Causal discovery from observational and interventional data across multiple environments

A fundamental problem in many sciences is the learning of causal structure underlying a system, typically through observation and experimentation. Commonly, one even collects data across multiple domains, such as gene sequencing from different labs, or neural recordings from different species. Although there exist methods for learning the equivalence class of causal diagrams from observational and experimental data, they are meant to operate in a single domain. In this paper, we develop a fundamental approach to structure learning in non-Markovian systems (i.e. when there exist latent confounders) leveraging observational and interventional data collected from multiple domains. Specifically, we start by showing that learning from observational data in multiple domains is equivalent to learning from interventional data with unknown targets in a single domain. But there are also subtleties when considering observational and experimental data. Using causal invariances derived from do-calculus, we define a property called S-Markov that connects interventional distributions from multiple-domains to graphical criteria on a selection diagram. Leveraging the S-Markov property, we introduce a new constraint-based causal discovery algorithm, S-FCI, that can learn from observational and interventional data from different domains. We prove that the algorithm is sound and subsumes existing constraint-based causal discovery algorithms.