Constraint-based causal discovery from multiple interventions over overlapping variable sets

Scientific practice typically involves repeatedly studying a system, each time trying to unravel a different perspective. In each study, the scientist may take measurements under different experimental conditions (interventions, manipulations, perturbations) and measure different sets of quantities (variables). The result is a collection of heterogeneous data sets coming from different data distributions. In this work, we present algorithm COmbINE, which accepts a collection of data sets over overlapping variable sets under different experimental conditions; COmbINE then outputs a summary of all causal models indicating the invariant and variant structural characteristics of all models that simultaneously fit all of the input data sets. COmbINE converts estimated dependencies and independencies in the data into path constraints on the data-generating causal model and encodes them as a SAT instance. The algorithm is sound and complete in the sample limit. To account for conflicting constraints arising from statistical errors, we introduce a general method for sorting constraints in order of confidence, computed as a function of their corresponding p-values. In our empirical evaluation, COmbINE outperforms in terms of efficiency the only pre-existing similar algorithm; the latter additionally admits feedback cycles, but does not admit conflicting constraints which hinders the applicability on real data. As a proof-of-concept, COmbINE is employed to co-analyze 4 real, mass-cytometry data sets measuring phosphorylated protein concentrations of overlapping protein sets under 3 different interventions.