Are Bayesian networks typically faithful?

Abstract

Faithfulness is a common 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. This does not directly imply that faithfulness is typical in restricted classes of Bayesian networks that 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 regularity conditions, the faithful parameters constitute a dense and open set, the unfaithful parameters have Lebesgue measure zero, and the induced faithful distributions are open and dense in the weak topology. This extends the existing results for linear Gaussian and discrete Bayesian networks. We also show for nonparametric classes of Bayesian networks with uniformly equicontinuous and uniformly bounded conditional densities that the faithful Bayesian networks are open and dense in the weak topology. All these results also hold for Bayesian networks with latent variables, if faithfulness is only required to hold with respect to the latent projection. Finally, for the considered conditional exponential family parametrisations and nonparametric conditional density models, the topological properties of conditional independence imply the existence of a consistent conditional independence test. Together with the topological properties of faithfulness, this implies that sound constraint-based causal discovery algorithms like PC and FCI are consistent on an open and dense -- and hence `typical' -- set of Bayesian networks.