If lottery tickets were bought after paying off debts and after loss of government benefits, no one who was in debt, or who was receiving government benefits, could buy lottery tickets. Unless I misunderstand.
A->B, C->B, C steals the evidence that C provides for A. You need to provide new D with A->D, C!->B. Where the implication from C to B is imperfect then B goes on providing some trickle of evidence to A but if the implications are equally strong then the trickle does not distinguish between A and C as opposed to other hypotheses and the prior odds win out.
I tried to explain in my previous comment why I think this is the wrong way of looking at it. You’re speaking as if B is a proposition with a truth-value that has a single cause. However, I think my explanation was not quite right either.
The weakest, most obviously true reply is that this is not a Boolean net; B does not have a single cause; and A ⇒ B and C ⇒ B can both be having an effect. It’s even possible, in the real-valued non-Boolean world, to have (remember this is not Boolean; this is more like a metabolic network) A > 0, C > 0, A ⇒ B, C ⇒ B, B < 0.
A reply that is a little stronger ( = has more consequences), and a little less clearly correct, is that your argument for C ⇒ B is not as good as my argument for A ⇒ B, so who’s stealing whose evidence?
The strongest, least-clear reply is that we have priors in favor of both A ⇒ B and C ⇒ B. Because they’re both just-so stories, and we have no quantitative expectations of how much of an increase in B either would provide; and, unlike when B is a truth-value, there’s no upper limit on how large B can get; A or C can’t steal much evidence from each other without some quantitative prediction. All the info you have is that A and C would both make B > 0, and B > 0. If C accounts for x points of B, and B = x + y, then this knowledge can increase the probability of A. C, C ⇒ B diminishes the probability of A in the absence of knowledge about the value of B and the value of B explained by C, but by so little compared to the priors, that presenting it as an argument against an argument from principles is misleading.
If lottery tickets were bought after paying off debts and after loss of government benefits, no one who was in debt, or who was receiving government benefits, could buy lottery tickets. Unless I misunderstand.
I tried to explain in my previous comment why I think this is the wrong way of looking at it. You’re speaking as if B is a proposition with a truth-value that has a single cause. However, I think my explanation was not quite right either.
The weakest, most obviously true reply is that this is not a Boolean net; B does not have a single cause; and A ⇒ B and C ⇒ B can both be having an effect. It’s even possible, in the real-valued non-Boolean world, to have (remember this is not Boolean; this is more like a metabolic network) A > 0, C > 0, A ⇒ B, C ⇒ B, B < 0.
A reply that is a little stronger ( = has more consequences), and a little less clearly correct, is that your argument for C ⇒ B is not as good as my argument for A ⇒ B, so who’s stealing whose evidence?
The strongest, least-clear reply is that we have priors in favor of both A ⇒ B and C ⇒ B. Because they’re both just-so stories, and we have no quantitative expectations of how much of an increase in B either would provide; and, unlike when B is a truth-value, there’s no upper limit on how large B can get; A or C can’t steal much evidence from each other without some quantitative prediction. All the info you have is that A and C would both make B > 0, and B > 0. If C accounts for x points of B, and B = x + y, then this knowledge can increase the probability of A. C, C ⇒ B diminishes the probability of A in the absence of knowledge about the value of B and the value of B explained by C, but by so little compared to the priors, that presenting it as an argument against an argument from principles is misleading.