion We have presented Topological iNference of Teleology (TNT) a theory of reasoning from structure to function in the domain of thermodynamic cycles. This theory describes a knowledge representation that enables efficient evidential reasoning from ....

F.V. Jensen, S.L. Lauritzen, K.G. Olesen, Bayesian updating in causal probabilistic networks by local computations, Computational Statistics Quarterly 5 (4) (1990) 269--282.

....by preprocessing; for other networks, huge reductions in their graph s size are obtained. 1 Introduction The currently most efficient algorithm for inference with a probabilistic network is the junctiontree propagation algorithm that builds upon a triangulation of a network s moralised graph [10, 8]. The running time of this algorithm depends on the specific triangulation used. In general, it is hard to find a triangulation for which this running time is minimal. As there is a strong relationship between the running time of the algorithm and the maximum of the triangulation s clique sizes, ....

F.V. Jensen, S.L. Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, vol. 4, pp. 269--282, 1990.

....and heuristic solutions. For example, in the class of conditioning algorithms, there are local conditioning [Di92] global conditioning [SAS94] dynamic conditioning [Da95] and recursive conditioning [Da01] in the class of clustering algorithms, there are Shnoey Shafer [SS90] Hugin [JLO90], and lazy propagation [MJ98] in the class of elimination, there are bucket elimination [De96] and general elimination [Co00] and so on. Besides these general exact inference algorithms, there are some exact special case inference algorithms including quickscore for two level networks with ....

F.V.Jensen,S.LauritzenandK.Olesen. Bayesian updating in causal probabilistic networks by local computation. Computational Statistics Quarterly 4:269-282, 1990.

....type of frequency assignment problems. Finally, tree decompositions are used to solve problems in the area of expert systems. The currently most ecient algorithm for the inference calculation in probabilistic (or Bayesian) networks builds upon a tree decomposition of a network s moralized graph [20, 24]. All these studies show that (dynamic programming) algorithms based on a path tree branch decomposition of the graph can be an alternative for integer programming techniques to solve hard (combinatorial) optimization problems. The procedure to solve an optimization problem with for instance ....

....seems to be dicult to nd a good lower bound, or a good upper bound, or both. The flat instances as well as the larger dsjc and dsjr instances are left out, since they have a very high density and thus are very time consuming. 2 4 6 8 10 12 14 16 18 0 1 2 3 4 5 6 7 8 9 10 [11,15] [16,20] [21,25] 26,30] 31,35] 36,40] MMD chromatic number (a) MMD chromatic number [6,10] 11,20] 21,30] 31,40] 41,50] 51,100] 101,150] 151,200] 201,250] 251,300] 301,350] Best MMD 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 cumultative (b) best upper bound ....

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F. V. Jensen, S. L. Lauritzen, and K. G. Olesen. Bayesian updating in causal probabilistic networks by local compuations. Computational Statistics Quarterly, 4:269-282, 1990.

....under uncertainty. The input to a probabilistic model is usually a Bayesian network [10] It may also consist of a set of potentials which define a Markov network [4] In this paper, we assume that the probabilistic model is described by a Markov network. For this model, the propagation method [5, 6, 7, 12, 13] can be conveniently applied to convert the potentials into marginal distributions. There is another important reason to characterize a probabilistic model by a Markov network, as it has been shown that such a network can be represented as a generalized relational database [14, 15, 16] That is, ....

....Inference System In order to convert a probabilistic model into a relational model, first we need to be able to efficiently transform the input potentials into marginals. Since we assume that the hypergraph induced by the potentials is a hypertree, we can apply the propagation method [6, 12] to compute all their marginals. This process involves first moving backward along the hypertree construction ordering to find the marginal of the root, then moving forward from the root to the leaves for determining marginals of the other potentials. The next task is to transform a probability ....

F.V. Jensen, S.L. Lauritzen, and K.G. Olesen, "Bayesian updating in causal probabilistic networks by local computations," Computational Statistics Quarterly, vol. 4, 269--282, 1990.

....in general are coarser than their tight refinement, and thus more efficient. To deal with these issues, we need to develop abstractions that depend on the details of the of the inference procedure we use. We now address these issues within the context of cluster tree (aka clique tree) algorithms [7, 10, 14]. We start with a presentation of one variant of tree based algorithms. The other variants have slightly different details, but our algorithm can be easily adopted to deal with these. 5.1 Clique Tree Propagation Algorithm Assume that B is a fixed network. A cluster tree for B is a tree over k ....

