Algebraic Statistics in Model Selection

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.