U. Kjaerulff. "Triangulation of graphs - Algorithms giving small total state space," Technical report R 90-09, University of Aalborg, Denmark, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....This in turn depends on the elimination order that we choose. For example, if we first eliminate the variable C then we get a factor over W;D;C; I, which is larger than any factor we actually had. Unfortunately, although one can use some simple greedy heuristics in choosing the elimination order [Kja90, HD96] finding the optimal elimination order, i.e. the one resulting in the smallest factors, is NP hard [ACP87] Furthermore, in some networks, even the optimal elimination order leads to factors which are exponential in the size of the Bayes net. In these cases we cannot use exact inference ....

....However, one can use some simple heuristics which tend to produce good triangulation on many real life Bayes net. One such simple heuristic is triangulating the graph using an elimination order where at each time we greedily eliminate the variable that adds the smallest number of new edges to G [Kja90] To summarize, transforming a Bayes net into a clique tree can be done in following steps, illustrated in Figure 2.4: ffl Building the moralized graph GM . ffl Triangulating the moralized graph by using either variable elimination or some different algorithm. ffl Finding the maximal cliques ....

U. Kjaerulff. Triangulation of graphs -- algorithms giving small total state space. Technical Report TR R 90-09, Department of Mathematics and Computer Science, Strandvejen, Aalborg, Denmark, 1990.

....smaller than , and if so output a triangulation in which all cliques are of size less than [1] These algorithms are super exponential in , however, and thus are applicable only to problems with small treewidths. In practice we have had success using fast heuristic triangulation methods [11] that allow us to guarantee the existence of a junction tree with a maximal clique no larger than for a given model. This is a conservative technique that may occasionally throw out models that in fact have small treewidth) A critical bottleneck in the algorithm is the parameter estimation ....

U. Kjaerulff, Triangulation of graphs---algorithms giving small total state space, Technical Report R90-09, Dept. of Math. and Comp. Sci., Aalborg Univ., Denmark, 1990.

....parents of nodes that have incoming temporal arcs. In the case of an HHMM, the interface is all the Q nodes. The cost of doing inference in each jtree depends on the sizes of the cliques. Minimizing the maximal clique size is NP hard, so we used a standard one step look ahead (greedy) algorithm [Kja90]. The resulting cliques are hard to interpret, but we can still analyze the complexity. Let NQ (c; D) be the number of Q nodes in clique c, let NF (c; D) be the number of F nodes, and let N c (D) be the number of cliques. Then the cost of inference in a jtree is proportional to Nc (D) X c=1 ....

U. Kjaerulff. Triangulation of graphs -- algorithms giving small total state space. Technical Report R-90-09, Dept. of Math. and Comp. Sci., Aalborg Univ., Denmark, 1990.

....G i will denote the subgraph of G induced by S i . F i = fv k 2 Adj(vn Gammai ) j v n Gammai v k g where is the precedence relation of the given order. Observe that since the elimination order is perfect, the subgraph of G induced by F i is complete. The material cited below is taken from [ Kjaerulff, 1990 ] A perfect elimination order can be found in O(n e) time using the maximum cardinality search algorithm. A nontriangulated graph can always be transformed into a triangulated one by adding edges. Finding a minimal triangulation, where every edge is necessary for the graph to be triangulated can ....

U. Kjaerulff. Triangulation of graphs -- algorithms giving small total state space. Research Report R-90-09, Aalborg University, Denmark, 1990.

....described in [Kush 95] There were several interesting problems we have touched during implementation phase. One of them is triangulation and we want to give attention to some triangulation algorithms together with tests of new designed one. A description of some of them can be found in [Jir 92, Kjaer 90] There are interesting models for handling uncertainty using conditional probabilities called Bayesian Nets within a frame of probabilistic models in AI. Introduction to their theory is in [Jen 93] Bayesian Nets use decomposable model which give them availability to do computation of ....

Kjaerulff U., 1990, Triangulation of Graphs - Algorithm Giving Small Total State Space, Institute of Electronic Systems, Aarlborg.

....necessarily minimal. This means that some edges that were added (fill in edges) might possibly be removed and the resulting graph remains triangulated. Kjaerulff provides an algorithm that, given a triangulation of a graph G and an ordering on its vertices, produces a minimal triangulated graph [Kj90]. We use Kjaerulff s algorithm with an ordering that is determined as follows. First in the ordering are the simplicial vertices in the order in which they are removed from G. The order of the remaining vertices is determined recursively while running Triangulate; In each level of the recursion, ....