....(s l;m j ; e) Cluster tree algorithm compute such a posterior for each cluster. This can be done efficiently by dynamic programming: for each separator we only need to compute two messages. By appropriate use of dynamic programming all of these messages can be computed in two passes over the tree [7, 14]. 5.2 Clique Tree Abstractions Suppose we are given a cluster tree and an evidence e. Can we abstract the values of cliques and separators The ideas from the previous section can be applied here in a straightforward fashion. Let S l;m be a separator. An abstraction l of Val(S l;m ) is safe ....

F. V. Jensen, S. L. Lauritzen, and K. G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, 5(4):269--282, 1990.

....actions influence the trajectory of the system being monitored. Finally, we motivated the notion of an alert s urgency, and we captured this notion within our framework: We provided an action based definition of urgency, Methods for exact inference include conditioning [10, 37, 39] clustering [21, 22, 27, 48], and symbolic probabilistic inference [29, 46] 15 and we developed, from our temporally extended decision model, a quantitative measure of this urgency. We propose that our alarm framework can guide the design of effective alarm systems in a variety of domains, and that such systems can be ....

F. V. Jensen, S. L. Lauritzen, and K. G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, 4:269--282, 1990.

....databases. Both models use nonembedded dependencies in practice, i.e. the Markov network and acyclic join dependency representations are both defined over the classes of nonembedded dependencies. The same conclusions have been reached regarding query processing in acyclic hypergraphs [4] [19], 35] and as to whether a set of pairwise consistent distributions (relations) are indeed marginal distributions from the same joint probability distribution [4] 10] Even the recent attempts to generalize the standard Bayesian database model, including horizontal independencies [6] 44] ....

F. V. Jensen, S. L. Lauritzen, and K. G. Olesen, "Bayesian updating in causal probabilistic networks by local computation," Comput. Stat. Quarterly, vol. 4, pp. 269--282, 1990.

....property of com14 piling a theory into a backtrack free (i.e. greedy) theory, and their complexity is dependent on the induced width graph parameter. The algorithms are variations on known algorithms, and, for the most part, are not new in the sense that the basic ideas have existed for some time [8, 34, 31, 50, 28, 39, 32, 3, 45, 46, 48, 47]. Definition 2 (graph concepts) A directed graph is a pair, G = fV; Eg, where V = fX 1 ; Xng is a set of elements and E = f(X i ; X j )jX i ; X j 2 V; i 6= jg is the set of edges. If (X i ; X j ) 2 E, we say that X i points to X j . For each variable X i , the set of parent nodes of X i , ....

F.V. Jensen, S.L Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computation. Computational Statistics Quarterly, 4:269--282, 1990.

.... by now is beginning to illustrate its worth in complex domains: practical applications are being developed for example for medical diagnosis and prognosis [Andreassen et al. 1987, Heckerman et al. 1992] for probabilistic information retrieval [Bruza van der Gaag, 1994] and in computer vision [Jensen et al. 1990b] This paper gives a tutorial introduction to the belief network framework and highlights some issues of ongoing research in applying the framework for real life problem solving. In Section 2 we briefly sketch the historical background of applying probability theory in knowledgebased systems. ....

....part is a decomposable, or chordal, graph; the quantitative part is a set of marginal distributions on the vertex sets of the cliques of this graph. The computational architecture for the algorithm derives from the qualitative part of this decomposable belief network and is termed a junction tree [Jensen at al. 1990a] In the junction tree architecture, the cliques of the decomposable graph are the autonomous objects and the clique intersections give rise to the communication channels. From a decomposable belief network, a probability of interest is computed by further marginalisation of an appropriate ....

F.V. Jensen, S.L. Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, vol. 4, 1990, pp. 269 - 282.

....optimally just by pre processing; for other networks, huge reductions in size are obtained. 1 Introduction The currently most e#cient algorithm for inference with a probabilistic network is the junctiontree propagation algorithm that builds upon a triangulation of a network s moralised graph [1, 2]. The running time of this algorithm depends on the triangulation used. In general, it is hard to find a triangulation for which this running time is minimal. As there is a strong relationship between the running time of the algorithm and the maximum of the triangulation s clique sizes, for ....

F.V. Jensen, S.L. Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, vol. 4, pp. 269--282, 1990.

.... 1996] or bucket elimination for belief assessment, BEBA [Dechter, 1996] that is based on the ideas of SPI [Shachter et al. 1990] This is a query oriented algorithm that exploits network structure for efficient inference, similarly to clique tree propagation [Lauritzen and Spiegelhalter, 1988; Jensen et al. 1990] One difference is the factors represent conditional probabilities rather than the marginal probabilities the cliques represent. Suppose we want to determine the probability of variable given evidence which is the conjunction of assignments to some variables , namely ....

