Learning equivalence classes of acyclic models with latent and selection variables from multiple datasets with overlapping variables

While there has been considerable research in learning probabilistic graphical models from data for predictive and causal inference, almost all existing algorithms assume a single dataset of i.i.d. observations for all variables. For many applications, it may be impossible or impractical to obtain such datasets, but multiple datasets of i.i.d. observations for different subsets of these variables may be available. Tillman et al. [2009] showed how directed graphical models learned from such datasets can be integrated to construct an equivalence class of structures over all variables. While their procedure is correct, it assumes that the structures integrated do not entail contradictory conditional independences and dependences for variables in their intersections. While this assumption is reasonable asymptotically, it rarely holds in practice with finite samples due to the frequency of statistical errors. We propose a new correct procedure for learning such equivalence classes directly from the multiple datasets which avoids this problem and is thus more practically useful. Empirical results indicate our method is not only more accurate, but also faster and requires less memory.