The geometry of independence tree models with hidden variables
Zwiernik, Piotr and Smith, J. Q. (2010) The geometry of independence tree models with hidden variables. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. Working papers, Vol.2010 (No.3).
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Official URL: http://www2.warwick.ac.uk/fac/sci/statistics/crism...
In this paper we investigate the geometry of undirected discrete
graphical models of trees when all the variables in the system are binary, where
leaves represent the observable variables and where the inner nodes are unobserved.
We obtain a full geometric description of these models which is given
by polynomial equations and inequalities. We also give exact formulas for their
parameters in terms of the marginal probability over the observed variables.
Our analysis is based on combinatorial results generalizing the notion of cumulants
and introduce a novel use of Mobius functions on partially ordered sets.
The geometric structure we obtain links to the notion of a tree metric considered
in phylogenetic analysis and to some interesting determinantal formulas
involving hyperdeterminants of 2 x 2 x 2 tables as defined in .
|Item Type:||Working or Discussion Paper (Working Paper)|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Statistics|
|Library of Congress Subject Headings (LCSH):||Trees (Graph theory), Multivariate analysis -- Graphic methods|
|Series Name:||Working papers|
|Publisher:||University of Warwick. Centre for Research in Statistical Methodology|
|Place of Publication:||Coventry|
|Number of Pages:||26|
|Status:||Not Peer Reviewed|
|Access rights to Published version:||Open Access|
 E. S. Allman and J. A. Rhodes, Phylogenetic ideals and varieties for the general Markov
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