Are Bayesian networks typically faithful?

Abstract

Faithfulness is a ubiquitous assumption in causal inference, often motivated by the fact that the faithful parameters of linear Gaussian and discrete Bayesian networks are typical, and the folklore belief that this should also hold for other classes of Bayesian networks. We address this open question by showing that among all Bayesian networks over a given DAG, the faithful Bayesian networks are indeed `typical': they constitute a dense, open set with respect to the total variation metric. However, this does not imply that faithfulness is typical in restricted classes of Bayesian networks, as are often considered in statistical applications. To this end we consider the class of Bayesian networks parametrised by conditional exponential families, for which we show that under mild regularity conditions, the faithful parameters constitute a dense, open set and the unfaithful parameters have Lebesgue measure zero, extending the existing results for linear Gaussian and discrete Bayesian networks. Finally, we show that the aforementioned results also hold for Bayesian networks with latent variables.

Tasks

Reproductions