Partial identification bounding methods use two steps:
Step 1: Starting from the expression of interest, unobserved quantities are first decomposed using the Law of Total Probability to introduce quantities observed in the data and reduce the role of the remaining unobserved quantities.
Step 2: One uses well-founded assumptions that allow to further bound the unobserved quantities that have shown up during the decomposition.
In the following, these steps are applied to the ATE of SNAP on health outcomes.

Abstracting from Measurement Error

Because of binary outcome variables the formula for the ATE is simplified: