  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 effiective 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.
