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Probabilities of Causation (PoC) play a fundamental role in decision-makingin law, health care and public policy. Nevertheless, their point identificationis challenging, requiring strong assumptions, in the absence of which onlybounds can be derived. Existing work to further tighten these bounds byleveraging extra information either provides numerical bounds, symbolic boundsfor fixed dimensionality, or requires access to multiple datasets that containthe same treatment and outcome variables. However, in many clinical,epidemiological and public policy applications, there exist external datasetsthat examine the effect of different treatments on the same outcome variable,or study the association between covariates and the outcome variable. Theseexternal datasets cannot be used in conjunction with the aforementioned bounds,since the former may entail different treatment assignment mechanisms, or evenobey different causal structures. Here, we provide symbolic bounds on the PoCfor this challenging scenario. We focus on combining either two randomizedexperiments studying different treatments, or a randomized experiment and anobservational study, assuming causal sufficiency. Our symbolic bounds work forarbitrary dimensionality of covariates and treatment, and we discuss theconditions under which these bounds are tighter than existing bounds inliterature. Finally, our bounds parameterize the difference in treatmentassignment mechanism across datasets, allowing the mechanisms to vary acrossdatasets while still allowing causal information to be transferred from theexternal dataset to the target dataset.

Tightening Bounds on Probabilities of Causation By Merging Datasets