F. V. Jensen, S. L. Lauritzen, and K. G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quaterly, 4:269--282, 1990.

....complexity of its running time. Furthermore the approximation algorithms for the weighted vertex feedback set have applications in areas of computer science other than AI. The leading inference algorithm for Bayesian networks is the clique tree algorithm [LS88] which has been further developed in [JOA90, JLO90]. In fact, SAS94] have recently shown that the weight of the largest clique is bounded by the weight of the loop cutset union the the largest parent set of a vertex in a Bayesian network implying that the clique tree algorithm is superior to the conditioning algorithm. We should note that despite ....

Jensen F. V., Lauritzen S. L. and Olesen K. G., Bayesian updating in causal probabilistic networks by local computations, Computational Statistics Quarterly, 4 (1990), 269--282.

....The tree may be directed or undirected, the messages may be passed in parallel or sequentially 3 , and the computation of the messages may or may not involve a division operation. For instance, Pearl s algorithm [Pea88] was formulated for directed trees without division; the Hugin JLO algorithm [JLO90] was formulated for undirected trees with division; and belief propagation [YFW01] was formulated for undirected networks without division. All of these algorithms are essentially equivalent. The advantage of the message passing algorithms arises when one is interested in computing all marginals ....

F. Jensen, S. L. Lauritzen, and K. G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, 4:269-282, 1990.

.... is widely used in physics, in particular, in statistical mechanics [3, 4] and has found a number of applications in other areas as well [5, 6, 7, 8] We present MFT in a generic way in the context of graphical models, which are a general framework for dealing with uncertainty in dependency models [1, 9, 10, 11]. The use of MFT in the context of graphical models was pioneered by Jordan, Saul and Jaakola [12, 13] In our paper we develop this approach in two new directions. First, in contrast to previous work we develop a systematic approach to MFT without reference to a particular model but instead work ....

....graphs since, as discussed previously, only neighboring nodes have to communicate. This locality is the appealing point of MFT. There is no necessity to compile the original graph to a tree like cover model as it is done by the junction tree algorithm by means of moralization and triangulation [9, 10]. Loops in the original graph may lead to an exponential complexity for exact inference methods (as, e.g. in our illustration in section 5.2) however, are of minor relevance for MFT. In particular in the case of Bayesian networks, mean eld inference exhibits further simplications. An additional ....

F. V. Jensen, S. L. Lauritzen, and K. G. Olsen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quaterly, 4:296--282, 1990.

....optimally just by pre processing; for other networks, huge reductions in size are obtained. 1 Introduction The currently most efficient algorithm for inference with a probabilistic network is the junction tree propagation algorithm that builds upon a triangulation of a network s moralised graph [1, 2]. The running time of this algorithm depends on the triangulation used. In general, it is hard to find a triangulation for which this running time is minimal. As there is a strong relationship between the running time of the algorithm and the maximum of the triangulation s clique sizes, for ....

F.V. Jensen, S.L. Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, vol. 4, pp. 269-- 282, 1990.

....7 we discuss the extend to which our results can be improved. 2 The Junction Tree Algorithm The junction tree algorithm is currently the most practical inference method for Bayesian networks. In this section we provide the relevant highlights of the junction tree algorithm. For details, consult [LS88, JLO90]. Definition A directed acyclic graph (DAG) is a graph with no directed cycles. In a DAG, pa(v) denotes the set of parents of a vertex v. A Bayesian network is a DAG such that with each vertex v we associate a finite set D(v) called the state space of v and a probability distribution P (vjpa(v) ....

Jensen F. V., Lauritzen S.L., and Olesen K.G., Bayesian updating in causal probabilistic networks by local computations, Computational Statistics Quarterly 4 (1990), 269--282.

....problems, and a common set of solutions. Implementations of Bayesian networks have been placed into three classes [Pearl, 1988; Henrion, 1990] 1. Exact methods that exploit the structure of the network to allow efficient propagation of evidence [ Pearl, 1988; Lauritzen and Spiegelhalter, 1988; Jensen et al. 1990] 2. Stochastic simulation methods that give estimates of probabilities by generating samples of instantiations of the network (e.g. Henrion, 1988; Pearl, 1987] 3. Search based approximation techniques that search through a space of possible values to estimate probabilities (e.g. ....

