On the toric algebra of graphical models

• 成果类型: Article
• 署名作者: Geiger, Dan; Meek, Christopher; Sturmfels, Bernd
• 署名单位: Technion Israel Institute of Technology; Microsoft; University of California System; University of California Berkeley
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053606000000263
• 发表日期: 2006
• 页码: 1463-1492
• 关键词: multidimensional contingency-tables; conditional-independence; geometry; systems; bases

摘要

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.