D. M. Chickering. A transformational characterization of equivalent Bayesian network structures. In Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence, pages 87--98, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... algorithms seek out the ADG models with highest posterior probability, and subsequent inference proceeds conditionally on these selected models [7, 3, 11, 17] Non Bayesian model selection methods are similar, replacing posterior model probabilities by, for example, penalized maximum likelihoods [4]. Heckerman et al. [11] highlighted a fundamental problem with this general approach. Several different ADGs may determine the same statistical model, i.e. may determine the same set of conditional independence restrictions among a given set of random variates, hence cannot be distinguished on the ....

D.M. Chickering, A transformational characterization of equivalent Bayesian network structures, in: Uncertainty in Artificial Intelligence: Proc. of the Eleventh Conf. (Morgan Kaufmann, San Francisco), 1995, pp. 87-98.

....sake of brevity,we often blur this distinction) Thus, for example, the sum over network structure hypotheses in Equation 33 should be replaced with a sum over equivalence class hypotheses. An efficient algorithm for identifying the equivalence class of a given network structure can be found in Chickering (1995). We note that hypothesis equivalence holds provided weinterpret Bayesian network structure simply as a representation of conditional independence. Nonetheless, stronger definitions of Bayesian networks exist where arcs have a causal interpretation (see Section 15) Heckerman et al. 1995b) and ....

....identifying network structures with high scores by some criterion. Consider the problem of finding the best network from the set of all networks in whicheach node has no more than k parents. Unfortunately, the problem for k 1 is NP hard even when we use the restrictive prior given by Equation 43 (Chickering et al. 1995). Thus, researchers have used heuristic search algorithms, including greedy search, greedy search with restarts, best first search, and Monte Carlo methods. One consolation is that these search methods can be made more efficient when the model selection criterion is separable. Given a network ....

Chickering, D. (1995). A transformational characterization of equivalent Bayesian network structures. In Proceedings of Eleventh ConferenceonUncertainty in Artificial Intelligence, Montreal, QU, pages 87--98. Morgan Kaufmann.

....sake of brevity, we often blur this distinction) Thus, for example, the sum over network structure hypotheses in Equation 33 should be replaced with a sum over equivalence class hypotheses. An efficient algorithm for identifying the equivalence class of a given network structure can be found in Chickering (1995). We note that hypothesis equivalence holds provided we interpret Bayesian network structure simply as a representation of conditional independence. Nonetheless, stronger def initions of Bayesian networks exist where arcs have a causal interpretation (see Section 15) Heckerman et al. 1995b) ....

....network structures with high scores by some criterion. Consider the problem of finding the best network from the set of all networks in which each node has no more than k parents. Unfortunately, the problem for k ) 1 is NP hard even when we use the restrictive prior given by Equation 43 (Chickering et al. 1995). Thus, researchers have used heuristic search algorithms, including greedy search, greedy search with restarts, best first search, and Monte Carlo methods. 33 One consolation is that these search methods can be made more efficient when the model selection criterion is separable. Given a network ....

Chickering, D. (1995). A transformational characterization of equivalent Bayesian network structures. In Proceedings of Eleventh Conference on Uncertainty in Artificial Intelligence, Montreal, QU, pages 87-98. Morgan Kaufmann.

....X Y both imply the same set of independencies (i.e. Ind(G) We say that two graphs G and G 0 are equivalent if Ind(G) Ind(G 0 ) This notion of equivalence is crucial, since when we examine observations from a distribution, we cannot distinguish between equivalent graphs. Results of [7, 32] show that we can characterize equivalence classes of graphs using a simple representation. In particular, these results establish that equivalent graphs have the same underlying undirected graph but might disagree on the direction of some of the arcs. Theorem 2.1 [32] Two graphs are equivalent ....

....that all members of the equivalence class contain the arc X Y ; an undirected edge X Y denotes that some members of the class contain the arc X Y , while others contain the arc Y X . Given a directed graph G, the PDAG representation of its equivalence class can be constructed efficiently [7]. 2.3 Learning Bayesian Networks The problem of learning a Bayesian network can be stated as follows. Given a training set D = fx 1 ; x N g of independent instances of X , find a network B = hG; Thetai that best matches D. More precisely, we search for an equivalence class of ....

