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Explicit, identical maximum likelihood estimates for some cyclic Gaussian and cyclic Ising models
Author(s) -
Marchetti Giovanni M.,
Wermuth Nanny
Publication year - 2017
Publication title -
stat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.61
H-Index - 18
ISSN - 2049-1573
DOI - 10.1002/sta4.155
Subject(s) - mathematics , ising model , exponential family , graphical model , conditional independence , gaussian , markov chain , simple (philosophy) , covariance , combinatorics , statistical physics , statistics , philosophy , physics , epistemology , quantum mechanics
Cyclic models are a subclass of graphical Markov models with simple, undirected probability graphs that are chordless cycles. In general, all currently known distributions require iterative procedures to obtain maximum likelihood estimates in such cyclic models. For exponential families, the relevant conditional independence constraint for a variable pair is given all remaining variables, and it is captured by vanishing canonical parameters involving this pair. For Gaussian models, the canonical parameter is a concentration, that is, an off‐diagonal element in the inverse covariance matrix, while for Ising models, it is a conditional log‐linear, two‐factor interaction. We give conditions under which the two different likelihood functions, that is, one for continuous and one for binary variables, permit nevertheless explicit maximum likelihood estimates, and we show that their estimated correlation matrices are identical, provided the relevant starting correlation matrices coincide. Copyright © 2017 John Wiley & Sons, Ltd.