You have evidence E and possible explanations A and B. A implies payoff 1 soon, and B implies payoff 0.
When you have utility calculation, estimated p(A) gets multiplied by value of A, estimated p(B) gets multiplied by value of B and you care about difference between to estimations.
Why difference? Well, you should be very happy that you are a human with cognitive capabilities and not a vivisected frog. Or you could be very unhappy that you are not guaranteed to be immortal. Adding either of these large constants to evaluations of all outcomes doesn’t change the difference of two possible situations here and now.
So you calculate p(A)v(A)+p(B)v(B), and if you take a bit of p(A) and give it to p(B) you get Δp(A)(v(A)-v(B)) change.
If you talk about credence, it is another story. You can relatively easily calculate p(E|A) and p(E|B). It is hard to find out p(A) and p(B). Bayesian approach hints that maybe you should just try to find a lot of independent evidence pieces so that their voice can drown the prior difference. But how is evidence summarized? You want to find (prior(A)*p(E|A))/(p(E|A)+p(E|B)) = prior(A)/(1+p(E|B)/p(E|A)). There is ratio of p(E|A) and p(E|B), so you need to look at Δ(p(E|A)/p(E|B)), and this depends on absolute values of probabilities in addition to change.
Ah, ok. For EU you care about the actual difference in probability, whereas for strength of evidence you care about the amount of information you would have to receive in order to change your credence that much, and this is compressed near the ends of the probability range.
Edit: What was your question again? At first I thought that you did not understand what you were talking about, and thought that your answer re the context would be a link or quote where somebody else brought it up. But what you gave me is almost a set of proofs, and the explanation is clear from your use of dB in the evidence/information case.
I was aware of the difference (what you note in your first paragraph), but could not articulate a good explanation of why there is a difference. I found this comment by vi21maobk9vp to be a good explanation, though I feel it could be more clear and concise. I’m not actually sure I could have given you the answer in the grandparent in so few words before reading vi21maobk9vp’s comment.
Can someone concisely explain why this is true:
for expected utility, the difference between 50% and 51% is the same as the difference between 80% and 81%.
for credence, the difference between 50% and 51% is much smaller than the difference between 80% and 81%.
It is measuring different things.
You have evidence E and possible explanations A and B. A implies payoff 1 soon, and B implies payoff 0.
When you have utility calculation, estimated p(A) gets multiplied by value of A, estimated p(B) gets multiplied by value of B and you care about difference between to estimations.
Why difference? Well, you should be very happy that you are a human with cognitive capabilities and not a vivisected frog. Or you could be very unhappy that you are not guaranteed to be immortal. Adding either of these large constants to evaluations of all outcomes doesn’t change the difference of two possible situations here and now.
So you calculate p(A)v(A)+p(B)v(B), and if you take a bit of p(A) and give it to p(B) you get Δp(A)(v(A)-v(B)) change.
If you talk about credence, it is another story. You can relatively easily calculate p(E|A) and p(E|B). It is hard to find out p(A) and p(B). Bayesian approach hints that maybe you should just try to find a lot of independent evidence pieces so that their voice can drown the prior difference. But how is evidence summarized? You want to find (prior(A)*p(E|A))/(p(E|A)+p(E|B)) = prior(A)/(1+p(E|B)/p(E|A)). There is ratio of p(E|A) and p(E|B), so you need to look at Δ(p(E|A)/p(E|B)), and this depends on absolute values of probabilities in addition to change.
Thanks!
I wonder if it would be worth having a Less Wrong Q&A site, or if that purpose is better served by existing Q&A sites.
Anyone?
What is the context of these claims? I don’t really see why either of them would be necessary.
They are necessary.
For expected utility,
.51v-.50v ==.01 v == .81v-.80v
When evaluating the strength of evidence,
50%
is0dB
,51%
is0.17dB
(diff0.17dB
),80%
is6.02dB
, and81%
is6.30dB
(diff0.28dB
).Please offer a correction if I’ve made a mistake there.
Ah, ok. For EU you care about the actual difference in probability, whereas for strength of evidence you care about the amount of information you would have to receive in order to change your credence that much, and this is compressed near the ends of the probability range.
Edit: What was your question again? At first I thought that you did not understand what you were talking about, and thought that your answer re the context would be a link or quote where somebody else brought it up. But what you gave me is almost a set of proofs, and the explanation is clear from your use of dB in the evidence/information case.
I was aware of the difference (what you note in your first paragraph), but could not articulate a good explanation of why there is a difference. I found this comment by vi21maobk9vp to be a good explanation, though I feel it could be more clear and concise. I’m not actually sure I could have given you the answer in the grandparent in so few words before reading vi21maobk9vp’s comment.