....Bayesian network. In this Bayesian network the random variable s q 2 is a binary random variable that has two values s P meaning that 4 There is actually an efficient algorithm for such an example using a clique hypertree representation [Lauritzen and Spiegelhalter, 1988; Jensen et al. 1990] This exploits the local nature of the propagation, which we do not exploit. These would not work so well when the structure cannot be exploited as well as for the cascaded adders, for example, if we add to the circuit another circuit to find the parity of the resulting bits. We chose this ....

F. V. Jensen, S. L. Lauritzen, andK.G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quaterly, 4:269--282, 1990.

....more sophisticated method (multi link lookahead) is proposed in [15] and is improved in [5] for learning decomposable Markov networks (DMNs) from data. DMNs are less expressive than Bayesian networks (BNs) However, DMNs are the runtime representation of several algorithms for inference with BNs [8, 6, 10], and can be the intermediate results for learning BNs. For example, learning PI models needs multi link lookahead and the search space for DAGs is much larger than that of chordal graphs. Learning DMNs first can then restrict the search for DAGs to a much smaller space, improving the efficiency. ....

F.V. Jensen, S.L. Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, (4):269--282, 1990.

....in computer systems [1] circuit diagnosis [12, 41] fraud detection [10] and plan recognition [36] Implementations of Bayesian networks have been placed into three classes [30, 17] 1. Exact methods that exploit the structure of the network to allow efficient propagation of evidence (e.g. [30, 25, 23]) 2. Stochastic simulation methods that give estimates of probabilities by generating samples of instantiations of the network (e.g. 16, 29, 21, 11, 22] 3. Search based approximation techniques that search through a space of possible values to estimate probabilities (e.g. 18, 4] The ....

....and Pearl [12] The circuit is a sequence of one bit adders, cascaded to form a multiple bit adder. We chose this example as it is simple to extend to large systems and also because it was used in [7] Note that there is an efficient algorithm for such an example using clique tree propagation [25, 23] that exploits the structure of the network to allow local propagation of conditioning information. A slight variant of the example would make clique tree propagation not work nearly as well. For example, if we add another circuit to the output of the adders, the algorithm in this paper would work ....

F. V. Jensen, S. L. Lauritzen, and K. G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quaterly, 4:269--282, 1990.

....type of frequency assignment problems. Finally, tree decompositions are used to solve problems in the area of expert systems. The currently most e#cient algorithm for the inference calculation in probabilistic (or Bayesian) networks builds upon a tree decomposition of a network s moralized graph [20, 24]. All these studies show that (dynamic programming) algorithms based on a path tree branch decomposition of the graph can be an alternative for integer programming techniques to solve hard (combinatorial) optimization problems. The procedure to solve an optimization problem with for instance ....

....seems to be di#cult to find a good lower bound, or a good upper bound, or both. 1 The flat instances as well as the larger dsjc and dsjr instances are left out, since they have a very high density and thus are very time consuming. 18 0 2 4 6 8 10 12 14 16 18 0 1 2 3 4 5 6 7 8 9 10 [11,15] [16,20] [21,25] 26,30] 31,35] 36,40] MMD chromatic number frequency (a) MMD chromatic number 0 1 2 3 4 5 [6,10] 11,20] 21,30] 31,40] 41,50] 51,100] 101,150] 151,200] 201,250] 251,300] 301,350] Best MMD .0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 cumultative (b) ....

[Article contains additional citation context not shown here]

F. V. Jensen, S. L. Lauritzen, and K. G. Olesen. Bayesian updating in causal probabilistic networks by local compuations. Computational Statistics Quarterly, 4:269--282, 1990.

.... work Probabilistic reasoning using Belief networks, computing the probability of one or more events given some evidence, is known to be NP hard [Cooper1990] However most commonly used exact algorithms for probabilistic inference such as join tree clustering [Lauritzen and Spiegelhalter1988, Jensen et al..1990] or variable elimination [Dechter1999, Zhang and Poole1994] can exploit the networks structure. These algorithms are time and space exponential in a graph parameter capturing the density of the network called inducedwidth, and therefore, when the induced width is small inference is efficient. ....

....satisfies the properties of tree decomposition. Therefore our derivation using cluster trees is immediately applicable to join trees. 3.1 Tree clustering The most common algorithm for belief updating is Join Tree Clustering. There are a few variants for processing join trees for belief updating [Jensen et al..1990, Shafer and Shenoy1990] The variant which we present here is applicable to treedecompositions in general and is geared toward space savings. We will call it tree clustering (TC) It is a message passing algorithm (either two phase message passing, or in asynchronous mode) Algorithm TC for ....

