Positivity violation in Judea Pearl's Smoking -> Tar -> Lung Cancer front-door adjustment example: P(tar|no smoking) = 0? I'm reading Judea Pearl's smoking and lung cancer example illustrating the front-door criterion for counterfactual analysis (p. 225 from Pearls' The Book of Why: The New Science of Cause and Effect).

One of the assumptions is the mediator variable measuring tar deposits in subject's lungs is exclusively caused by smoking and no other factors.
This means $P(\text{tar}|\text{no smoking}) = 0$. Wouldn't this constitute a positivity violation, making front-door adjustment impossible?
The same issue arises in front-door adjustment healthcare models where we have Healthcare intervention -> Skill/experience of nurse or doctor -> Health outcome. Those subjects not participating do not receive care from a nurse or doctor. The Skill/experience of nurse or doctor effectively equals 0 for the entire control group.
 A: As you've stated it, yes. However, I highly doubt that non-smokers would have zero tar deposits in their lungs, particularly on average, given the presence of second-hand smoke, possibly wood fires, etc. Keep in mind that tar deposits is going to be a continuous variable, and for any continuous variable $Z,$ $P(Z=z)=0.$ In Pearl, Glymour, and Jewell's book Causal Inference in Statistics: A Primer, pp. 66ff., the authors go into a bit more detail of exactly that scenario. The sham data they use has far lower numbers for the tar deposits for non-smokers, but they're not identically zero.
You are quite correct in your healthcare scenario. What that means is that you cannot use the frontdoor criteria as-is. You would need to change your population, or your variables, or your whole approach (perhaps backdoor would work, here?). What's different in the healthcare scenario is that Healthcare Intervention is not a completely continuous variable, so that the probability of it being zero for certain subgroups can be identically zero.