....In 16 instances the algorithms produced equal output. The second line records the same information regarding the logarithm of the total state space. The two algorithms were run on Medianus I with state sizes randomly selected from the range 3 to 21 with an average of approximately 6 (as in [Kj90]) One hundred random runs were made. In 68 runs our algorithm has outperformed the enhanced minimum weight heuristics in both optimality criteria. Figure 3 shows that the averaged improvement of T was 0:64 and the maximum improvement was 2:37 which amounts to a reduction of storage by a factor of ....

Kjaerulff U., Triangulation of graph-algorithms giving small total state space. Technical Report R 90-09, Department of Mathematics and Computer Science, Aalborg university, Denmark, March 1990.

....smaller than # , and if so output a triangulation in which all cliques are of size less than # [1] These algorithms are super exponential in # , however, and thus are applicable only to problems with small treewidths. In practice we have had success using fast heuristic triangulation methods [9] that allow us to guarantee the existence of a junction tree with a maximal clique no larger than # for a given model. This is a conservative technique that may occasionally throw out models that in fact have small treewidth) This approach is particularly reasonable given that exact inference ....

U. Kjaerulff, Triangulation of graphs---algorithms giving small total state space, Technical Report R90-09, Dept. of Math. and Comp. Sci., Aalborg Univ., Denmark, 1990.

....but different orderings lead to different computational loads. Unfortunately, the construction of an optimal ordering is known to be a NP complete problem, so we should settle for a good, not necessarily optimal, ordering. Several heuristics are known to produce efficient orderings in practice [7, 14]. The variable algorithm can also be used to compute maximum a posteriori configurations, essentially by turning some of the summations into maximizations [3] There is To appear, Workshop on Probabilistic Reasoning in Artificial Intelligence, Atibaia, Brazil, November 20, 2000 no need to ....

U. Kjaerulff. Triangulation of graphs --- algorithms giving small total state space. Technical Report R-9009, Department of Mathematics and Computer Science, Aalborg University, Denmark, March 1990.

....steps in order. ffl Moralize the original graph G, i.e. connect together all parents who share a common child, and then drop the directionality of the arcs. This will result in an undirected graph, GM . ffl Choose an elimination ordering , e.g. according to the heuristics discussed in [Kja90]. ffl Let al..l nodes be initially unmarked. For each node in order , mark it and join all its unmarked neighbors. This will result in a triangulated graph, G T . See [BG96] for more effective ways to triangulate a graph. ffl Find the maximal cliques in GT ; call them C. ffl Build an undirected ....

U. Kjaerulff. Triangulation of graphs -- algorithms giving small total state space. Technical Report R90 -09, Dept. of Math. and Comp. Sci., Aalborg Univ., Denmark, 1990. Available at www.cs.auc.dk/ uk.

....works in two stages: in the first stage a junction tree is constructed, and in the second stage, messages are propagated through the junction tree. We will give a short review of triangulation and junction tree construction; for more detail see e.g. Jensen, 1988 ] Almond and Kong, 1991 ] Kjaerulff, 1990 ] Jensen and Jensen, 1994 ] Junction trees have traditionally been constructed by the following method: 1. Moralize the belief network by adding arcs connecting every pair of parents of any variable in the network, and dropping the directions on all arcs. 2. Triangulate the (now ....

.... trees of radically differing cost; minimiz ing the cost of a junction tree is NP complete [ Arnborg et al. 1987 ] The most effective known heuristic is the minimum weight heuristic, which eliminates the variable whose non eliminated neighbors form the clique with the smallest potential [ Kjaerulff, 1990 ] One difficulty with node elimination, or with triangulation in general, is that it is very difficult to understand. The original graphical structure of the belief network doesn t help much: it is generally very difficult to determine visually if a graph is triangulated, and seeing the cliques ....

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Uffe Kjaerulff. Triangulation of graphs---algorithms giving small total state space. Technical Report R 90-09, Department of Mathematics and Computer Science, Aalborg University, Denmark, March 1990.

....of the adaptive reasoning scheme, we need to pay an equal amount of attention to its limitations. First, this adaptive reasoning scheme inherits the limitation of RLCM. That is, it is restricted to singly connected networks. Several techniques, such as clustering, conditioning, and triangulation [12,26,27], exist to transform a multiply connected network into a singly connected one. Unfortunately, this will cause the spatial size, thus the representational complexity, of a Bayesian network to increase exponentially. Nevertheless, this provides a means to deal with an arbitrary network topology ....

