Continuity problem is that the 1⁄2 answer is independent of the ratio of expected number of wakings in the two branches of the experiment
Why is this a problem? I’m perfectly comfortable with that property. Since you really just have one random variable in each arm. You can call them different days of the week, but with no new information they are all just the same thing
By D do you mean W?
What’s happened is closer to E(H|D) = E(D|H) E(H) / E(D), over one run of the experiment, and this yields 1⁄3 immediately.
Is this how you came up with the 1⁄3 solution? If so, I think it requires more explanation. Such as what D is precisely.
Continuity problem is that the 1⁄2 answer is independent of the ratio of expected number of wakings in the two branches of the experiment
Why is this a problem?
The next clause of the sentence is the problem
unless the ratio is 0 (or infinite) at which point special case logic is invoked to prevent the trivially absurd claim that credence of Heads is 1⁄2 when you are never woken under Heads.
The problem is special casing out the absurdity, and thus getting credences that are discontinuous in the ratio. On the other hand, you seem to take 1/21in PSB (ie you do let it depend on the ratio) but deviate from 1⁄21 when multiple runs of PSB aggregate, which is not what I had expected...
D was used in the comment I was replying to as an “event” that was studiously avoiding being W.
The problem as I see it with W is that it’s not a set of outcomes, it’s really a multiset. That’s fine in it’s way, but it gets confusing because it no longer bounds probabilities to [0,1]. Your approach is to quash multiple membership to get a set back.
Why is this a problem? I’m perfectly comfortable with that property. Since you really just have one random variable in each arm. You can call them different days of the week, but with no new information they are all just the same thing
By D do you mean W?
Is this how you came up with the 1⁄3 solution? If so, I think it requires more explanation. Such as what D is precisely.
The next clause of the sentence is the problem
The problem is special casing out the absurdity, and thus getting credences that are discontinuous in the ratio. On the other hand, you seem to take 1/21in PSB (ie you do let it depend on the ratio) but deviate from 1⁄21 when multiple runs of PSB aggregate, which is not what I had expected...
D was used in the comment I was replying to as an “event” that was studiously avoiding being W.
http://lesswrong.com/lw/28u/conditioning_on_observers/201l shows multiple ways I get the 1/3 solution; alternatively betting odds taken on awakening or the long run frequentist probability, they all cohere, and yield 1/3.
The problem as I see it with W is that it’s not a set of outcomes, it’s really a multiset. That’s fine in it’s way, but it gets confusing because it no longer bounds probabilities to [0,1]. Your approach is to quash multiple membership to get a set back.