detecting bias in causal ML models given varying definitions of treatment

I am evaluating a policy change where full information is replaced by partial information and I want to understand whether or not it adversely affects my causal models. Suppose that a firm sends marketing communications to customers via smartphone and in past years if the customers engage with those communications, event data is sent back to the firm in all regions. However, now in some regions, the engagement data is not sent back to the firm. Consider the following variable definitions: $a_{i, t}$ : action(s) taken by customer i in period t $d_{i, t}$ : communications delivered to customer i in period t $c_{i, t}$ : engagement (clicks) by customer i in period t $X_{i,t}$ : salient characteristics (covariates) of customer i in period t The full signal model: $P(a|c,d,X)$ partial signal model: $P(a|d,X)$ Clearly, the missing variable is the click-through rate, $\pi = P(c|d)$ . And so, I can construct an "apples-to-apples" of regions treated (delivery information only) to untreated regions (click information available) by adjusting for the CTR. The high level idea is that the full signal effect estimate divided by CTR equals the partial signal effect estimate. More precisely: $\hat{a} = \frac{P(a|c,d,X)}{\pi} = P(a|d,X)$ And so I might use a relatively simple causal model, like inverse propensity weighting (IPW) to adjust the effect estimate for treatment propensity, conditioned on $X$ . Does the customer background predict a treated region. If no effect exists then $\frac{P(…