How to determine interventional distributions from observational data? How do we compute/query interventional distributions from observational data (i.e. without knowledge of the causal graph such as a Structural Causal Model (SCM))?
 A: In general, observational data is not sufficient to obtain the interventional distributions. You will "only" obtain the Markov equivalence class (e.g. with the pcalg R-package). E.g., think of two random variables $A$ and $B$, where your observation consists only of an $n\times 2$ table of $n$ observations of the pairs $(a_i, b_i)$. This table can only tell you (approximately) whether they are dependent or not. It doesn't contain any information about whether $A$ causes $B$, i.e. $A\to B$, or $B$ causes $A$, i.e. $A\leftarrow B$.
However, those equivalence classes are often not too large (see the linked paper above and this paper, section 3), so this can already contain lots of helpful information.
The standard methods of getting (some of) the interventional distributions via adjustment sets (e.g. the backdoor method) all presume some extra knowledge such as that the treatment does cause the outcome and the knowledge of all the confounders. But that information is often available, at least approximately.
Another approach is that of additive noise models: it often suffices to have only a little bit of qualitative information about the distributions to be able to identify the causal model and the interventional distributions.
Finally, also addressing the comments to the question, note that there is nothing that the method of potential outcomes can do that Pearl's causal graph approach cannot, and vice versa. It is just that some tasks are sometimes easier to tackle with one method than with the other; it is also often a question of subjective taste. For instance, the backdoor method is part of both approaches, and both deal with counterfactuals, too.