U. Kjaerulff, "Triangulation of Graphs --- Algorithms Giving Small Total State Space," Internal Technical Report, R-90-09, Institute of Electronic System, University of Aalborg, Denmark, March 1990.

....would be needed to transform the networks into polytree hypergraphs which are akin to singly connected networks. Unfortunately, finding the optimal cutsets, and obtaining the optimal hypergraph (with respect to the number of states) using the triangulation process are both proven to be NP hard [20][28,29] There are some attempts to extend the query capability of a probabilistic inference to include composite hypotheses (or multiple causes) 2] and particularly in the domain of diagnosis [4,16] However, the attention is focused on the special cases such as the bipartite networks with a ....

U. Kjaerulff, "Triangulation of Graphs --- Algorithms Giving Small Total State Space," Internal Technical Report, R-90-09, Institute of Electronic System, University of Aalborg, Denmark, March 1990.

....= AC. Notice that when either the number of the non evidential variables in a hypothesis (type Gamma 1 Gamma complexity) or the number of irrelevant variables (type Gamma 2 Gamma complexity) increases, the computational load soon becomes prohibitive. Various methods were proposed elsewhere [2,7] for efficient calculations of the likelihood of simple hypotheses. However, the aforementioned complexity problems which are relevant to (local) composite hypotheses still remain unsolved. It is the objective of this research to develop an efficient reasoning scheme which addresses the complexity ....

U. Kjaerulff, "Triangulation of Graphs --- Algorithms Giving Small Total State Space," Internal Technical Report, R-90-09, Institute of Electronic System, University of Aalborg, Denmark, March 1990.

.... UNIVERSITY Optimal Decomposition of Probabilistic Networks by Simulated Annealing by Uffe Kjaerulff December 1991 To be referenced as: Kjaerulff, U. 1992) Optimal decomposition of probabilistic networks by simulated annealing, Statistics and Computing 2: 7 17. Institute for Electronic Systems Department of Mathematics and Computer Science Fredrik Bajers Vej 7E DK 9220 Aalborg Denmark Tel. 45 98 15 85 22 TELEX 69 790 aub ....

....a decomposition of the probabilistic network. As indicated above, triangulating a graph is essentially a problem of establishing an elimination ordering. Several algorithms have been suggested for establishing elimination orderings which have certain desired properties (Fujisawa Orino 1974, Kjaerulff 1990, Rose, Tarjan Lueker 1976, Tarjan Yannakakis 1984) As the task of establishing an elimination ordering may be perceived as a sorting problem, such algorithms are quite often referred to as sorting algorithms. Since most of the algorithms apply some sort of search strategy, the term search ....

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Kjaerulff, U. (1990). Triangulation of graphs --- algorithms giving small total state space, Research Report R-90-09, Department of Mathematics and Computer Science, Aalborg University, Denmark.

....It can be proven [8] that for a constant number of values r v per node, the minimal triangulations with respect to J F are also local minima for J and J . Even if most of the local minima found by lexicographic search were good enough (something that is not supported by practical experience [6]) the problem with this algorithm is that it takes into account only topological information, ignoring the values of r v . As our simulation will show, this is an important drawback. Kjaerulff introduced the minimum weight (MW) heuristic [6] a greedy minimization method for J (thus taking r v ....

....(something that is not supported by practical experience [6] the problem with this algorithm is that it takes into account only topological information, ignoring the values of r v . As our simulation will show, this is an important drawback. Kjaerulff introduced the minimum weight (MW) heuristic [6], a greedy minimization method for J (thus taking r v into account) and later a simulated annealing approach [7] that explicitly optimizes J . Becker [3] introduced recently a triangulation algorithm which is not based on node elimination. The algorithm minimizes the cliquewidth J C , ....

[Article contains additional citation context not shown here]

U. Kjaerulff. Triangulation of graphs--algorithms giving small total state space. Technical Report R 90-09, Department of Mathematics and Computer Science, Aalborg University, Denmark, 1990.

....steps in order. ffl Moralize the original graph G, i.e. connect together all parents who share a common child, and then drop the directionality of the arcs. This will result in an undirected graph, GM . ffl Choose an elimination ordering , e.g. according to the heuristics discussed in [Kja90]. ffl Let al..l nodes be initially unmarked. For each node in order , mark it and join all its unmarked neighbors. This will result in a triangulated graph, G T . ffl Find the maximal cliques in GT ; call them C. ffl Build an undirected weighted graph G J whose nodes are the cliques C and where ....

