R. D. Shachter. An ordered examination of influence diagrams. Networks, 20:535--563, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....namely the operation of merging value nodes. This approach does not separate BN inferences, and it requires arc reversals (see the fifth row of Table 7.1) 7.2.3. Shachter [27] Let d be an SD candidate node in an influence diagram N , and let v be the only value node in N . Shachter [27] and [28] has noticed that optimal decision functions ffi d of d can be obtained through d E[vj d = fi; d = ff] 1) for each fi d . Further in this direction, Shachter and Peot [29] first half) proposes a way to scale the value function v and change v into a observed random node, denoted ....

....direction, Shachter and Peot [29] first half) proposes a way to scale the value function v and change v into a observed random node, denoted by u (see also Cooper [3] Formula (1) is transformed into d P ( d = fi; d = ffju = 1) 2) Thus, this approach separates BN inferences. Even though [28] points out the possibility that the conditional expectation E[vj d = fi; d = ff] can be computed in one portion of the original network, the algorithm proposed by the paper does not work in a divide and conquer fashion. After the optimal decision function for d is computed, the decision node d is ....

R. Shachter (1990), An ordered examination of influence diagrams, Networks, Vol. 20, 5, pp. 535-564.

....an improvement of the algorithm suitable for belief updating only. The beliefupdating task has special semantics which allows restricting the computation to relevant portions of the belief network. These restrictions are already available in the literature in the context of the existing algorithms [27, 44]. Since summation over all values of a probability function is 1, the recorded functions of some buckets will degenerate to the constant 1. If we can predict these cases in advance, we can avoid needless computation by skipping some buckets. If we use a topological ordering of the belief ....

R. D. Shachter. An ordered examination of influence diagrams. Networks, 20:535--563, 1990.

....The amount of space needed to specify a decision rule for the current decision increases exponentially with the number of observed variables. Thus, there has been considerable work on identifying irrelevant parents of decision nodes in single agent influence diagrams [Howard and Matheson, 1984; Shachter, 1990; 1998] However, the multi agent case raises subtleties that are absent in the single agent case. This is another problem we plan to address in future work. Acknowledgements This work was supported by Air Force contract F30602 00 2 0598 under DARPA s TASK program and by ONR MURI ....

R. D. Shachter. An ordered examination of influence diagrams. Networks, 20:535--563, 1990.

.... buckets and then compute a function Q1 p = max Xp Pi 2Q1 and a function by: 1 p = maxXp Q i 2Q1 i P k2lp k maxXp Q i 2Q1 i (10) If P k2lp k is still too high dimensional and cannot fit into one mini bucket, we can further partition Q 1 into mixed mini buckets fQ 11 ; Q 1 j ; Q 1t g and compute separate functions for each subset Q 1 j Q 1 . By exchanging summation and multiplication we get: 1 p P j maxXp [ P k2Q1j k Q i 2Q1 i ] maxXp Q i 2Q1 i (11) Therefore, it will allow computing for each Q 1 j Q 1 , the function 1 ....

.... Q 1 fits into one mini bucket of an (i,m) partitioning compute a component using the full bucket decision rule: Q1 p = maxXp Pi 2Q1 P 2Q1 else, move all functions not defined on X p to lower buckets, then apply an arbitrary (i,m) partitioning for the rest of the functions called Q 11 ; Q 1n . For any mixed Q 1 j Q 1 , compute: 1 j p = maxXp Q i 2Q1 i P k2Q1 j k maxXp Q i 2Q1 i (13) Add and to the bucket of the largest index in their argument list. 3. Forward: Return an upper bound on the maximum expected utility computed in the first bucket. ....

