  We consider the problem of learning causal Directed Acyclic Graphs (DAGs)
using combinations of observational and interventional experimental data.
Current methods tailored to this setting assume that interventions either
destroy parent-child relations of the intervened (target) nodes or only alter
such relations without modifying the parent sets, even when the intervention
targets are unknown. We relax this assumption by proposing a Bayesian method
for causal discovery from general interventions, which allow for modifications
of the parent sets of the unknown targets. Even in this framework, DAGs and
general interventions may be identifiable only up to some equivalence classes.
We provide graphical characterizations of such interventional Markov
equivalence and devise compatible priors for Bayesian inference that guarantee
score equivalence of indistinguishable structures. We then develop a Markov
Chain Monte Carlo (MCMC) scheme to approximate the posterior distribution over
DAGs, intervention targets and induced parent sets. Finally, we evaluate the
proposed methodology on both simulated and real protein expression data.