U. Kjaerulff. Triangulation of graphs -- algorithms giving small total state space. Technical Report R90 -09, Dept. of Math. and Comp. Sci., Aalborg Univ., Denmark, 1990. Available at www.cs.auc.dk/ uk.

....NP complete, good heuristics exist. The following one steplook ahead (Olmstead 1983, Kong 1986) usually gives good results: eliminate repeatedly a node not requiring fill ins and if this is not possible, eliminate a node yielding the smallest elimination domain. For alternatives, see for example Kjaerulff (1990, 1992) Back to our task. Suppose now that we have a triangulation G 0 of the domain graph G. Proposition 5 yields that it is quite easy to determine an elimination sequence yielding the present triangulation. Let the elimination sequence be x 1 ; xn . The elimination sequence provides ....

Kjaerulff, U.: 1990, Triangulation of graphs --- algorithms giving small total state space, Research Report R 90-09, Department of Mathematics and Computer Science, Aalborg University.

....able to display the obvious triangulation, or we will be able to lower bound the size of cliques in a triangulated graph by considering the cliques in the moral graph. Thus we will not need to consider specific algorithms for triangulation (for discussion of triangulation algorithms, see, e.g. Kjaerulff, 1990). 3. Examples In this section we present examples of graphical models in which exact inference is generally infeasible. Our first example involves a diagnostic system in which a fixed graphical model is used to answer queries. The remaining examples involve estimation problems in which a ....

Kjaerulff, U. (1990). Triangulation of graphs---algorithms giving small total state space. Research Report R-90-09, Department of Mathematics and Computer Science, Aalborg University, Denmark.

....sequence that generates the strong triangulation. However, the triangulation problem is simpler than for ordinary probability propagation, since the set of admissible elimination sequences is smaller; at this stage, it appears that simple adaptations of the heuristic algorithms described by Kjaerulff (1990) work very well. Moreover, even given a triangulation, there might exist several strong junction trees for the collection of cliques. Besides the use of the strong junction tree for computation of expected utilities and optimal decision policies, it should be possible to exploit the junction tree ....

Kjaerulff, U. (1990). Triangulation of graphs --- algorithms giving small total state space. Research Report R-90-09, Department of Mathematics and Computer Science, Aalborg University, Denmark.

....NP hard (Wen, 1991) We carry out experiments with distinct crossover and mutation operators and with different population sizes, mutation rates and selection biasses. The results are analyzed statistically. They turn out to improve the results obtained with most other known triangulation methods (Kjaerulff, 1990) and are comparable to the ones obtained with simulated annealing (Kjaerulff, 1990; Kjaerulff, 1992) Keywords: Bayesian networks, genetic algorithms, optimal decomposition, graph triangulation, moral graph, NP hard problems, statistical analysis. 1 Introduction The Bayesian networks constitute ....

....operators and with different population sizes, mutation rates and selection biasses. The results are analyzed statistically. They turn out to improve the results obtained with most other known triangulation methods (Kjaerulff, 1990) and are comparable to the ones obtained with simulated annealing (Kjaerulff, 1990; Kjaerulff, 1992) Keywords: Bayesian networks, genetic algorithms, optimal decomposition, graph triangulation, moral graph, NP hard problems, statistical analysis. 1 Introduction The Bayesian networks constitute a reasoning method based on probability theory. A Bayesian network consists of a ....

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Kjaerulff, U. (1990) Triangulation of graphs -- Algorithms giving small total state space. Technical Report R 90-09, University of Aalborg, Denmark.

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U. Kjaerulff. "Triangulation of graphs - Algorithms giving small total state space," Technical report R 90-09, University of Aalborg, Denmark, 1990.

No context found.

U. Kjaerulff. Triangulation of graphs -- algorithms giving small total state space. Technical Report R-90-09, Dept. of Math. and Comp. Sci., Aalborg Univ., Denmark, 1990.

No context found.

U. Kjaerulff. Triangulation of graphs --- algorithms giving small total state space. Technical Report R-9009, Department of Mathematics and Computer Science, Aalborg University, Denmark, March 1990.

No context found.

U. Kjaerulff. Triangulation of graphs - algorithms giving small total space, 1990.

No context found.

U. Kjaerulff. Triangulation of graphs -- algorithms giving small total state space. Technical Report R-90-09, Dept. of Math. and Comp. Sci., Aalborg Univ., Denmark, 1990.

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