R. D. Shachter. An ordered examination of influence diagrams. Networks, 20:535--563, 1990.

....topological properties of these algorithms, and in particular ties its time and space complexity to the induced width of an underlying graph. In principle, the algorithm is similar to the variable elimination algorithm proposed by Shachter and others (Shachter 1986; 1988; 1990; Tatman Shachter 1990; Shachter Peot 1992; Shenoy 1992; Zhang 1998; F. Jensen Dittmer 1994) and in particular, it is analogous to the join tree clustering algorithm for evaluating influence diagrams (F. Jensen Dittmer 1994) However, the new exposition using the bucket datastructure unifies the algorithm with a ....

....: ffi m ) consisting of one rule for each decision variable. To evaluate an influence diagram is to find an optimal policy that maximizes the expected utility (meu) and to compute the 1 The original definition of ID had only one reward node. We allow multiple rewards as discussed in (Tatman Shachter 1990) optimal expected utility. Assume that x is an assignment over both chance variables and decision variables x = x 1 ; xn ; d 1 ; dm ) The meu task is to compute E = max Delta= ffi 1 ; ffi m ) X x1 ; x n Pi x i P (x i ; ejx Delta pa i )u(x Delta ) 1) where x ....

Shachter, R. D. 1990. An ordered examination of influence diagrams. Networks 20:535--563.

....of the algorithm suitable 25 for belief updating only. The belief updating task has special semantics which allows restricting the computation to relevant portions of the belief network. TheseSuch restrictions are already available in the literature in the context ofthe existing algorithms [28, 44]. Since summation over all values of a probability function is 1, the recorded functions of some buckets will degenerate to the constant 1. If we can predict these cases in advance, we can avoid needless computation by skipping some buckets. If we use a topological ordering of the belief ....

R. D. Shachter. An ordered examination of influence diagrams. Networks, 20:535--563, 1990.

....the computation of the conditionals and for computation of an optimal strategy. In the last decade, there have been many improvements to the influence diagram representation and solution technique. Ezawa [1986] has examined efficient deletion sequences for solving influence diagrams. Tatman and Shachter [1990] describe an extension of the influence diagram technique for a decomposition of the joint utility function. Shenoy [1992] has proposed a generalization of influence diagrams, called valuation networks, which allow for representation of any probability model (whereas influence diagrams allow for ....

....and Peot [1992] Zhang et al. 1994] Jensen et al. 1994] Cowell [1994] Goutis [1995] and Qi and Poole [1995] have proposed modifications to the symmetric influence diagram technique to make the representation and solution more efficient. Call and Miller [1990] Smith et al. 1993] Fung and Shachter [1990], Covaliu and Oliver [1995] and Qi et al. 1994] have proposed modifications to the influence diagram technique for representing and solving asymmetric decision problems. Finally, Shenoy [1993] has proposed a generalization of the symmetric valuation network technique for asymmetric decision ....

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Shachter, R. D. (1990), "An ordered examination of influence diagrams," Networks, 20, 535--563.

....did not really use the direction of the links in the network. By standard we mean the Lauritzen Spiegelhalter (Lauritzen Spiegelhalter 1988) the Shafer Shenoy (Shafer Shenoy 1990) and the Hugin (Jensen, Lauritzen Olesen 1990) algorithms and the various variations over these algorithms ( (Shachter 1990) and (Jensen 1995) These algorithms build a secondary structure (a junction tree or a join tree) by triangulating the (moralized) network. This structure can be used for propagation for all information scenaria. Therefore, the algorithms do not exploit independences induced by the evidence. ....

Shachter, R. (1990), `An ordered examination of influence diagrams', Networks 20(5), 535--563.

....for performing probabilistic inference in Bayesian networks. The more commonly known inference algorithms are the LauritzenSpiegelhalter (Lauritzen and Spiegelhalter, 1988) the Shafer Shenoy (Shafer and Shenoy, 1990) and the Hugin algorithms and variations over these methods such as for example (Shachter, 1990; Jensen, 1995) These methods were developed to efficiently calculate the posterior probability distributions for all variables in the Bayesian network. If the reasoning, on the other hand, is focused on a small subset of the variables other more efficient algorithms exist. The SPI algorithm (Li ....

