### Citations

5131 | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images.
- Geman, Geman
- 1984
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Citation Context ...lt value. Figure 2b shows an example where variables are contained in gates – this example is described in the following section. 2.1 Examples of models with gates Figure 2a shows a line process from =-=[7]-=-. The use of gates makes clear the assumption that two neighboring image pixels xi and xj have a dependency between their intensity values, unless there is an edge eij between them. An opaque three-wa... |

449 |
Expectation propagation for approximate Bayesian inference.
- Minka
- 2001
(Show Context)
Citation Context ... performing message passing in some example mixture models. In this section, we describe how gate notation can be generally incorporated into Variational Message Passing [10], Expectation Propagation =-=[11]-=- and Gibbs Sampling [7] to allow each of these algorithms to support context-specific independence. For reference, Table 1 shows the messages needed to apply standard EP or VMP using a fully factorize... |

381 |
Mapping Mendelian factors underlying quantitative traits using RFLP linkage maps.
- Lander, Botstein
- 1989
(Show Context)
Citation Context ...e. Gates can also be used to test for independence. In this case the selector variable is connected only to the gate, as shown in the example of figure 2b. This is a model used in functional genomics =-=[8]-=- where the aim is to detect associations between a genetic variant gn and some quantitative trait xn (such as height, weight, intelligence etc.) given data from a set of N individuals. The binary sele... |

337 | Context-specific Independence in Bayesian Networks, in:
- Boutilier, Friedman, et al.
- 1996
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Citation Context .... However, there is no widely used notation for representing context-specific dependencies, that is, dependencies which are present or absent conditioned on the state of another variable in the graph =-=[2]-=-. Such a notation would be necessary not only to represent and communicate context-specific dependencies, but also to be able to exploit context-specific independence to achieve efficient and accurate... |

276 | Operations for learning with graphical models",
- Buntine
- 1994
(Show Context)
Citation Context ...of the gates. Notice that in this example, an integer selector variable is used and the key values are the integers 1,2,3. For large homogeneous mixtures, gates can be used in conjunction with plates =-=[6]-=-. For example, figure 1d shows how a mixture of N Gaussians can be represented by placing the gate, Gaussian factor and mean/precision variables inside a plate, so that they are replicated N times. Ga... |

51 | Variational approximations between mean field theory and the junction tree algorithm.
- Wiegerinck
- 2000
(Show Context)
Citation Context ...e a gate. The model is: p(x,c,m1,m2,y) = p(c)p(m1)p(m2) ∏ k (f1(x|y)f2(y|mk)) δ(c−k) . If we use a structured variational approximation where y is conditioned on c, then the fixed-point equations are =-=[9]-=-: ( ∑ q(c = k) ∝ p(c = k)exp q(x) ∑ ) q(y|c = k)log f1(x|y) x y ( ∑ exp q(y|c = k) ∑ ) ( q(mk)log f2(y|mk) exp − ∑ ) q(y|c = k)log q(y|c = k) y mk ( ) ( ) ∑ ∑ q(y|c = k) ∝ exp q(x)log f1(x|y) exp q(mk... |

41 | Case-factor diagrams for structured probabilistic modeling.
- McAllester, Collins, et al.
- 2004
(Show Context)
Citation Context ...oit context-specific independence to achieve efficient and accurate inference. A number of notations have been proposed for representing context-specific dependencies, including: case factor diagrams =-=[3]-=-, contingent Bayesian networks [4] and labeled graphs [5]. None of these has been widely adopted, raising the question: what properties would a notation need, to achieve widespread use? We believe it ... |

36 | Factor graphs and algorithms
- Frey, Kschischang, et al.
- 1997
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Citation Context ... model. We present general equations for expectation propagation and variational message passing in the presence of gates. 1 Introduction Graphical models, such as Bayesian networks and factor graphs =-=[1]-=-, are widely used to represent and visualise fixed dependency relationships between random variables. Graphical models are also commonly used as data structures for inference algorithms since they all... |

35 | Robust Bayesian mixture modelling,
- Svensen, Bishop
- 2005
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Citation Context ...d for each case, but it may increase the cost. For example, Archambeau and Verleysen [14] suggested a structured approximation for Student-t mixture models, instead of the factorised approximation of =-=[13]-=-. Their modification can be viewed as a gate enlargement (figure 3). By enlarging the gate block to include unm, the blurring between the multiplication factor and unm is removed, increasing accuracy.... |

25 | Approximate inference for infinite contingent Bayesian networks
- Milch, Marthi, et al.
- 2005
(Show Context)
Citation Context ...to achieve efficient and accurate inference. A number of notations have been proposed for representing context-specific dependencies, including: case factor diagrams [3], contingent Bayesian networks =-=[4]-=- and labeled graphs [5]. None of these has been widely adopted, raising the question: what properties would a notation need, to achieve widespread use? We believe it would need to be: • simple to unde... |

14 | Robust Bayesian clustering.
- Archambeau, Verleysen
- 2007
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Citation Context ...uivalently placed inside each gate of G. This increases accuracy since a separate set of messages will be maintained for each case, but it may increase the cost. For example, Archambeau and Verleysen =-=[14]-=- suggested a structured approximation for Student-t mixture models, instead of the factorised approximation of [13]. Their modification can be viewed as a gate enlargement (figure 3). By enlarging the... |

11 | Gates: A graphical notation for mixture models
- Minka, Winn
- 2008
(Show Context)
Citation Context ...variable to uniform factors in gates for any missing values of c. After balancing, each gate is part of a gate block – a set of gates activated by different values of the same condition variable. See =-=[12]-=- for details. 4.1 Variational Message Passing with gates VMP can be augmented to run on a gate model in normalised form, by changing only the messages out of the gate and by introducing messages from ... |

5 |
Labeled Graph Notations for Graphical Models: Extended Report
- Mjolsness
- 2004
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Citation Context ...d accurate inference. A number of notations have been proposed for representing context-specific dependencies, including: case factor diagrams [3], contingent Bayesian networks [4] and labeled graphs =-=[5]-=-. None of these has been widely adopted, raising the question: what properties would a notation need, to achieve widespread use? We believe it would need to be: • simple to understand and use, • flexi... |