D. M. Chickering. A transformational characterization of equivalent Bayesian network structures. In Proc. Eleventh Conference on Uncertainty in Artificial Intelligence (UAI '95), pp. 87--98. 1995.

....both imply the same set of independencies (i.e. Ind(G) We say that two graphs G and G 0 are equivalent if Ind(G) Ind(G 0 ) This notion of equivalence is crucial, since when we examine observations from a distribution, we often cannot distinguish between equivalent graphs. Results of (Chickering 1995, Pearl Verma 1991) show that we can characterize equivalence classes of graphs using a simple representation. In particular, these results establish that equivalent graphs have the same underlying undirected graph but might disagree on the direction of some of the edges. Moreover, an ....

....that all members of the equivalence class contain the edge X Y ; an undirected edge X Y denotes that some members of the class contain the edge X Y , while others contain the edge Y X . Given a directed graph G the PDAG representation of its equivalence class can be constructed efficiently (Chickering 1995). 4 2.4 Learning Bayesian Networks The problem of learning a Bayesian network can be stated as follows. Given a training set D = fx 1 ; x N g of independent instances of X , find a network B = hG; i that best matches D. The common approach to this problem is to introduce a ....

Chickering, D. M. (1995), A transformational characterization of equivalent Bayesian network structures, in Besnard & Hanks (1995), pp. 87--98.

.... selected models (Cooper and Herskovits (1990) Buntine (1994) Spiegelhalter et al. (1993) Heckerman et al. (1994) Madigan and Raftery (1994) Non Bayesian model selection methods proceed in a similar manner, replacing posterior model probabilities by, for example, penalized maximum likelihoods (Chickering (1995)) Heckerman et al. (1994) highlighted a fundamental problem with this general approach. Because several different ADGs may determine the same statistical model, i.e. may determine the same set of conditional independence restrictions among a given set of random variates, the collection of all ....

....analysts and expert system builders can overcome several difficulties associated with ADG models. Three such difficulties were listed in Section 1 here we examine these in more detail and indicate how the introduction of essential graphs can help to overcome them. 1. Heckerman et al. (1994) and Chickering (1995) argue that statistical inference for ADG models should be score equivalent : in the absence of a priori causal knowledge, Markov equivalent ADGs should have identical posterior model probabilities (Bayesian) or identical penalized likelihoods (non Bayesian) Under this criterion, therefore, ....

[Article contains additional citation context not shown here]

Chickering, D. M. (1995). A transformational characterization of equivalent Bayesian network structures. In Proceedings of the Eleventh Annual Conference on Uncertainty in Artificial Intelligence, Philippe Besnard and Steve Hanks, eds., pp. 87-98. Morgan Kaufmann, San Mateo.

....[8] does exactly that. It measures the mutual information (formally defined below) between all pairs of variables and selects a maximal spanning tree as the required network. We aim to use a similar argument for finding networks that are not necessarily trees. Here, the general problem is NP hard [5]. However, a seemingly reasonable heuristic is to select pairs (X; Y ) with high dependency between them and create a network with these edges. This approach however, does not take more complex interactions into account. For example, if the true structure includes a substructure of the form X ....

D. M. Chickering. A transformational characterization of equivalent Bayesian network structures. In UAI '95, pp. 87--98. 1995.

....induced network, but rather, on the features in the class of networks that are equivalent to it. Two Bayesian network structures G and G 0 are equivalent, if they imply exactly the same set of independence statements. The characterization of Bayesian network equivalence classes is studied in [3, 18, 19, 20]. Results in these papers establish that equivalent networks agree on the connectivity between variables, but might disagree on the direction of the arcs. These results also show that each equivalence class of network structures can be represented by a partially directed graph (PDAG) where a ....

....X Y denotes that some members of the class contain the arc X Y , and some contain the arc Y X . The score in [13] is structure equivalent in the sense that equivalent networks receive the same score. In our experiments, we learn network structures and then use the procedure described in [3] to convert them to to PDAGs. 3 Bootstrap for Confidence Estimation Let G be a network structure. A feature of interest in this structure might be the existence of an X Y in the PDAG that corresponds to G. Another feature of interest might be that X precedes Y in the PDAG that corresponds to ....

