  Within the field of causal inference, it is desirable to learn the structure
of causal relationships holding between a system of variables from the
correlations that these variables exhibit; a sub-problem of which is to certify
whether or not a given causal hypothesis is compatible with the observed
correlations. A particularly challenging setting for assessing causal
compatibility is in the presence of partial information; i.e. when some of the
variables are hidden/latent. This paper introduces the possible worlds
framework as a method for deciding causal compatibility in this difficult
setting. We define a graphical object called a possible worlds diagram, which
compactly depicts the set of all possible observations. From this construction,
we demonstrate explicitly, using several examples, how to prove causal
incompatibility. In fact, we use these constructions to prove causal
incompatibility where no other techniques have been able to. Moreover, we prove
that the possible worlds framework can be adapted to provide a complete
solution to the possibilistic causal compatibility problem. Even more, we also
discuss how to exploit graphical symmetries and cross-world consistency
constraints in order to implement a hierarchy of necessary compatibility tests
that we prove converges to sufficiency.
