Alexander analyzes the difference between p1 and p2 in terms of the famous “explaining away” effect. Alexander supposes that p2 has learned some “causes”:
The reason is that the probability mass provided by examples x₁, …, xₙ such that ϕ(xᵢ) holds is now distributed among the universal statement ∀x:ϕ(x) and additional causes Cⱼ known to the more powerful agent that also imply ϕ(xᵢ). Consequently, ∀x:ϕ(x) becomes less “necessary” and has less relative explanatory power for the more informed agent.
An implication of this perspective is that if the weaker agent learns about the additional causes Cⱼ, it should also lower its credence in ∀x:ϕ(x).
Postulating these causes adds something to the scenario. One possible view is that Alexander is correct so far as Alexander’s argument goes, but incorrect if there are no such Cj to consider.
However, I do not find myself endorsing Alexander’s argument even that far.
If C1 and C2 have a common form, or are correlated in some way—so there is an explanation which tells us why the first two sentences, ϕ(x1) and ϕ(x2), are true, and which does not apply to n>2 -- then I agree with Alexander’s argument.
If C1 and C2 are uncorrelated, then it starts to look like a coincidence. If I find a similarly uncorrelated C3 for ϕ(x3), C4 for ϕ(x4), and a few more, then it will feel positively unexplained. Although each explanation is individually satisfying, nowhere do I have an explanation of why all of them are turning up true.
I think the probability of the universal sentence should go up at this point.
So, what about my “conditional probabilities also change” variant of Alexander’s argument? We might intuitively think that ϕ(x1) and ϕ(x2) should be evidence for the universal generalization, but p2does not believe this—its conditional probabilities indicate otherwise.
I find this ultimately unconvincing because the point of Paul’s example, in my view, is that more accurate priors do not imply more accurate posteriors. I still want to understand what conditions can lead to this (including whether it is true for all notions of “accuracy” satisfying some reasonable assumptions EG proper scoring rules).
Another reason I find it unconvincing is because even if we accepted this answer for the paradox of ignorance, I think it is not at all convincing for the problem of old evidence.
What is the ‘problem’ in the problem of old evidence?
(continued..)
Explanations?
Alexander analyzes the difference between p1 and p2 in terms of the famous “explaining away” effect. Alexander supposes that p2 has learned some “causes”:
Postulating these causes adds something to the scenario. One possible view is that Alexander is correct so far as Alexander’s argument goes, but incorrect if there are no such Cj to consider.
However, I do not find myself endorsing Alexander’s argument even that far.
If C1 and C2 have a common form, or are correlated in some way—so there is an explanation which tells us why the first two sentences, ϕ(x1) and ϕ(x2), are true, and which does not apply to n>2 -- then I agree with Alexander’s argument.
If C1 and C2 are uncorrelated, then it starts to look like a coincidence. If I find a similarly uncorrelated C3 for ϕ(x3), C4 for ϕ(x4), and a few more, then it will feel positively unexplained. Although each explanation is individually satisfying, nowhere do I have an explanation of why all of them are turning up true.
I think the probability of the universal sentence should go up at this point.
So, what about my “conditional probabilities also change” variant of Alexander’s argument? We might intuitively think that ϕ(x1) and ϕ(x2) should be evidence for the universal generalization, but p2 does not believe this—its conditional probabilities indicate otherwise.
I find this ultimately unconvincing because the point of Paul’s example, in my view, is that more accurate priors do not imply more accurate posteriors. I still want to understand what conditions can lead to this (including whether it is true for all notions of “accuracy” satisfying some reasonable assumptions EG proper scoring rules).
Another reason I find it unconvincing is because even if we accepted this answer for the paradox of ignorance, I think it is not at all convincing for the problem of old evidence.
What is the ‘problem’ in the problem of old evidence?
… to be further expanded later …