Layman's explanation for Finest Fully Randomized Causally Interpretable Structure Tree Graph (FFRCISTG) and NPSEM-IE I am reading Single World Intervention Graphs (SWIGs): A Unification of the Counterfactual and Graphical Approaches to Causality, and they describe both Finest Fully Randomized Causally Interpretable Structure Tree Graph (FFRCISTG) and Non-Parametric Structural Equation Models with Independent Errors
(NPSEM-IE), but feel I am struggling to understand. I was wondering if someone could explain these in layman's term, and how they improve upon the DAG? Is it just that now nodes are intervened upon and its expressed explicitly in the graph, while counterfactual distributions were not stated in DAGs?
 A: As for the difference between FFRCISTG and NPSEM-IE, they don't seem to be defined explicitly in the paper, and it is assumed that the reader is already familiar with them based on previous knowledge/reading. Specifically that you are familiar with at least either the work of Pearl or of Robins.
If you look at the first footnote, I think the point that they are making is that what Pearl calls "NPSEM", they think should be called "NPSEM-IE" because it makes an additional assumption about independent errors ("IE") that isn't really necessary to define a non-parametric structural equation model (NPSEM). I think they use the term FFRCISTG to refer to what they think Pearl should have called an NPSEM.
Basically I think a more introductory reference it would be useful to compare to is Pearl's 2009 Causality, cf. his website http://bayes.cs.ucla.edu/BOOK-2K/ , specifically section 7.1 (discussing non-parametric structural models). At least I personally found that the definitions given therein was overly vague, and I think one of the goals of the SWIGs paper is to be more explicit, especially about what Pearl calls "submodels" and what maybe corresponds to "single worlds" or "graph templates" in the SWIGs paper.
The 1997 paper Axioms of Causal Relevance by Pearl and his former student David Galles is also a useful comparison, it contains much of the same material as section 7.1 in Pearl's Causality but explains more.
With regards to "the DAG", I think (even setting aside interventions for a moment), another source of confusion is that there are two different DAGs, i.e. they are defined and constructed differently, and then they only turn out to be the same as the result of a theorem. Cf. the section "SWIGs for NPSEMs with FFRCISTG independence", p.16 in particular, of the SWIGs paper.
Basically, you have both

*

*a graph resulting from the structural equations connecting the variables ("NPSEM")

*a graph encoding (via the Markov blanket / d-separation criterion) conditional independencies of the (joint distribution of the) variables

and then the "factorization property" mentioned in the SWIGs paper is the theorem that these two graphs will be the same (at least if the first one is a DAG, and probably also given some other assumptions).
As far as I understand, Pearl's proposed graph formalism basically tries to reduce everything to computations on a single DAG, even though every possible intervention leads to a distinct NPSEM (and thus "morally" a distinct graph). Thus Pearl's formalism may be seen as "sweeping this under the rug".
The section "A new graphical view of the back-door formula" in the SWIGs paper seems to argue for example that this "sweeping under the rug" makes the back-door formula more complicated than it needs to be. They also appear to suggest in footnote 20 for a different reason that Pearl's approach is an oversimplification.
Obviously I don't fully understand the SWIGs paper either myself, but this answer is too long for a comment, and I thought that sharing my partial insights might help others along their way to achieving full insights.