Shachter, R. D. (1990). An ordered examination of influence diagrams. Networks, 20(5):535--563.

....did not really use the direction of the links in the network. By standard we mean the LauritzenSpiegelhalter [Lauritzen and Spiegelhalter, 1988] the Shafer Shenoy [Shafer and Shenoy, 1990] and the Hugin [Jensen et al. 1990] algorithms and the various variations over these algorithms ( [Shachter, 1990] and [Jensen, 1995] These algorithms build a secondary structure (a junction tree or a join tree) by triangulating the (moralized) network. This structure can be used for propagation for all information scenaria. Therefore, the algorithms do not exploit independences induced by the evidence. ....

....A is instantiated and no evidence has been entered to DAG 4 , then it is only necessary to sent messages down to DAG 4 . We may relax the requirement to the updating algorithm such that we are only interested in updated probabilities for a very small set of variables. In that case the SPI method [Shachter et al. 1990] and the bucket sort algorithm [Dechter, 1996] can utilize specific independences, as they consist of a collect operation only, where the variables are successively eliminated by multiplying the functions involving A (say) and marginalizing A out of this product. These methods, however, are not ....

Shachter, R. D. (1990). An ordered examination of influence diagrams. Networks, 20(5):535--563.

....on the vertical links. The E step of the learning algorithm for HMM s involves calculating the posterior probabilities of the hidden (unshaded) variables given the observed (shaded) variables. case of generic algorithms for calculating posterior probabilities on directed graphs (see, e.g. Shachter, 1990). 2.2 Hidden Markov models In the graphical model formalism a hidden Markov model (HMM; Rabiner, 1989) is represented as a chain structure as shown in Figure 2.1. Each state node is a multinomial random variable z t . The links between the state nodes are parameterized by the transition matrix ....

Shachter, R. (1990). An ordered examination of influence diagrams. Networks, 20, 535--563.

....relevance, that will provide an even stronger criterion for focusing reasoning. A node n is computationally relevant to target nodes T given evidence E if we cannot compute the posterior marginal distribution of T unless we know the conditional probability distribution of n [ Shachter, 1988, Shachter, 1990, Suermondt, 1992 ] Computational Relevance: Structure The class of computationally relevant nodes excludes one type of nodes that are structurally relevant, known as barren nodes. Barren nodes are uninstantiated child less nodes in the graph. They depend on the evidence, but do not ....

....the belief in cat, dog, allergy, and sneezing, and paw marks is a barren node. Barking is another possible example of a barren node. Unobserved, it is not computationally relevant to any other nodes in the network. Barren nodes can be removed by a variant of an efficient algorithm proposed by Shachter [ 1988, 1990 ] Judea Pearl suggested in a private communication applying the algorithm of Geiger et al. 1990 ] to reduce barren nodes. By attaching to each node n in the graph a dummy parent node representing the conditional probability of n given its direct ancestors and performing the algorithm, one can ....

Shachter, Ross D. 1990. An ordered examination of influence diagrams. Networks 20(5):535--563.

....value is known. ffl Both network predictions (m i ) and the actual responses (o i ) are represented. The DAG also models the error as a Gaussian. SS #55 GRAPHICAL MODELS Other problems can be represented in graphical models using: ffl known and unknown variables, ffl deterministic nodes [88], ffl standard probabilities functions at nodes (Gaussian, multinomial, Dirichlet, logistic, etc. ffl mixed directed and undirected arcs [32] ffl optional arcs (indicating alternative models) 14] ffl plates representing samples [14] Graphical models offer a unified framework for ....

R.D. Shachter. An ordered examination of influence diagrams. Networks, 20:535--563, 1990.

