## Optimal Structure Identification with Greedy Search (2002)

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### BibTeX

@MISC{Chickering02optimalstructure,

author = {David Maxwell Chickering and Craig Boutilier},

title = {Optimal Structure Identification with Greedy Search},

year = {2002}

}

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### Abstract

In this paper we prove the so-called "Meek Conjecture". In particular, we show that if a is an independence map of another DAG then there exists a finite sequence of edge additions and covered edge reversals in such that (1) after each edge modification and (2) after all modifications H.

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Citation Context ... that each variable is independent of its non-descendants given its parents. That is, any other independence constraint that holds can be derived from the Markov conditions (see, 509sCHICKERING e.g., =-=Pearl, 1988-=-). We use A⊥⊥GB|S to denote the assertion that DAG G imposes the constraint that A is independent of B given set S. When the DAG G is clear from context we use A⊥⊥B|S. When S = /0,weuseA⊥⊥ GB (or A⊥⊥B... |

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(Show Context)
Citation Context ...a PDAG can constrain the status of other edges. An edge is compelled (reversible) in a PDAG if the corresponding edge is compelled (reversible) in a consistent extension of that PDAG. Proposition 32 (=-=Chickering, 2002-=-) Let P be any PDAG that admits a consistent extension and contains a compelled edge X → Y . If there is an edge, either directed or undirected, between Y and some node Z such that Z and X are not adj... |

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A transformational characterization of Bayesian network structures
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(Show Context)
Citation Context ...cal parents, with the exception that X is not a parent of itself. That is, X → Y is covered in G if Pa G Y = PaGX ∪ X. The significance of covered edges is evident from the following result: Lemma 2 (=-=Chickering, 1995-=-) Let G be any DAG model, and let G ′ be the result of reversing the edge X → YinG. ThenG ′ is a DAG that is equivalent to G if and only if X → Y is covered in G. The following transformational charac... |

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1 |
there are edges labeled “unknown
- While
(Show Context)
Citation Context ...such an active path in G ′ : (1) there must exist at least one such path that includes the edge X → Y , (2) neither Z nor any descendant of Z (including D) inG can belong to the conditioning set, and =-=(3)-=- Y is not in the conditioning set. The first conclusion we make from these three properties (see Figure 5a) is that there must be a descendant E of Y in G ′ —and hence E is also a descendant of Y in G... |

1 |
Y be the lowest ordered edge that is labeled “unknown
- Let
(Show Context)
Citation Context ...ls. In particular, Haughton (1988) shows that Equation 3 for curved exponential models can be approximated using Laplace’s method for integrals, yielding SB(G, D) = log p(D|ˆθ, G h ) − d log m + O(1) =-=(4)-=- 2 where ˆθ denotes the maximum-likelihood values for the network parameters, d denotes the dimension (i.e., number of free parameters) of G, and m is the number records in D. 519sChickering The first... |

1 | Y and every edge incident into Y with “compelled - Label |

1 | Y and all “unknown” edges incident into Y with “compelled - Label |

1 | Y and all “unknown” edges incident into Y with “reversible - Label |