There is no problem with what the doctor is doing. The doctor is trying to minimize the number of deaths given that (s)he measures C, as you said.
The question is, how do we quantify what the effect of medicine A on death is? In other words, how do you answer the question “does medicine A help or hurt?” given that you know p(Y,A,C). This is where you don’t want to use p(Y | A). This is because sicker people will die more and get medicine more, hence you might be mislead into thinking that giving people A increases death risk.
Often, but not always (one common issue is the size of C can be very large). Even if you measure all the symptoms, and are interested in the effect of the medicine conditional on these symptoms (what they call “effect modification” in epidemiology) there is the question of confounders you did not measure that would prevent p(Y | A, C) from being equal to the effect you want, which is p(Y | do(A), C).
I don’t get the relevance of this p(Y|A).
In EDT, you contition on both action and observations.
In this case, the EDT doctor prescribes argmin_A p(Y | A & C)
What’s the problem with that?
There is no problem with what the doctor is doing. The doctor is trying to minimize the number of deaths given that (s)he measures C, as you said.
The question is, how do we quantify what the effect of medicine A on death is? In other words, how do you answer the question “does medicine A help or hurt?” given that you know p(Y,A,C). This is where you don’t want to use p(Y | A). This is because sicker people will die more and get medicine more, hence you might be mislead into thinking that giving people A increases death risk.
Only if you ignore the symptoms.
In medicine you want answer questions of the type “given symptoms C, does medicine A help or hurt?”
Often, but not always (one common issue is the size of C can be very large). Even if you measure all the symptoms, and are interested in the effect of the medicine conditional on these symptoms (what they call “effect modification” in epidemiology) there is the question of confounders you did not measure that would prevent p(Y | A, C) from being equal to the effect you want, which is p(Y | do(A), C).
I suppose that’s what randomized trials are for.
Or you can read my dissertation if you want to answer these types of questions but can’t randomize :).