Estimation of means in graphical Gaussian models with symmetries

Gehrmann, Helene; Lauritzen, Steffen

doi:10.1214/12-aos991

Cited by 8 publications

(12 citation statements)

References 30 publications

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“…Note that the graphical models (9) and (11) are sub-models of (14) and (17). So if we prove that the MLE does not exist for one observation on the graphs (9) and (11), this follows also for the graphs (14) and (17).…”

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confidence: 74%

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Geometry of maximum likelihood estimation in Gaussian graphical models

Uhler¹

2012

Ann. Statist.

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We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. An algebraic elimination criterion allows us to find exact lower bounds on the number of observations needed to ensure that the maximum likelihood estimator (MLE) exists with probability one. This is applied to bipartite graphs, grids and colored graphs. We also study the ML degree, and we present the first instance of a graph for which the MLE exists with probability one, even when the number of observations equals the treewidth.Comment: Published in at http://dx.doi.org/10.1214/11-AOS957 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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confidence: 74%

“…The case where the mean is unknown or partially known is more complex, since mean and covariance matrix can generally not be estimated independently. Gehrmann and Lauritzen [11] describe symmetry relations on the underlying graph which ensure estimability of the mean vector independently from the true covariance matrix Σ.…”

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confidence: 99%

Geometry of maximum likelihood estimation in Gaussian graphical models

Uhler¹

2012

Ann. Statist.

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“…These are known as RCOP models (see Definition 2.2), and were introduced by Højsgaard and Lauritzen in [HL08]. RCOP models have been studied further in works such as [GL12,Uhl11].…”

Section: Introductionmentioning

confidence: 99%

Symmetrically colored Gaussian graphical models with toric vanishing ideal

Coons¹,

Maraj²,

Misra³

et al. 2021

Preprint

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A colored Gaussian graphical model is a linear concentration model in which equalities among the concentrations are specified by a coloring of an underlying graph. The model is called RCOP if this coloring is given by the edge and vertex orbits of a subgroup of the automorphism group of the graph. We show that RCOP Gaussian graphical models on block graphs are toric in the space of covariance matrices and we describe Markov bases for them. To this end, we learn more about the combinatorial structure of these models and their connection with Jordan algebras.

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“…Symmetry restrictions for the multivariate Gaussian distribution have a long history dating back to Wilks [1946] and the reader is referred to Gehrmann & Lauritzen [2012] for a complete list of references. Graphical Gaussian models with symmetry restrictions were first considered by Hylleberg el al.…”

Section: Introductionmentioning

confidence: 99%

“…In this article we consider graphical Gaussian models with symmetry constraints. Symmetry restrictions for the multivariate Gaussian distribution have a long history dating back to Wilks () and the reader is referred to Gehrmann & Lauritzen () for a complete list of references. Graphical Gaussian models with symmetry restrictions were first considered by Hylleberg, Jensen, & Ørnbøl (): the symmetry restrictions in that article could be described by a group action.…”

Section: Introductionmentioning

confidence: 99%

Approximate Bayesian estimation in large coloured graphical Gaussian models

Gao

Massam

2017

Can J Statistics

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Distributed estimation methods have recently been used to compute the maximum likelihood estimate of the precision matrix for large graphical Gaussian models. Our aim, in this paper, is to give a Bayesian estimate of the precision matrix for large graphical Gaussian models with, additionally, symmetry constraints imposed by an underlying graph which is coloured. We take the sample posterior mean of the precision matrix as our estimate. We study its asymptotic behaviour under the regular asymptotic regime when the number of variables p is fixed and under the double asymptotic regime when both p and n grow to infinity. We show in particular, that when the number of parameters of the local models is uniformly bounded, the standard convergence rate we obtain for the asymptotic consistency, in the Frobenius norm, of our estimate of the precision matrix compares well with the rates in the current literature for the maximum likelihood estimate.

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Estimation of means in graphical Gaussian models with symmetries

Cited by 8 publications

References 30 publications

Geometry of maximum likelihood estimation in Gaussian graphical models

Geometry of maximum likelihood estimation in Gaussian graphical models

Symmetrically colored Gaussian graphical models with toric vanishing ideal

Approximate Bayesian estimation in large coloured graphical Gaussian models

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