## Dependency networks for inference, collaborative filtering, and data visualization (2000)

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Venue: | Journal of Machine Learning Research |

Citations: | 156 - 10 self |

### BibTeX

@ARTICLE{Heckerman00dependencynetworks,

author = {David Heckerman and David Maxwell Chickering and Christopher Meek and Robert Rounthwaite and Carl Kadie and Pack Kaelbling},

title = {Dependency networks for inference, collaborative filtering, and data visualization},

journal = {Journal of Machine Learning Research},

year = {2000},

volume = {1},

pages = {49--75}

}

### Years of Citing Articles

### OpenURL

### Abstract

We describe a graphical model for probabilistic relationships|an alternative to the Bayesian network|called a dependency network. The graph of a dependency network, unlikeaBayesian network, is potentially cyclic. The probability component of a dependency network, like aBayesian network, is a set of conditional distributions, one for each node given its parents. We identify several basic properties of this representation and describe a computationally e cient procedure for learning the graph and probability components from data. We describe the application of this representation to probabilistic inference, collaborative ltering (the task of predicting preferences), and the visualization of acausal predictive relationships.

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Citation Context ...ere to consider continuous variables) such as methods using a probabilistic decision tree (e.g., Buntine, 1991), a generalized linear model (e.g., McCullagh and Nelder, 1989), a neural network (e.g., =-=Bishop, 1995-=-), a probabilistic support-vector machine (e.g., Platt, 1999), or an embedded regression /classification model (Heckerman and Meek, 1997). This observation suggests a simple, heuristic approach for le... |

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Citation Context ... for probabilistic inference in the latter representation---for example, the junction tree algorithm of Jensen, Lauritzen, and Olesen (1990). Alternatively, we can use Gibbs sampling (e.g., Geman and =-=Geman, 1984-=-; Neal, 1993; Besag, Green, Higdon, and Mengersen, 1995; Gilks, Richardson, and Spiegelhalter, 1996), which we examine in some detail. First, let us consider the use of Gibbs sampling for recovering t... |

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Citation Context ...bbs sampler. The sequence x 1 � x 2 �:::can be viewed as samples drawn from a homogenous Markov chain with transition matrix M having elements M jji = p(x t+1 = jjx t = i). (We use the terminology=-= of Feller, 1957.) It-=- is not di cult to see that M is the product M 1 ::: M n , where M k is the \local" transition matrix describing the resampling of Xk according to the local distribution p(xkjpa k). The positivit... |

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Citation Context ...lities. We have found the latter to be significantly easier to interpret. The proof of Theorem 1 appears in the Appendix, but it is essentially a restatement of the Hammersley-Clifford theorem (e.g., =-=Besag, 1974-=-). This correspondence is no coincidence. As is discussed in Besag (1974), several researchers who developed the Markov-network representation did so by initially investigating a graphical representat... |

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Citation Context ... functions, one for each of the c maximal cliques in U , such that joint distribution has the form p(x) = c Y i=1 OE i (x i ); (2) where X i are the variables in clique i, i = 1; : : : ; c (e.g., see =-=Lauritzen, 1996-=-). The following theorem shows that consistent dependency networks and Markov networks have the same representational power. Theorem 1: The set of positive distributions that can be encoded by a consi... |

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Citation Context ... probabilistic decision tree (e.g., Buntine, 1991), a generalized linear model (e.g., McCullagh and Nelder, 1989), a neural network (e.g., Bishop, 1995), a probabilistic support-vector machine (e.g., =-=Platt, 1999-=-), or an embedded regression /classification model (Heckerman and Meek, 1997). This observation suggests a simple, heuristic approach for learning the structure and probabilities of a dependency netwo... |