F.V. Jensen, S.L Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computation. Computational Statistics Quarterly, 4:269--282, 1990.

....probabilistic reasoning. The most popular variants are join tree clustering algorithms, also called junction trees. The schemes vary somewhat in their graph definition as well as in the way tree decompositions are processed [Maier, 1983, Dechter and Pearl, 1989, Lauritzen and Spiegelhalter, 1988, Jensen et al. 1990, Georg Gottlob and Scarello, 1999, Shenoy, 1996, Schmidt and Shenoy, 1998] They all involve a decomposition of a hypergraph into a hypertree. To allow a coherent discussion and extension of these methods we find it necessary to introduce a unifying perspective. We present a unifying ....

F.V. Jensen, S.L Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computation. Computational Statistics Quarterly, 4:269--282, 1990.

.... case, all known exact algorithms for multiply connected network rely on the construction of an equivalent singly connected networks, the junction tree, by clustering the original variables, according to the cliques of the corresponding triangulated moral graph [34, 29, 37] with refinements in [26]. A similar algorithm for the estimation of the most probable configuration of the variables X i is given in [14] Schachter et al. 38] show that all the known exact inference algorithms are equivalent in some sense to the algorithms in [26] and [14] A number of well known algorithms in very ....

.... moral graph [34, 29, 37] with refinements in [26] A similar algorithm for the estimation of the most probable configuration of the variables X i is given in [14] Schachter et al. 38] show that all the known exact inference algorithms are equivalent in some sense to the algorithms in [26] and [14] A number of well known algorithms in very different fields, originally derived using specialized techniques have been shown to be special cases of belief propagation and typically of Pearl s algorithm. Such reductions usually have simplified and illuminated the original derivation. ....

F. V. Jensen, S. L. Lauritzen, and K. G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Comput. Statist. Quart., 4:269--282, 1990.

....by many di erent communities many years ago, e.g. genetics (linkage analysis) speech recognition (HMMs) tracking (Kalman tering) data compression (density estimation) channel coding (turbocodes) etc. The general framework was developed by Pearl [Pea88] and various European researchers [JLO90, CDLS99] who used it to make probabilistic expert systems. Many of these systems were used for medical diagnosis. 4 For example, consider a missile tracking an airplane, where the goal is to minimize the squared distance between itself and the target. 16 The same kind of diagnosis technology ....

F. Jensen, S. L. Lauritzen, and K. G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, 4:269-282, 1990.

....agent through communication. Then H is connected. 4 On hypertree organization The difficulty of coherent inference in multiply connected (with loops) graphical models of probabilistic knowledge is well known and many inference algorithms have been proposed. Those based on message passing, e.g. [13, 9, 5, 15], all convert a multiply connected network into a tree. However, no formal arguments can be found, e.g. in [13, 4, 11, 1] which demonstrate convincingly that message passing cannot be made coherent in multiply connected networks. This leaves the question whether it is impossible to construct ....

F.V. Jensen, S.L. Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, (4):269--282, 1990.

....employed for uncertainty management in artificial intelligence. For instance, many algorithms exist for learning a Bayesian network [3, 12, 13, 15, 21] or a Markovnetwork [27,28] Moreover, efficient inference techniques exist for query processing in Bayesian networks [17] and Markov networks [14,16, 20]. In this discussion, we promote the harmonization of the IR and Bayesian network communities by directly adopting the proven learning and query processing techniques already implemented in probabilistic reasoning systems. Wewould liketomake it clear that the work here is quite ....

....the conditional independencies defined in equation (6) 2.2 MarkovNetworks Even though Bayesian networks provide an economical representation of a joint probability distribution, it may still be difficult to compute marginal distributions. Thus, several efficient local computation algorithms [14,16, 20] were developed for computing marginal distributions in Markov networks [11] It should be noted that a Markov network defined by Hajek, Havranek and Jirousek [11]iscalledadecom posable Markov network byPearl [17] That is, the definition of Markov network in [17] is differentfrom the one used ....

[Article contains additional citation context not shown here]

F.V. Jensen, S.L. Lauritzen and K.G. Olesen, Bayesian updating in causal probabilistic networks by local computation. Computational Statistics Quarterly, 4, 269-282, 1990.