....parents are all known, by virtue of their determinism, they are also known. They could also have been unshaded, since the fact that they are known can be deduced from their parents. The analysis of deterministic nodes in Bayesian networks and, more generally, in influence diagrams is considered by (Shachter, 1990). Deterministic nodes cannot have any neighbors, meaning they do not occur in undirected subgraphs. To analyze these nodes, we need to extend the usual definition of a parent and a child for a graph. Definition 9 The non deterministic children of a node x, denoted ndchildren(x) are the set of ....

Shachter, R. (1990). An ordered examination of influence diagrams. Networks, 20, 535--563.

....= f(x 1 ; xn ) 0 otherwise . for some function f not specified. A DAG constructed entirely of double ovals is equivalent to a data flow graph where the inputs are shaded. The analysis of deterministic nodes in Bayesian networks and more generally, in influence diagrams is considered by [Sha90] The conjunctive form of each rule in Figure 7(a) is not expressed in the graph, and presumably would be given in the formulas accompanying the graph, however the basic functional structure of the rule set exists. In Figure 7(b) a node has been labeled with its functional type. The functional ....

....because they are the building blocks on which methods given later in the paper will be built. We look at arc reversal, arc removal, and Gibbs sampling. Many more sophisticated variations and combinations of these algorithms exist in the literature, including the handling of deterministic nodes [Sha90] and those handling chain graphs and undirected graphs [Fry91] 5.1.1 Arc reversal The arc reversal operator interchanges the order of two nodes connected by a directed arc [Sha86] This operator corresponds to Bayes theorem and is used, for instance, to automate the derivation of Equation (12) ....

R.D. Shachter. An ordered examination of influence diagrams. Networks, 20:535--563, 1990.

....upward and downward in the tree, in which the posterior probability at a given nonterminal is the sum of posterior probabilities associated with its children. The recursion can be viewed as a special case of generic algorithms for calculating posterior probabilities on directed graphs (see, e.g. Shachter, 1990). 1 Throughout the paper we restrict ourselves to three levels for simplicity of presentation. z 1 x y z 2 z 3 Figure 1: The hierarchical mixture of experts as a graphical model. The E step of the learning algorithm for HME s involves calculating the posterior probabilities of the hidden ....

Shachter, R. (1990). An ordered examination of influence diagrams. Networks, 20, 535--563.

....complexity is dependent on the same 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 (Cannings et al. 1978; Pearl, 1988; Lauritzen and Spiegelhalter, 1988; Tatman and Shachter, 1990; Jensen et al. 1990; R.D. Shachter and Favro, 1990; Bacchus and van Run, 1995; Shachter, 1986; Shachter, 1988; Shimony and Charniack, 1991; Shenoy, 1992) What we are presenting here is a syntactic and uniform exposition emphasizing these algorithms form BUCKET ELIMINATION 3 Algorithm ....

....and bound may be viewed as conditioning algorithms. The complexity of conditioning algorithms is exponential in the conditioning set, however, their space complexity is only linear. Our resulting hybrid of conditioning with elimination which trade off time for space (see also (Dechter, 1996b; R. D. Shachter and Solovitz, 1991) are applicable to all algorithms expressed within this framework. The work we present here also fits into the framework developed by Arnborg and Proskourowski (Arnborg, 1985; Arnborg and Proskourowski, 1989) They present table based reductions for various NP hard graph ....

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R. D. Shachter. An ordered examination of influence diagrams. Networks, 20:535--563, 1990.

....reason for abandoning the two phase approach might be that people believe that direct evaluation is more efficient. However, all those algorithms that evaluate influence diagrams directly suffer from a common shortcoming in handling asymmetric decision problems (Covaliu and Oliver 1992, Fung and Shachter 1990, Phillips 1990, Shachter 1986, Smith et al. 1993) Decision problems are usually asymmetric in the sense that the set of possible outcomes of a random variable may vary depending on different conditioning states, and the set of legitimate alternatives of a decision variable may vary depending on ....

