Algebraic Geometry of Bayesian Networks

We develop the necessary theory in algebraic geometry to place Bayesian networks into the realm of algebraic statistics. This allows us to create an algebraic geometry--statistics dictionary. In particular, we study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification, in terms of primary decomposition of polynomial ideals, is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties. Moreover, a complete algebraic classification, in terms of generating sets of polynomial ideals, is given for Bayesian networks on at most three random variables and one hidden variable. The relevance of these results for model selection is discussed.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2004

Author dc:creator

• Garcia-Puente, Luis David

Chair dc:contributor.committeechair

• Laubenbacher, Reinhard C.

Committee members dc:contributor.committeemember

• Brown, Ezra A.
• Farkas, Daniel R.
• Green, Edward L.
• Shimozono, Mark M.

• statistical modelling
• algebraic geometry
• bayesian networks
• computational commutative algebra
• statistics

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Dc Identifier Other

etd-04162004-172816

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/11133