Type of Document Dissertation Author Garcia-Puente, Luis David Author's Email Address lgarcia@math.vt.edu URN etd-04162004-172816 Title Algebraic Geometry of Bayesian Networks Degree PhD Department Mathematics Advisory Committee

Advisor Name Title Laubenbacher, Reinhard C. Committee Chair Brown, Ezra A. Committee Member Farkas, Daniel R. Committee Member Green, Edward L. Committee Member Shimozono, Mark M. Committee Member Keywords

- statistical modelling
- algebraic geometry
- bayesian networks
- computational commutative algebra
- statistics
Date of Defense 2004-04-01 Availability unrestricted AbstractWe develop the necessary theory in algebraic geometry to placeBayesian 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.

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