On the toric algebra of graphical models
By Dan Geiger, Christopher Meek, Bernd Sturmfels. 2006
209 citations recorded by the source corpus.
We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a log-linear model, or other more general exponential models. For decomposable graphical models these conditions are equivalent to a set of conditional independence statements similar to the Hammersley-Clifford theorem; however, we show that for nondecomposable graphical models they are not. We also show that nondecomposable models can have nonrational maximum likelihood estimates. These results are used to give several novel characterizations of decomposable graphical models.
Subfields: Artificial Intelligence, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/900175ec07667b73a2c0c857855bb0df1df50c8d
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Artificial Intelligence
- Mathias Drton
- Marloes H. Maathuis
- Alessandro Rinaldo
- Stephen E. Fienberg
- Lior S. Pachter
- Bernd Sturmfels
- Amit Khetan
- F. Catanese
In Computational Theory and Mathematics
- Bernd Sturmfels
- Mathias Drton
- Seth Sullivant
- Thomas Kahle
- Alessandro Rinaldo
- Stephen E. Fienberg
- Lior S. Pachter
- Marloes H. Maathuis
Citations in the corpus
- The maximum likelihood degree (2004, methodology): “Chor, Khetan and Snir [7] showed that the ML degree of a phylogenetic model equals 9, and Geiger, Meek and Sturm-fels [15] proved that an undirected graphical model has ML degree one if and only if it is decomposable.”
- Algebraic Statistics for Computational Biology: Preface (2005, methodology): “Our goal here is to introduce the log-linear Markov random field model using algebraic notation (for a more general exposition of MRFs in algebraic statistics, we refer to [Geiger et al., 2005]).”
- Maximum Likelihood Estimation in Log-Linear Models: Theory and Algorithms (2011, methodology): “In our analysis, we rely on the key notion of facial sets, originally introduced in a slightly different form by Geiger et al. (2006).”
- Algebraic statistical models (2007, background): “Note that some work in algebraic statistics has focused on discrete distributions corresponding to the boundary of the probability simplex (Geiger et al., 2006).”
- Structure Learning in Graphical Modeling (2016, background): “We remark that factorization of nonpositive distributions for categorical variables is treated by Geiger et al. (2006).”
- Decompositions of binomial ideals (2009, background): “22 D-04103 Leipzig Germany E-mail: kahle@mis.mpg.de is central in the field of algebraic statistics, where closure of discrete exponential families, such as graphical or hierarchical models, have been recognized to be nonnegative real parts of toric varieties (Geiger et al 2006).”