Confusion about Appendix J

[adamrupe]

It is unclear why J.1 talks about interventions on $X^{\mathcal{R}}$; for example the last line has $q_i^{a|x} (a ; \mathrm{do}(x))$. Is this just a mistake / typo? Just below in J.2 puts the do operator on the full conditional $q_\star^{\mathcal{L}|\mathcal{R}}$ variable distribution. The conditional intervention $q_\star^{\mathcal{L}|\mathcal{R}}$ described in J.2 is in line with the rest of the paper, e.g., $q_\star^{a|x}$ as given on page 4. So it seems like the statements in J.1 are a mistake? For an unconditional intervention, $q^a_\star$, wouldn't $\mathcal{R}$ in (69) be empty? But then there would be an "intervention" on the empty set.

[djinnome]

It doesn't completely solve the mystery of equation (69), but if I wanted to generate an augmentation variable such that its only dependencies were on its direct subunit ancestors, then $q_i^{\mathcal{L}|\mathcal{R}} (\mathcal{L} ; \mathrm{do}(\mathcal{R}))$ would be consistent with that goal, since $\mathrm{do}(\mathcal{R})$ cuts off any dependencies between $\mathcal{L}$ and all its indirect subunit ancestors.