Margins of discrete Bayesian networks

In this paper we provide a complete algebraic characterization of the model implied by a Bayesian network with latent variables when the observed variables are discrete. We show that it is algebraically equivalent to the so-called nested Markov model, meaning that the two are the same up to inequality constraints on the joint probabilities. The nested Markov model is therefore the best possible approximation to the latent variable model whilst avoiding inequalities, which are extremely complicated in general. Latent variable models also suffer from difficulties of unidentifiable parameters and non-regular asymptotics; in contrast the nested Markov model is fully identifiable, represents a curved exponential family of known dimension, and can easily be fitted using an explicit parameterization.