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Scientific practice typically involves repeatedly studying a system, eachtime trying to unravel a different perspective. In each study, the scientistmay 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 fromdifferent data distributions. In this work, we present algorithm COmbINE, whichaccepts a collection of data sets over overlapping variable sets underdifferent experimental conditions; COmbINE then outputs a summary of all causalmodels indicating the invariant and variant structural characteristics of allmodels that simultaneously fit all of the input data sets. COmbINE convertsestimated dependencies and independencies in the data into path constraints onthe data-generating causal model and encodes them as a SAT instance. Thealgorithm is sound and complete in the sample limit. To account for conflictingconstraints arising from statistical errors, we introduce a general method forsorting constraints in order of confidence, computed as a function of theircorresponding p-values. In our empirical evaluation, COmbINE outperforms interms of efficiency the only pre-existing similar algorithm; the latteradditionally admits feedback cycles, but does not admit conflicting constraintswhich hinders the applicability on real data. As a proof-of-concept, COmbINE isemployed to co-analyze 4 real, mass-cytometry data sets measuringphosphorylated protein concentrations of overlapping protein sets under 3different interventions.

Constraint-based Causal Discovery from Multiple Interventions over Overlapping Variable Sets