Neural Information Processing Systems http://nips.cc/

R be a polynomial with at most s non-zero real coefficients.

The challenge of learning the causal structure underlying a certain phenomenon is undertaken by connecting the set of conditional independences (CIs) readable from the observational data, on the one side, with the set of corresponding constraints implied over the graphical structure, on the other, which are tied through a graphical criterion known as d-separation (Pearl, 1988). In this paper, we investigate the more general setting where multiple observational and experimental distributions are available. We start with the simple observation that the invariances given by CIs/dseparation are just one special type of a broader set of constraints, which follow from the careful comparison of the different distributions available. Remarkably, these new constraints are intrinsically connected with do-calculus (Pearl, 1995) in the context of soft-interventions. We then introduce a novel notion of interventional equivalence class of causal graphs with latent variables based on these invariances, which associates each graphical structure with a set of interventional distributions that respect the do-calculus rules. Given a collection of distributions, two causal graphs are called interventionally equivalent if they are associated with the same family of interventional distributions, where the elements of the family are indistinguishable using the invariances obtained from a direct application of the calculus rules. We introduce a graphical representation that can be used to determine if two causal graphs are interventionally equivalent. We provide a formal graphical characterization of this equivalence. Finally, we extend the FCI algorithm, which was originally designed to operate based on CIs, to combine observational and interventional datasets, including new orientation rules particular to this setting.

We consider the problem of experimental design for learning causal graphs that have a tree structure. We propose an adaptive framework that determines the next intervention based on a Bayesian prior updated with the outcomes of previous experiments, focusing on the setting where observational data is cheap (assumed infinite) and interventional data is expensive. While information greedy approaches are popular in active learning, we show that in this setting they can be exponentially suboptimal (in the number of interventions required), and instead propose an algorithm that exploits graph structure in the form of a centrality measure. If each intervention yields a very large data sample, we show that the algorithm requires a number of interventions less than or equal to a factor of 2 times the minimum achievable number. We show that the algorithm and the associated theory can be adapted to the setting where each performed intervention yields finitely many samples. Several extensions are also presented, to the case where a specified set of nodes cannot be intervened on, to the case where K interventions are scheduled at once, and to the fully adaptive case where each experiment yields only one sample. In the case of finite interventional data, through simulated experiments we show that our algorithms outperform different adaptive baseline algorithms.

Neural Information Processing Systems http://nips.cc/