D. M. Chickering. A transformational characterization of equivalent Bayesian network structures. In UAI '95, pp. 87-- 98. 1995.

....partially directed graph (PDAG) that describes the equivalence class of the learned network. These edges can be either directed, denoting that the edge direction is the same in all equivalent networks, or undirected, denoting that either direction is possible in some equivalent network. Results of [2, 8] describe the relationship between such PDAGs and equivalence classes of Bayesian networks. In particular, every equivalence class can be represented by a unique PDAG. The rest of the paper is organized as follows: In Section 2, we briefly review the definition of Bayesian networks and the methods ....

....on the existence of an arc in the induced network, but rather, on the existence of an arc in the equivalent class of networks. As discussed in the introduction, two Bayesian network structures G and G 0 are equivalent, if they imply exactly the same set of independence statements. The results in [2, 8] establish that two equivalent networks structures must agree on the connectivity between variables, but might disagree on the direction of the arcs. These results also show that each equivalent class of network structures can be represented by a partially directed graph (PDAG) where a directed X ....

[Article contains additional citation context not shown here]

D. M. Chickering. A transformational characterization of equivalent Bayesian network structures. In UAI '95, pages 87--98. 1995.

....if they have the same structure ignoring arc directions and the same v structures (Verma and Pearl, 1990) A vstructure is an ordered tuple (x; y; z) such that there is an arc from x to y and from z to y, but no arc between x and y. Using this characterization of network structure equivalence, Chickering (1995) has created an efficient algorithm for identifying all Bayesian network structures that are equivalent to a given network structure. Given that B h S is the assertion that the physical probabilities for the joint space of U can be encoded in the network structure B S , it follows that the ....

....forest for which P w(x i ; x j ) is a maximum. This search can be done using a maximum spanning tree algorithm. Now, let us consider the case where we find the best network from the set of all networks in which each node has no more than k parents. Unfortunately, the problem for k 1 is NP hard (Chickering et al. 1995). Therefore, it is appropriate to use heuristic search algorithms. Most of the commonly discussed search methods for learning Bayesian networks make successive arc changes to the network, and employ the property of decomposability to evaluate the merit of each change. The possible changes that can ....

Chickering, D. (1995). A transformational characterization of equivalent Bayesian network structures. In Proceedings of Eleventh Conference on Uncertainty in Artificial Intelligence, Montreal, QU. Morgan Kaufmann.

....DAG model, or sets of DAG models, given data, posses a serious computational challenge, because the number of DAG models grows faster than exponential in n. Methods for searching through the space of model structures are discussed (e.g. by Cooper and Herskovits (1992) Heckerman, Geiger, and Chickering (1995a) and Friedman and Goldszmidt (1997) From a statistical viewpoint, an important question which needs to be addressed is how to specify the quantities p(m h ) p(dj m ; m h ) p( m jm h ) needed for evaluating p(m h jd) for every DAG model m that could conceivably be considered by a ....

....X i and X k in either direction. Verma and Pearl show that two structures for X are independence equivalent if and only if they have identical edges and identical v structures. This characterization makes it easy to identify independence equivalent structures. An alternative characterization by Chickering (1995) is useful for proving our claim that independence equivalent structures have the same marginal likelihood. An arc reversal is a transformation from one structure to another, in which a single arc between two nodes is reversed. An arc between two nodes is said to be covered if those two nodes ....

[Article contains additional citation context not shown here]

Chickering, D. (1995). A transformational characterization of equivalent Bayesian networks structures. In Proceedings of Eleventh Conference on Uncertainty in Artificial Intelligence, Montreal, QU, pages 87--98. Morgan Kaufmann.

.... two causal models are statistically equivalent if and only if they have the same skeleton (undirected graph) and they have the same v structures (nodes that are the children of two parents which are themselves nonadjacent) Such models cannot be distinguished on the basis of sample data alone [ Chi95 ] , so the discovery of one is as good as the discovery of another in this experiment. For TETRAD II, in Figure 3, undirected arcs 10 50 100 200 500 1000 2000 5000 0 2 5 10 50 100 20 30 40 60 70 80 90 Sample Size Model 4 Model 5 Model 6 3 2 2 1 1 1 1 10 50 100 200 500 1000 2000 5000 0 2 4 5 10 50 ....

