Algebraic Statistics in Model Selection

TL;DR

This paper applies computational algebraic geometry to Bayesian networks, establishing a framework that links algebraic and statistical dimensions, and deriving constraints for model selection in algebraic statistics.

Contribution

It introduces an algebraic statistics framework for Bayesian networks, connecting algebraic geometry with statistical modeling and extending previous theoretical work.

Findings

Link between effective and algebraic dimensions of Bayesian networks

Derivation of independence constraints via polynomial ideals

Extension of algebraic statistical methods to model selection

Abstract

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.