I side with you on this issue. It irks me all the time when the Bayesian foundations are vaguely criticized with an air of superiority, as if dismissing them is a sign of having transcended to some higher level of existence (neorationalists, I’m looking at you). On the other hand, I could accept tool-boxing, in accordance to the principle of “one truth, many methods to find it” if and only if:
it effectively showed better results than the Bayesian methods
it wouldn’t suddenly forget the pluri-decennial findings on the fallibility of human intuitions.
On the other hand:
Should you treat “evidence” for a hypothesis, or “data”, as having probability 1?
That point was mostly referring to when you perform the “Bayesian update”, the rule you use can be either strict conditionalization (P(H) = P(H|E)), which assumes P(E) = 1, or Jeffreys’ conditionalization, (P(H) = P(H|E)P(E) + P(H|~E)P(~E)). The latter seems to be the most intuitively correct rule, but I guess there are some subtle issues with using that rule that I need to dive deeper into to really understand.
The latter seems to be the most intuitively correct rule
So if I extract an red ball from an urn, should I condition the probability of finding a black ball in the next turn on not having extracted a red ball?
Besides, P(H) is most definitely not equal to P(H|E). P(H) is on the other hand demonstrably equal to P(H|E)P(E)+P(H|-E)P(-E), the usual decomposition of unity. I think we are talking about two completely different things here.
I’m talking about the following issue, found at this link:
A. The problem of uncertain evidence. The Simple Principle of Conditionalization requires that the acquisition of evidence be representable as changing one’s degree of belief in a statement E to one — that is, to certainty. But many philosophers would object to assigning probability of one to any contingent statement, even an evidential statement, because, for example, it is well-known that scientists sometimes give up previously accepted evidence. Jeffrey has proposed a generalization of the Principle of Conditionalization that yields that principle as a special case. Jeffrey’s idea is that what is crucial about observation is not that it yields certainty, but that it generates a non-inferential change in the probability of an evidential statement E and its negation ~E (assumed to be the locus of all the non-inferential changes in probability) from initial probabilities between zero and one to Pf(E) and Pf(~E) = [1 − Pf(E)]. Then on Jeffrey’s account, after the observation, the rational degree of belief to place in an hypothesis H would be given by the following principle:
Principle of Jeffrey Conditionalization:
Pf(H) = Pi(H/E) × Pf(E) + Pi(H/~E) × Pf(~E) [where E and H are both assumed to have prior probabilities between zero and one]
Counting in favor of Jeffrey’s Principle is its theoretical elegance. Counting against it is the practical problem that it requires that one be able to completely specify the direct non-inferential effects of an observation, something it is doubtful that anyone has ever done. Skyrms has given it a Dutch Book defense.
I side with you on this issue. It irks me all the time when the Bayesian foundations are vaguely criticized with an air of superiority, as if dismissing them is a sign of having transcended to some higher level of existence (neorationalists, I’m looking at you).
On the other hand, I could accept tool-boxing, in accordance to the principle of “one truth, many methods to find it” if and only if:
it effectively showed better results than the Bayesian methods
it wouldn’t suddenly forget the pluri-decennial findings on the fallibility of human intuitions.
On the other hand:
This is provably true: P(X|X) = 1.
P(X) = P(X /\ X) = P(X|X)P(X) ⇔ P(X|X) = 1.
That point was mostly referring to when you perform the “Bayesian update”, the rule you use can be either strict conditionalization (P(H) = P(H|E)), which assumes P(E) = 1, or Jeffreys’ conditionalization, (P(H) = P(H|E)P(E) + P(H|~E)P(~E)). The latter seems to be the most intuitively correct rule, but I guess there are some subtle issues with using that rule that I need to dive deeper into to really understand.
So if I extract an red ball from an urn, should I condition the probability of finding a black ball in the next turn on not having extracted a red ball?
Besides, P(H) is most definitely not equal to P(H|E). P(H) is on the other hand demonstrably equal to P(H|E)P(E)+P(H|-E)P(-E), the usual decomposition of unity. I think we are talking about two completely different things here.
I’m talking about the following issue, found at this link: