Appendix Proof of Lemma 1 I provide a proof only for sharp bounds on P$_{1}$ ( y, 0 | z ). Sharp bounds on P$_{0}$ ( y, 1 | z ) are obtained similarly. The model (1) under M. 1 − M. 4 is uninformative about the counterfactual distribution term P [ Y$_{1}$ ≤ y | p < U ] . Therefore by plugging 0 and 1 into the term, bounds on P [ Y$_{1}$ ≤ y, 0 | z ] can be obtained as follows: where 33 P [ Y$_{1}$ ≤ y, 0 | z ] = P [ Y$_{1}$ ≤ y, p ( z ) < U ] = P [ Y$_{1}$ ≤ y, p ( z ) < U ≤ p ] + P [ Y$_{1}$ ≤ y, p < U ] = lim p ( z ) → p P ( y | 1 , z ) p − P ( y | 1 , z ) p ( z ) + P [ Y$_{1}$ ≤ y | p < U ] (1 − p ) . P [ Y$_{1}$ ≤ y, 0 | z ] ∈ $^{[}$L wst 10 ( y, z ) , U wst 10 ( y, z ) $^{]}$, L wst 10 ( y, z ) = lim p ( z ) → p P ( y | 1 , z ) p − P ( y | 1 , z ) p ( z ) , U wst 10 ( y, z ) = lim p ( z ) → p P ( y | 1 , z ) p − P ( y | 1 , z ) p ( z ) + 1 − p.