  Probabilities of Causation (PoC) play a fundamental role in decision-making
in law, health care and public policy. Nevertheless, their point identification
is challenging, requiring strong assumptions, in the absence of which only
bounds can be derived. Existing work to further tighten these bounds by
leveraging extra information either provides numerical bounds, symbolic bounds
for fixed dimensionality, or requires access to multiple datasets that contain
the same treatment and outcome variables. However, in many clinical,
epidemiological and public policy applications, there exist external datasets
that examine the effect of different treatments on the same outcome variable,
or study the association between covariates and the outcome variable. These
external datasets cannot be used in conjunction with the aforementioned bounds,
since the former may entail different treatment assignment mechanisms, or even
obey different causal structures. Here, we provide symbolic bounds on the PoC
for this challenging scenario. We focus on combining either two randomized
experiments studying different treatments, or a randomized experiment and an
observational study, assuming causal sufficiency. Our symbolic bounds work for
arbitrary dimensionality of covariates and treatment, and we discuss the
conditions under which these bounds are tighter than existing bounds in
literature. Finally, our bounds parameterize the difference in treatment
assignment mechanism across datasets, allowing the mechanisms to vary across
datasets while still allowing causal information to be transferred from the
external dataset to the target dataset.
