### Citations

1584 |
Graphical Models
- Lauritzen
- 1996
(Show Context)
Citation Context ...te that variable coding in uninfluential here. The directed acyclic graph in Figure 2 is a possible data generating process for the above bidirected graph model, because the directed Markov property (=-=Lauritzen, 1996-=-, § 3.2.2) implies, among others, the same marginal independencies and, moreover, it is associated with the recursive factorization pr(XV = xV , U = u) = pr(x2 | x1, u)pr(x3 | x4, u)pr(x1)pr(x4)pr(u),... |

800 |
Constrained Optimization and Lagrange Multiplier Methods
- Bertsekas
- 1982
(Show Context)
Citation Context ...uation ∂`(ω;n) ∂ω + ∂g(ω) ∂ω τ = 0 together with the constraint equation g(ω) = 0. If ω̂ is a local maximum of `(ω;n) subject to g(ω) = 0, and ∂g(ω)/∂ω is a full rank matrix, then a classical result (=-=Bertsekas, 1982-=-) guarantees that there exists a unique τ̂ such that the gradient equation is satisfied by (ω̂, τ̂). In the following we assume that the maximum likelihood estimate of interest is a local (constrained... |

140 | Multivariate dependencies: Models, analysis and interpretation. - Cox, Wermuth - 1996 |

85 |
Maximum Likelihood Estimation of Parameters Subject to Constraint
- AITCHISON, J, et al.
- 1958
(Show Context)
Citation Context ...ion is obtained using a first order expansion of s(ω;n) and g(ω) about ωt; see Evans and Forcina (2011) for details. The matrix inversion in the updating equation can be solved block-wise as follows (=-=Aitchison and Silvey, 1958-=-): [ F(ωt) −G(ωt) −G(ωt)T 0 ]−1 = [ R Q QT −P−1 ] , where P = G(ωt)TF(ωt)−1G(ωt), Q = −F(ωt)−1G(ωt)P−1, R = F(ωt)−1 + F(ωt)−1G(ωt)QT. Then, introducing the relative score vector e(ωt;n) = F(ωt)−1s(ωt;... |

64 |
Multivariate logistic models.
- Glonek, McCullagh
- 1995
(Show Context)
Citation Context ...he discrete case, two alternative parameterizations of bidirected graph models are available: the Möbius parameterization (Drton and Richardson, 2008) and the multivariate logistic parameterization (=-=Glonek and McCullagh, 1995-=-; Lupparelli et al., 2009). Our approach avoids some disadvantages of both these parameterizations: log-mean linear interactions can be interpreted as measures of association, which allows one to spec... |

64 | Markov properties for acyclic directed mixed graphs.
- Richardson
- 2003
(Show Context)
Citation Context ...ph model is the family of probability distributions for XV satisfying a given Markov property with respect to a bidirected graph G. The distribution of XV satisfies the connected set Markov property (=-=Richardson, 2003-=-) if, for every disconnected set D, the subvectors corresponding to its connected components XC1 , . . . , XCr are mutually independent; in symbols XC1 ⊥⊥XC2 ⊥⊥ · · · ⊥⊥XCr . We denote by B(G) the bid... |

60 | Linear dependencies represented by chain graphs. - Cox, Wermuth - 1993 |

51 | Marginal models for categorical data. - Bergsma, Rudas - 2002 |

25 | Binary models for marginal independence.
- Drton, Richardson
- 2008
(Show Context)
Citation Context ...t graphical models of marginal independence are log-mean linear models. In the discrete case, two alternative parameterizations of bidirected graph models are available: the Möbius parameterization (=-=Drton and Richardson, 2008-=-) and the multivariate logistic parameterization (Glonek and McCullagh, 1995; Lupparelli et al., 2009). Our approach avoids some disadvantages of both these parameterizations: log-mean linear interact... |

22 | Marginal regression analysis of a multivariate binary response - Ekholm, Smith, et al. - 1995 |

20 | Additive and multiplicative models and interactions - Darroch, Speed - 1983 |

18 | Maximum likelihood methods for a generalized class of log-linear models. - Lang - 1996 |

18 | Chain graph models of multivariate regression type for categorical data. Bernoulli, - Marchetti, Lupparelli - 2010 |

13 | Association models for a multivariate binary response - Ekholm, McDonald, et al. - 2000 |

11 | Marginal models: For dependent, clustered, and longitudinal categorial data - Bergsma, Croon, et al. - 2009 |

11 | Marginal log-linear Parameters for Graphical Markov Models. - Evans, Richardson - 2013 |

10 | Parameterizations and fitting of bi-directed graph models to categorical data. Available at arXiv:0801.1440.
- Lupparelli, Marchetti, et al.
- 2008
(Show Context)
Citation Context ...ative parameterizations of bidirected graph models are available: the Möbius parameterization (Drton and Richardson, 2008) and the multivariate logistic parameterization (Glonek and McCullagh, 1995; =-=Lupparelli et al., 2009-=-). Our approach avoids some disadvantages of both these parameterizations: log-mean linear interactions can be interpreted as measures of association, which allows one to specify interesting sub-model... |

8 | Marginal log-linear models for categorical data - Bergsma, Rudas - 2002 |

7 | Marginal log-linear parameterization of conditional independence models. - Rudas, Bergsma, et al. - 2010 |

5 | Two algorithms for fitting constrained marginal models - Evans, Forcina - 2013 |

5 | Marginal parameterizations of discrete models defined by a set of conditional independencies - Forcina, Lupparelli, et al. - 2010 |

4 | The Marke-Nyman temperament scale: an English translation - Coppen |