Sue's reasoning is right for Sue, but wrong for Jane (and vice versa), given that 2/3 of sitters will sit in on a 100 Heads interview but 1/2^100 of participants will have a 100 Heads interview. — Michael
So if heads then woken once on Monday and twice on Tuesday, otherwise woken twice on Monday and once on Tuesday.
Sue tells Jane that it's Monday.
What is Jane's credence that the coin landed heads?
I say 1/2.
It's exactly the same reasoning as before.
Sue should reason as if she is randomly selected from the set of all sitters, and 1/3 of sitters sitting in a Monday room are sitting in a heads room.
Jane should reason as it she is randomly selected from the set of all participants, and 1/2 of participants in a Monday room are sitting in a heads room. — Michael
Although you linked to my most recent post, I assume you intended to respond to this one. — Pierre-Normand
No, I was just trying to rephrase your secret mission example into a way that I could understand better. Did I misinterpret it? — Michael
I would argue that Jane should update her credence in the same way in light of the same information. — Pierre-Normand
Was my rephrasing of it wrong? I'm treating DZ#1 as Monday and DZ#2 as Tuesday. If twice at DZ#1 then twice on Monday, if once at DZ#2 then once on Tuesday. If you know what it's DZ#1 then you know that it's Monday. — Michael
Jane shouldn't update her credence to match Sue and Sue shouldn't update her credence to match Jane. — Michael
But we are agreed on the validity of Sue's credences in both scenarios, right? — Pierre-Normand
Neither the sitter nor the participant should update their credence to match the other's. — Michael
Isn't this problem a consequence of asking the wrong question, because the question is ambiguous? Given the setup, and that you'd be woken twice if it is tails, what's the likelihood now that you're awake, that the coin shows tails? That seems to resolve the ambiguity or am I missing something? — Benkei
Since you're still a committed Thirder, here's what bothers me:
(1) When a coin is tossed it has only two proper outcomes. (No leaners, no edges, and the usual ceteris paribus.) SB, upon being awakened, finds herself in one of three possible situations. But that doesn't work. A coin toss cannot, by itself, engender three distinct situations for her to be in. For instance, "for all she knows," this is her second tails interview; but there is no way to go from the coin toss directly to the second tails interview. Not without passing through the first interview. So SB's intuition that three things can happen to her requires there to be some other determinant of her situation; what is that other factor? — Srap Tasmaner
(2) The setup confounds wagering arguments. That won't matter much to a lot of people, but it's uncomfortable. Annoying. Ramsey used Dutch book arguments from the beginning, and despite their limitations they can be clarifying. Each time I've tried to construct a sane payoff table I've failed. I've wondered lately if there might be a conditional wager that comes out rational, but I can work up enough hope of success to bother. Partial beliefs, within suitable limits, ought to be expressible as wagers, but not in this case, and that blows.
I still think it's a language issue and it revolves around the verb "shows". I agree with Srap it doesn't make sense to pretend a coin has three states. The coin can't "show" three options but the "coin when shown to SB" can and these are simply different things. It's just that mathematicians only have a hammer that makes this look like a nail. — Benkei
The original statement of the problem fails to specify what constitutes an individual act of verification of her credence, though, such that we can establish the target ratio unambiguously. As I've previously illustrated with various examples, different pragmatic considerations can lead to different verification methods, each yielding different values for P(H), aligning with either the Halfer or Thirder stance. — Pierre-Normand
The precise effect of the drug is to reset your belief-state to what it was just before you were put to sleep at the beginning of the experiment. If the existence of such a drug seems fanciful, note that it is possible to pose the problem without it — all that matters is that the person put to sleep believes that the setup is as I have described it.
If you ask, what's the likelihood you will see Heads if I bring you the coin, then the number of times she could have been woken up becomes relevant. — Benkei
(2) The setup confounds wagering arguments. That won't matter much to a lot of people, but it's uncomfortable. Annoying. Ramsey used Dutch book arguments from the beginning, and despite their limitations they can be clarifying. Each time I've tried to construct a sane payoff table I've failed. I've wondered lately if there might be a conditional wager that comes out rational, but I can work up enough hope of success to bother. Partial beliefs, within suitable limits, ought to be expressible as wagers, but not in this case, and that blows. — Srap Tasmaner
The problem (and Elga's solution) have nothing to do with how to "verify" one's credence. It simply asks what a rational person should/would believe were they told the rules of the experiment, woken up, and asked their credence. — Michael
To ask how probable it is that the coin landed on heads involves a tacit reference to the counterfactual circumstances where you are presently facing a (hidden) coin that didn't land the way it actually did. — Pierre-Normand
When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?
...
I've just argued that when you are awakened on Monday, that credence ought to change to 1/3.
...
But you were also certain that upon being awakened on Monday you would have credence 1/3 in H.
Not exactly, because if it's Monday the coin hasn't been flipped at all. It's only hidden if today is Tuesday and the coin is tails. — Michael
This is an ambiguous claim. It is true that if you randomly select a seeing from the set of all possible seeings then it is twice as likely to be a tails-seeing, but the experiment doesn't work by randomly selecting a seeing from the set of all possible seeings and then "giving" it to Sleepy Beauty. It works by tossing a coin, and then either she sees it once or she sees it twice. — Michael
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