Learning directed graphical models with latent variables

Abstract

We consider the problem of learning directed graphical models with latent variables, represented by directed maximal ancestral graphs, from a conditional independence oracle. We show that given a set of separation statements from some directed maximal ancestral graph G* = (V*,E*), we can map posets with ground set V* to minimal IMAPs of G* such that the sparsest of these minimal IMAPs is Markov equivalent to G*. We give a diagrammatic interpretation of these minimal IMAPs in terms of the Hasse diagram of the poset of posets. Namely, the Hasse diagram of these minimal IMAPs corresponds to the Hasse diagram of the poset of posets after identifying posets that map to the same minimal IMAP. We show that moving between these minimal IMAPs using legitimate mark changes corresponds to covering relations in the poset obtained after identification. Finally, we conjecture that a greedy search to minimize sparsity over this contracted space by moving between minimal IMAPs using legitimate mark changes converges to G*.