Then just suppose there are some medical diagnostics with these probabilities, etc. But see below for a comparison of this with your coin scenario.
More generally, let a=P(A)=1/4, b=P(B), c=P(C), x=P(BC), y=P(AC), z=P(AB), t=P(ABC).
The “before and after independence” assumption [BAI] is that x=bc and at=yz
The “independent tests” assumption [IT] is that (x-t)=(b-z)(c-y) and at=yz.
(2) How the “independent tests” assumptions are “stable”?
As I stated earlier, the condition A or the condition ~A screens off P(A|X).
Neither [BAI] nor [IT] is stable with respect to the variable a=P(A), and I see no equation involving anything I’d call “screening off”, though there might be one somewhere.
In any case, that does not interest me, because your equation (7) has my attention:
A qualitative distinction between my would-be medical scenario [EG1] and your coin scenario is that medical diagnostics, and particularly their dependence/independence, are not always “causally justified”, but two coin flips can be seen as independent because they visibly just don’t interact.
I bet your (7) gives a good description of this somehow, or something closely related… But I still have to formalize my thoughts about this.
Let me think about it awhile, and I’ll post again if I understand or wonder something more precise.
Then just suppose there are some medical diagnostics with these probabilities, etc.
You just glossed over the whole point of this exercise. The problem is that values such as P(ABC) are combined facts about the population and both tests. Try defining the scenario using only facts about the population in isolation (P(A)), about the tests in isolation (P(B|A), P(C|A), P(B|~A), P(C|~A)), and the dependence between the tests (P(B|AC), P(B|A~C), P(B|~AC), P(B|~A~C), [EDIT: removed redundant terms, I got carried away when typing out the permutations]). The point is to demonstrate how you have to contrive certain properties of the dependence between the tests, the conditional probabilities given ~A, to make summary properties you care about work out for a specific separate fact about the population, P(A).
Here’s a simple example where pev(A,BC)=pev(A,B)pev(A,C):
[EG1] P(A)=1/4, P(B)=P(C)=1/2, P(BC)=1/4, P(AC)=P(BC)=1/16, P(ABC)=1/64.
Then just suppose there are some medical diagnostics with these probabilities, etc. But see below for a comparison of this with your coin scenario.
More generally, let a=P(A)=1/4, b=P(B), c=P(C), x=P(BC), y=P(AC), z=P(AB), t=P(ABC).
The “before and after independence” assumption [BAI] is that x=bc and at=yz
The “independent tests” assumption [IT] is that (x-t)=(b-z)(c-y) and at=yz.
Neither [BAI] nor [IT] is stable with respect to the variable a=P(A), and I see no equation involving anything I’d call “screening off”, though there might be one somewhere.
In any case, that does not interest me, because your equation (7) has my attention:
A qualitative distinction between my would-be medical scenario [EG1] and your coin scenario is that medical diagnostics, and particularly their dependence/independence, are not always “causally justified”, but two coin flips can be seen as independent because they visibly just don’t interact.
I bet your (7) gives a good description of this somehow, or something closely related… But I still have to formalize my thoughts about this.
Let me think about it awhile, and I’ll post again if I understand or wonder something more precise.
I assume you meant:
[EG1] P(A)=1/4, P(B)=P(C)=1/2, P(BC)=1/4, P(AC)=P(AB)=1/16, P(ABC)=1/64.
You just glossed over the whole point of this exercise. The problem is that values such as P(ABC) are combined facts about the population and both tests. Try defining the scenario using only facts about the population in isolation (P(A)), about the tests in isolation (P(B|A), P(C|A), P(B|~A), P(C|~A)), and the dependence between the tests (P(B|AC), P(B|A~C), P(B|~AC), P(B|~A~C), [EDIT: removed redundant terms, I got carried away when typing out the permutations]). The point is to demonstrate how you have to contrive certain properties of the dependence between the tests, the conditional probabilities given ~A, to make summary properties you care about work out for a specific separate fact about the population, P(A).