- I found Sec 5.1 and 5.2 difficult to read and I think clarity can be improved. What confused me initially was that you suggest fixing $P^*(U|x, a)$ but then the $\sup$ in Eq. 5 is also over the distributions $p(u|x, A)$. Reading it further, the sup is only for $A \neq a$ but I think clarifying that you only fix for the treatment $a$ that enters into $Q$ would be useful. Maybe this is obvious, but it will still make it easier to understand what is being optimized over in the $\sup$.
- It would also be nice to have some intuition of the proof of Theorem 1. Also, the invertible function $f^*$ would depend on the fixed $P^*$. Does certain distributions $P^*$ make it easier to determine $f^*$. In practice, how should you determine which $P^*$ to fix?
- It would also be nice to have some intuition of the proof of Theorem 1. Also, the invertible function $f^*$ would depend on the fixed $P^*$. Does certain distributions $P^*$ make it easier to determine $f^*$. In practice, how should you determine which $P^*$ to fix?