  Probabilities of Causation play a fundamental role in decision making in law,
health care and public policy. Nevertheless, their point identification is
challenging, requiring strong assumptions such as monotonicity. In the absence
of such assumptions, existing work requires multiple observations of datasets
that contain the same treatment and outcome variables, in order to establish
bounds on these probabilities. However, in many clinical trials and public
policy evaluation cases, there exist independent datasets that examine the
effect of a different treatment each on the same outcome variable. Here, we
outline how to significantly tighten existing bounds on the probabilities of
causation, by imposing counterfactual consistency between SCMs constructed from
such independent datasets ('causal marginal problem'). Next, we describe a new
information theoretic approach on falsification of counterfactual
probabilities, using conditional mutual information to quantify counterfactual
influence. The latter generalises to arbitrary discrete variables and number of
treatments, and renders the causal marginal problem more interpretable. Since
the question of 'tight enough' is left to the user, we provide an additional
method of inference when the bounds are unsatisfactory: A maximum entropy based
method that defines a metric for the space of plausible SCMs and proposes the
entropy maximising SCM for inferring counterfactuals in the absence of more
information.