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Citation Context ...istic inference in the latter representation---for example, the junction tree algorithm of Jensen, Lauritzen, and Olesen (1990). Alternatively, we can use Gibbs sampling (e.g., Geman and Geman, 1984; =-=Neal, 1993-=-; Besag, Green, Higdon, and Mengersen, 1995; Gilks, Richardson, and Spiegelhalter, 1996), which we examine in some detail. First, let us consider the use of Gibbs sampling for recovering the joint dis... |

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Citation Context ... cation techniques (or regression techniques, if we were to consider continuous variables) such as methods using a probabilistic decision tree (e.g., Buntine, 1991), a generalized linear model (e.g., =-=McCullagh and Nelder, 1989-=-), a neural network (e.g., Bishop, 1995), a probabilistic support-vector machine (e.g., Platt, 1999), or an embedded regression/classi cation model (Heckerman and Meek, 1997). This observation suggest... |

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Citation Context ...e estimated by any number of probabilistic classification techniques (or regression techniques, if we were to consider continuous variables) such as methods using a probabilistic decision tree (e.g., =-=Buntine, 1991-=-), a generalized linear model (e.g., McCullagh and Nelder, 1989), a neural network (e.g., Bishop, 1995), a probabilistic support-vector machine (e.g., Platt, 1999), or an embedded regression /classifi... |

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Citation Context ...nt ofXn Xi given Pai, i =1�:::�n. Because p is positive and these independencies comprise the global Markov property ofaMarkov network with Ai = Pai, i =1�:::�n, the Hammersley{Cli ord theorem=-= (e.g., Lauritzen, Dawid, Larsen, and Leimer, 1990-=-)) implies that p can be represented by this Markov network. 2 Theorem 4: A minimal consistent dependency network for a positive distribution p(x) must be bi-directional. Proof: Suppose the theorem is... |

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Citation Context ... be estimated by anynumber of probabilistic classi cation techniques (or regression techniques, if we were to consider continuous variables) such as methods using a probabilistic decision tree (e.g., =-=Buntine, 1991-=-), a generalized linear model (e.g., McCullagh and Nelder, 1989), a neural network (e.g., Bishop, 1995), a probabilistic support-vector machine (e.g., Platt, 1999), or an embedded regression/classi ca... |

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(Show Context)
Citation Context ...Xn, in this order, and resample each Xi according to p(xijx 1�:::�xi;1�xi+1�:::�xn) =p(xijpa i). We call this procedure an ordered Gibbs sampler. As described by the following theorem (also =-=proved in Hofmann, 2000-=-), this ordered Gibbs sampler de nes a joint distribution for X. Theorem 1: An ordered Gibbs sampler applied to a dependency network for X, where each Xi is discrete and each local distribution p(xijp... |

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(Show Context)
Citation Context ...own in Equation 6 are particularly useful. Here, we cite a potentially useful bound of this form given by Cho and Meyer (1999). These authors provide references to many other bounds as well. Theorem (=-=Cho and Meyer, 1999-=-): Let P and ~P = P + E be transition matrices for two homogenous, irreducible k-state Markov chains with respective stationary distributions and ~. Let kEk1 denote the in nity-norm of E, the maximum ... |

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(Show Context)
Citation Context ...ere to consider continuous variables) such as methods using a probabilistic decision tree (e.g., Buntine, 1991), a generalized linear model (e.g., McCullagh and Nelder, 1989), a neural network (e.g., =-=Bishop, 1995-=-), a probabilistic support-vector machine (e.g., Platt, 1999), or an embedded regression/classi cation model (Heckerman and Meek, 1997). This observation suggests a simple, heuristic approach for lear... |

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(Show Context)
Citation Context ...own in Equation 6 are particularly useful. Here, we cite a potentially useful bound of this form given by Cho and Meyer (1999). These authors provide references to many other bounds as well. Theorem (=-=Cho and Meyer, 1999-=-): Let P and ~P = P + E be transition matrices for two homogenous, irreducible k-state Markov chains with respective stationary distributions and ~. Let kEk1 denote the in nity-norm of E, the maximum ... |