  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.
