Even in the easiest cases, there are generally a whole host of particular assumptions necessary to make causal inferences about anything. But normally, we think that the benefit of experiments is that they help us to isolate causal effects by manipulating one thing at a time. That is a key strength of experiments: regardless of the underlying causal structure, by manipulating one variable we can nevertheless isolate causal effects.

Well, take a look at this:

in order to identify the causal structure by single-intervention experiments some additional parametric assumption beyond Markov, faithfulness and acyclicity is necessary. Alternatively, without additional assumptions, causal discovery requires a large set of very demanding experiments, each intervening on a large number of variables simultaneously.

Emphasis mine. This is from a fascinating short paper entitled “Experimental Indistinguishability of Causal Structures” by Frederick Eberhardt.

I’m still working through the implications. My initial thought is that for the narrowest interpretation of experiments, where we define the thing that we care about as the ATE, and the ATE as the difference between treatment and control, this critique does not much matter. We have defined the object of interest as whatever happens when we intervene.

But if we really believe that the goal of experiments is to intervene in order to learn about the causal structure of the world more generally—and I have to believe that that is really the goal that justifies experimentation—then the object of interest is the relationship between “whatever happens when we intervene” and “causal structure of the world.” In that case, the implications are profound for our ability to learn about that relationship using experiments.