Algebraic geometry of Bayesian networks
Journal of Symbolic Computation
Dimension correction for hierarchical latent class models
UAI'02 Proceedings of the Eighteenth conference on Uncertainty in artificial intelligence
Asymptotic model selection for naive Bayesian networks
UAI'02 Proceedings of the Eighteenth conference on Uncertainty in artificial intelligence
Graphical models and exponential families
UAI'98 Proceedings of the Fourteenth conference on Uncertainty in artificial intelligence
On the geometry of Bayesian graphical models with hidden variables
UAI'98 Proceedings of the Fourteenth conference on Uncertainty in artificial intelligence
Automated analytic asymptotic evaluation of the marginal likelihood for latent models
UAI'03 Proceedings of the Nineteenth conference on Uncertainty in Artificial Intelligence
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We develop the necessary theory in computational algebraic geometry to place Bayesian networks into the realm of algebraic statistics. We present an algebra-statistics dictionary focused on statistical modeling. In particular, we link the notion of effective dimension of a Bayesian network with the notion of algebraic dimension of a variety. We also obtain the independence and non-independence constraints on the distributions over the observable variables implied by a Bayesian network with hidden variables, via a generating set of an ideal of polynomials associated to the network. These results extend previous work on the subject. Finally, the relevance of these results for model selection is discussed.