## A new algorithm for maximum likelihood estimation in Gaussian graphical models for marginal independence (2003)

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Venue: | In U. Kjærulff and C. Meek (Eds.), Proceedings of the 19th Conference on Uncertainty in Artificial Intelligence |

Citations: | 15 - 7 self |

### BibTeX

@INPROCEEDINGS{Drton03anew,

author = {Mathias Drton},

title = {A new algorithm for maximum likelihood estimation in Gaussian graphical models for marginal independence},

booktitle = {In U. Kjærulff and C. Meek (Eds.), Proceedings of the 19th Conference on Uncertainty in Artificial Intelligence},

year = {2003},

pages = {184--191},

publisher = {Morgan Kaufmann}

}

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### Abstract

Graphical models with bi-directed edges (↔) represent marginal independence: the absence of an edge between two vertices indicates that the corresponding variables are marginally independent. In this paper, we consider maximum likelihood estimation in the case of continuous variables with a Gaussian joint distribution, sometimes termed a covariance graph model. We present a new fitting algorithm which exploits standard regression techniques and establish its convergence properties. Moreover, we contrast our procedure to existing estimation algorithms. 1

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Citation Context ...he key to prove convergence properties of the algorithm in Section 4.4 is to recognize that the algorithm consists of iterated partial maximizations over sections of the parameter space P(G) (compare =-=Lauritzen 1996-=-, Appendix A.4, and Meng and Rubin 1993). More accurately, we will consider the parameter space Θ = {Σ ∈ P(G) | ℓ(Σ) ≥ ℓ( ˆ Σ (0) )} (35) which of course contains the global maximizer of ℓ(Σ). The set... |

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Citation Context ...ties of the algorithm in Section 4.4 is to recognize that the algorithm consists of iterated partial maximizations over sections of the parameter space P(G) (compare Lauritzen 1996, Appendix A.4, and =-=Meng and Rubin 1993-=-). More accurately, we will consider the parameter space Θ = {Σ ∈ P(G) | ℓ(Σ) ≥ ℓ( ˆ Σ (0) )} (35) which of course contains the global maximizer of ℓ(Σ). The set Θ is obviously closed, and under the c... |

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Citation Context ... dominated by the solution of the systems of nsp(i) and sp(i) linear equations in (29) and (34), respectively. 5 EXAMPLE DATA Table 1 presents data on p = 4 variables measured on n = 39 patients (see =-=Cox and Wermuth 1993-=-, Table 7, and Kauermann 1996, Table 1). If we index the Table 1: Observed Marginal Correlations and Standard Deviations. W V X Y V 0.060 X −0.460 0.042 Y −0.071 −0.404 −0.334 SD 5.72 92.00 7.86 2.07 ... |

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Citation Context ... 2002). For a large enough sample size, a multimodal likelihood seems not to arise in practice assuming the model assumptions hold but might still arise if the model assumptions do not hold (see also =-=Cox and Wermuth 1996-=-, p. 102f). 3.2 THE LIKELIHOOD EQUATIONS The likelihood equations are the estimating equations obtained by setting the derivatives of the log-likelihood ℓ(Σ) with respect to σij, i = j or i ↔ j, to ze... |

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Citation Context ...ystems of nsp(i) and sp(i) linear equations in (29) and (34), respectively. 5 EXAMPLE DATA Table 1 presents data on p = 4 variables measured on n = 39 patients (see Cox and Wermuth 1993, Table 7, and =-=Kauermann 1996-=-, Table 1). If we index the Table 1: Observed Marginal Correlations and Standard Deviations. W V X Y V 0.060 X −0.460 0.042 Y −0.071 −0.404 −0.334 SD 5.72 92.00 7.86 2.07 variables in this data set by... |

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Citation Context ...imum which is then global. The model N(G) is a curved but not regular exponential family, thus, the log-likelihood need not be concave. In fact, the log-likelihood can have multiple local maxima (cf. =-=Drton and Richardson 2002-=-). For a large enough sample size, a multimodal likelihood seems not to arise in practice assuming the model assumptions hold but might still arise if the model assumptions do not hold (see also Cox a... |

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Citation Context ...here in the case where all edges are bi-directed. 2.2 RELATION TO DAG MODELS WITH HIDDEN VARIABLES Graphical models for marginal independence can be motivated by the following consideration (see also =-=Pearl and Wermuth, 1994-=-). Suppose that there is DAG D with vertex set V ∪ U, where the variables in V are observed, and those in U are unobserved. Suppose further that observed variables i ∈ V have no children in the graph,... |

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