....clique here is a maximal set of mutually connected nodes. 8 the cost is exponential in the size of the largest clique of a triangulated 2 graph (e.g. 28] The cliques of a triangulated graph can be arranged in a tree structure (the junction tree) where computations can be carried out eciently[29, 22]. The graph in Figure 1 is triangulated and its cliques already form a tree. 3.1 Directed graphical models The second type of graph models, Bayesian networks, are based on directed graphs. In directed graphs, the edges signify asymmetric relations between the variables, loosely speaking the ....

F. Jensen, S. Lauritzen, and K. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, 4:269-282, 1990.

....makes the overall model more compact (i.e. the total number of required parameters is reduced) P (X # = faulty)isgivenastheaprioriprobability for the component to have failed, i.e. the probability unconditioned on the equipment failure. After a propagation in the Bayesian network (see [21] for a description of how this is done) the posterior probability for a component failure given that the equipment is faulty (enforced by using the constraint node) can be read o# the node representing that component in the BN, and the probability for each MCS to be the actual MCS can be found in ....

....e i 1 ) Hence P (C # = faulty e i ) can be calculated by expanding the evidence iteratively. The first step of this procedure requires the aprioridistribution over the MCSs, P (C # = faulty e 0 ) This distribution should be calculated by a full propagation in the Bayesian network, see [21]; remember that this propagation can be performed o# line (i.e. before troubleshooting starts) The evidence e i is then incorporated by using Equation 7 until we obtain P (C # = faulty e i ) This means that calculating P (R(A) e i )isofcomplexityO(R) where R is the number of MCSs in the ....

F. V. Jensen, S. L. Lauritzen, K. G. Olesen, Bayesian updating in causal probabilistic networks by local computations, Computational Statistics Quarterly 4 (1990) 269--282.

.... problems is blocking Gibbs sampling (Jensen, Kong Kjaerulff 1995) This algorithm is based on the substantial amount of research in the area of exact inference in Bayesian networks and a particular exact scheme, the junction tree propagation method (Lauritzen Spiegelhalter 1988, Jensen, Lauritzen Olesen 1990). The algorithm effectively makes it possible to sample blocks of 4 variables simultaneously, with blocks usually containing more than 90 of the variables. Due to the joint updating of the majority of variables, this sampling scheme can have very fast mixing and avoid problems of reducibility ....

Jensen, F. V., Lauritzen, S. L. & Olesen, K. G. (1990). Bayesian updating in causal probabilistic networks by local computations, Computational Statistics Quarterly 4: 269--282.

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F. V. Jensen, S. L. Lauritzen, and K. G. Olesen. 1990. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quaterly, 4:269--282.

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F. V. Jensen, S. L. Lauritzen, K. G. Olesen, Bayesian updating in causal probabilistic networks by local computations, Computational Statistics Quarterly 4 (1990) 269--282.

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F. V. Jensen, S. L. Lauritzen, and K. G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quaterly, 4:269--282, 1990.

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Jensen, F.; Lauritzen, S.; and Olesen, K. 1990. Bayesian updating in causal probabilistic networks by local computation. Computational Statistics Quarterly 4:269--282.

No context found.

F.V. Jensen, S.L. Lauritzen and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, 4, 269--282, 1990.

No context found.

F.V. Jensen, S.L. Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computation. Computational Statistics Quarterly, 4:269--282, 1990.

No context found.

F.V. Jensen, S.L Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computation. Computational Statistics Quarterly, 4:269--282, 1990.

No context found.

F. Jensen, S. L. Lauritzen, and K. G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, 4:269--282, 1990.

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F. V. Jensen, S. Lauritzen and K. Olesen. Bayesian Updating in Causal Probabilistic Networks by Local Computation. Computational Statistics Quarterly 4:269-282, 1990.

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Finn V. Jensen, Ste#en L. Lauritzen, and Kristian G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quaterly, 4:269--282, 1990.

No context found.

F.V. Jensen, S.L. Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computations. Computational Statistics Quarterly, 4:269-282, 1990.

No context found.

F.V. Jensen, S.L Lauritzen, and K.G. Olesen. Bayesian updating in causal probabilistic networks by local computation. Computational Statistics Quarterly, 4:269--282, 1990.

No context found.

Jensen, F.V., Lauritzen, S.L. and K.G. Olesen. Bayesian Updating in Causal Probabilistic Networks by Local Computations. Computational Statistics Quarterly, Vol. 4, 1990, pp. 269-282.

No context found.

F. V. Jensen, S. L. Lauritzen, and K. G. Olesen. Bayesian Updating in Causal Probabilistic Networks by Local Computations. Computational Statistics Quarterly, 4:pp.269-282, 1990.

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