....d is removed by maximization A New Method for Influence Diagram Evaluation 11 Other developments. Influence diagrams are closely related to Bayesian nets (Pearl 1988) Quite a few algorithms have been developed in the literature (Jensen et al. 1990, Lauritzen and Spiegelhalter 1988, Pearl 1988, Shachter et al. 1990, Zhang and Poole 1992a) for computing marginal probabilities and posterior probabilities in Bayesian nets. Thus, it is natural to ask whether we can make use of these Bayesian net al..gorithms for influence diagram evaluation. This problem is examined in (Cooper 1988, Ndilikilikesha 1991, Shachter ....

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Shachter, R. D. 1990. An ordered examination of influence diagrams. Networks, 20:535--563.

....than graph union [19] Shachter considered seeking agreement about the direction of information flow through a BN by imposing a pre set ordering across the BNs nodes. Arcs that violated that ordering were reversed (accompanied by all other modifications necessary to perform a legal arc reversal) [130]. Finally, the authors of this survey introduced several algorithms that merge the DAGs underlying BNs provided by different contributors into a single consensus BN [96, 97, 98] 8 Summary This paper reviewed a recent wave of exciting work surrounding a family of graphical models of Bayesian ....

R.D. Shachter. An Ordered Examination of Influence Diagrams. Networks, 20:535--563, 1990.

....in time linear in the size of the graph (Geiger et al. 1989) Identification of requisite information can also be determined in time linear in the size of the graph. These algorithms have been generalized to deal with deterministic nodes in belief networks and influence diagrams (Geiger et al. 1990; Shachter 1990). This paper introduces the Bayes Ball algorithm, a simpler and more e#cient algorithm to identify conditional irrelevance and requisite information. For belief networks, Bayes Ball runs in time linear in the size of the active part of the graph, so it is considerably faster when most of a ....

....so it is considerably faster when most of a graphical knowledge base is irrelevant. It also corrects an error in the requisite information algorithm given in Geiger (1990) More significantly, for decision problems it runs in time linear in the size of the graph; up until now the fastest algorithm (Shachter 1990) has been O( number of decisions) graph size) Finally, the decision algorithm has been extended to allow multiple separable value nodes. The identification of irrelevant nodes and requisite information is a fundamental operation in any belief network or influence diagram processor. It allows ....

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Shachter, R. D. "An Ordered Examination of Influence Diagrams." Networks 20 (1990): 535-563.

....Although we will not explore methods to select the conditioning set, it is critical to the success of the method and a subject worthy of papers by itself. Having observed XK , we can cut the outgoing arcs from K in the belief network, which is equivalent to separation by K in the moral graph (Shachter 1990a) For example, the multiply connected graph shown in Figure 1a becomes the singly connected graph shown in Figure 4a after X 2 has been observed. We can compute the overall desired result by considering all possible cases for XK and weighting by their probabilities, using the Law of Total ....

....variables in the cluster, X S i , are conditionally independent of some of the conditioning variables, XK1 , given the evidence ffl and the remaining conditioning variables XK2 . This independence can be recognized in linear time in the size of the network (Geiger and others 1990; Shachter 1988; Shachter 1990b) If there is an efficient way to compute P rfXK1 jXK2 ; fflg, then we can save iterations by recognizing that P rfX S i [K ; fflg = P rfXK1 jXK2 ; fflgP rfX S i ; XK2 ; fflg : Finally, we have assumed throughout that the conditioning variables XK were taken through all their values. If, ....

Shachter, R. D. "An Ordered Examination of Influence Diagrams." Networks 20 (1990b): 535-563.

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R. D. Shachter. An ordered examination of influence diagrams. Networks, 20:535--563, 1990.

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R. Shachter (1990), An ordered examination of influence diagrams, Networks, Vol. 20, 5, pp. 535-564.

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Shachter, R. D. 1990. An ordered examination of influence diagrams. Networks 20:535--563. 86

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Buntine Shachter, R. (1990). An ordered examination of influence diagrams. Networks, 20, 535--563.

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