Decision-making in complex systems often requires understanding counterfactuals of general, potentially highdimensional, interventions with limited data. Collecting sufficient data for every counterfactual in complex systems may be near impossible due to cost or ethical reasons. With the recent growth in expressivity and power in generative modeling, generative models that can synthesize counterfactual outcomes under generalized interventions stand as a viable solution for supporting robust decision-making in real-world systems. In an ideal world, we may simply train a generative model with the data we have, and sample from the generator under the intervention of interest. Counterfactual generative modeling may fail with such an approach due to confounding bias. Correlations observed in the sampled data may be mistaken for true causal effects, yielding incorrect downstream decisions. For example, generating medical images under changes in intervention dose can help track disease progression and identify optimal dosing strategies. However, if the training data primarily consisted of those who were responsive to intervention (e.g., younger populations), then the generator would identify the ranges in the data as effective even if this does not hold for different populations (e.g.

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.