David M. Chickering. A transformational characterization of equivalent Bayesian network structures. In Proc. of the 11th Conference on Uncertainty in Artificial Intelligence, pages 87--98, 1995.

....relationships. e.g. two belief nets are equivalent if they have the same arcs and the same v structure. Therefore, unique belief net structures are actually acyclic graphs where compelled edges have an enforced direction, but the others can be chosen arbitrarily (but only once) Chickering [Chi 95] characterizes the equivalence classes of belief net structures, and [Chi 97] provides an O(n 2 e 3 ) algorithm for determining the class for a given structure. We decided not to push on this uniqueness issue, as it will not alter our results in any way. 7 The (empirical) description ....

D.M. Chickering, A Transformational Characterization of Equivalent Bayesian Network Structures, UAI'95, 1995.

....sake of brevity, we often blur this distinction) Thus, for example, the sum over network structure hypotheses in Equation 33 should be replaced with a sum over equivalence class hypotheses. An efficient algorithm for identifying the equivalence class of a given network structure can be found in Chickering (1995). We note that hypothesis equivalence holds provided we interpret Bayesian network structure simply as a representation of conditional independence. Nonetheless, stronger definitions of Bayesian networks exist where arcs have a causal interpretation (see Section 15) Heckerman et al. 1995b) and ....

....network structures with high scores by some criterion. Consider the problem of finding the best network from the set of all networks in which each node has no more than k parents. Unfortunately, the problem for k 1 is NP hard even when we use the restrictive prior given by Equation 43 (Chickering et al. 1995). Thus, researchers have used heuristic search algorithms, including greedy search, greedy search with restarts, best first search, and Monte Carlo methods. One consolation is that these search methods can be made more efficient when the model selection criterion is separable. Given a network ....

Chickering, D. (1995). A transformational characterization of equivalent Bayesian network structures. In Proceedings of Eleventh Conference on Uncertainty in Artificial Intelligence, Montreal, QU, pages 87--98. Morgan Kaufmann.

....example, network structures in which every pair of nodes is connected (complete network structures) are equivalent because they both encode all joint distributions for U . Another example is given in Figure 1. Characterizations of equivalent Bayesian networks for discrete variables are obtained in [Ch95, VP90]. We have said that, given a set of variables U and a network structure B, the hypothesis B h stands for the assertion that the joint distribution of U factors according to B. By this definition of network structure hypothesis, if follows immediately that if two network structures B 1 and B 2 ....

D. Chickering, A transformational characterization of equivalent Bayesian-network structures, Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence, Montreal QU, July 1995. Morgan Kaufmann.

....the same adjacencies as the path diagrams in the covariance equivalence class that it represents. In addition, an edge is oriented as X Z in the pattern if and only if it is oriented as X Z in every path diagram in the simple covariance equivalence class. Meek 1995, Andersson et al. 1995, and Chickering 1995 show how to generate a pattern from an acyclic graph in O(E) time (where E is the number of edges. In the case of acyclic path diagrams which may also contain latent variables, and the case of cyclic path diagrams which do not contain latent variables, there is an object analogous to a pattern ....

Chickering, D. (1995) A Transformational Characterization of Equivalent Bayesian Network Structures, Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence, Philippe Besnard and Steve Hanks (Eds.), Morgan Kaufmann Publishers, Inc., San Mateo, CA.

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D. M. Chickering. A transformational characterization of equivalent Bayesian network structures. In Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence, pages 87--98, 1995.

No context found.

Chickering, D. (1995). A transformational characterization of equivalent Bayesian network structures.

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D. Chickering. A transformational characterization of equivalent Bayesian networks. In P. Besnard and S. Hanks, editors, Proc. of the Conf. on Uncertainty in AI, pages 87--98, 1995.

No context found.

D. M. Chickering. A Transformational Characterization of Equivalent Network Structures. In Proc. of the Conf. on Uncertainty in Artificial Intelligence, pages 87--98, 1995.

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