Talk:Sleeping Beauty problem
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This article was nominated for deletion on September 15, 2006. The result of the discussion was keep. |
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[edit] re-formulation
I've mixed feelings about the re-formulation done by User:Gdr. I'm aware of the style guide preferring prose over lists, but I find my older version easier to follow. Also I'm totally unaware of using credence in a quantitative manner. But I'm not a native speaker. Pjacobi 18:06, 15 Jul 2004 (UTC)
- "Credence" isn't the word I would have used, but I think it's okay, and at any rate it seems to be the word the original problem used. I also preferred the list version of the article -- but then, the structure of the list may lead people to mentally categorize the outcomes in a way they're not "supposed" to -- so I don't know. Triskaideka 01:03, 16 Jul 2004 (UTC)
[edit] 1/2 vs. 1/3
This isn't a fair betting game, because it is equivalent to her making a 50-50 bet and winning $1 if she's right or losing $2 if she's wrong!
The betting argument may sound funny at first, because it sounds unintuitive (at least to some people).
The probability that the coin landed heads up is indeed always 1/2. However, if Beauty bets $1 during each interview that the coin landed heads up, then, after a large number of these experiments, we see that every time the coin lands heads up, Beauty bets $1 once and wins, whereas every time the coin lands tails up, Beauty bets $1 TWICE and loses EACH TIME.
-Dusik (2:40 PM EDT, Thursday, July 15, 2004)
- IMHO this paradox isn't actually that difficult and the heated discussion obscured this fact. 1/2 and 1/3 are both correct answers, but to different question: Q What's the probability of a heads experiment? A 1/2 -- Q What's the probability of Beauty being awakened after heads? A 1/3. Pjacobi 18:45, 15 Jul 2004 (UTC)
- The idea of betting is sort of thrown in at the last minute, as the entry stands now, and I think it's irrelevant to the original problem. The problem never states that Beauty gains anything by correctly guessing which way the coin landed. If she did, then certainly she should guess tails. But in fact she's never asked which way the coin landed at all. She's asked how probable it is that the coin landed heads. And since the problem specifies that she has no way of getting additional information about the coin toss, the question she's being asked, in any interview, is really just "What is the probability that a fair coin that was tossed at some point in the past landed on its head?" Of course the answer to this is 1/2. In fact, I don't think this is a paradox, just an easily misunderstood problem. Triskaideka 19:27, 15 Jul 2004 (UTC)
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- Most mathematical paradoxes are, since mathematics tends not to contradict itself. I think you've summed up the counter-intuitive factor in this one nicely Daibhid C
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- I edited the article, added a paragraph about how the solution depends on the reader's interpretation, and also added a link to a web page that says about the same thing. People may disagree on just what the interview question means. Triskaideka 01:03, 16 Jul 2004 (UTC)
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- Thanks for all the feedback, guys, and for elaborating on the gambling aspect of the problem in the article. I think I've got a clearer idea of the issue at hand now, and my conclusion is actually that there is only one "correct" answer to the question: 1/2. The answer of 1/3 is not the correct one, but rather the more profitable one, precisely for the reason of being right twice 50% of the time, just because the question is *asked* twice. So, in other words, guessing the probability correctly is not the goal of the contest. So, in conclusion, the paradox here is that a good percentage of the people hearing this paradox will perceive this as a gambling scenario and look for the best rewarded answer. It's one of those instances where "common sense" may get in the way of mathematics. Another reason why it's a paradox is because it's unintuitive, in many cases I would guess because the reader will not completely understand the question logically, but use (untrained) intuition to guess the answer, which should commonly yield answers of both 1/2 and 1/3. Am I making any sense? Dusik 15:55, 11 July 2005 (UTC)
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[edit] Edit 00:30, 19 Jul 2004 81.152.235.34
I find the "solution" and "analogy" of 81.152.235.34 slightly misleading. I'm still of the opinion, that more than anything else, this paradox highlightens the importance of clearly formulated questions, as 1/2 and 1/3 are both correct answers, but to different questions. In addition I would like to illustrate the possible questios and answers by fair odds when betting, as quite a number of peoply enjoy clearer thinking, if money is involved.
Comments?
Pjacobi 09:42, 19 Jul 2004 (UTC)
- With today's additions, I'm quite satisfied. Pjacobi 09:55, 19 Jul 2004 (UTC)
I restored the gambling argument for ⅓ since I think people do think more clearly about probability when it's expressed as a gamble. I also cut a big chunk of waffle by User:Triskaideka and 81.152.235.34 and replaced it with a concise solution and a spoiler warning. Gdr 23:55, 2004 Jul 22 (UTC)
- I reiterate my protest against this usage of the idea of gambling. I would rather see it used to help the reader understand an argument that has already been made. As it stands, the idea of gambling is abruptly inserted, after the problem statement but before the solution, and it seems to create a new solution that wouldn't exist without the idea of gambling, which isn't present in the problem statement.
- I also don't understand the objection to my and 81.152.235.34's text, which I thought attempted to explain in plain language how the two answers could be arrived at. I'm not particular about how we approach the explanation, but I do think that User:Gdr's revisions have removed some of that explanation in favor of text that laypeople won't find as accessible. Brevity is a good thing, but not if it comes at the expense of the reader's understanding. Triskaideka 16:03, 23 Jul 2004 (UTC)
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- I don't like the current version best. The explanation What is the conditional probability of heads, given that Beauty has been awakened and interviewed? The rules for conditional probability give ⅓ as the answer is incompatible with the definition of Conditional probability or needs a completely other formulation, as the probability of Beaty being awakened and interviewed is 1.
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- But I wasn't fully satisfied with any prior version, so I don't know how to proceed.
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- Pjacobi 16:36, 23 Jul 2004 (UTC)
[edit] by the Pseudonymous 81.152.235.34 - now registered as BernardSumption
Having read the disagreement on the (huge) usenet thread it seems to me that there will never be a total resolution. This is why when suggesting the answer to be 1/2, I made clear the assumption that this result hinges on. The edits by myself and Triskaideka were an attempt to put both sides of the argument in laymen's terms, and Gdr's edits were probably more correct and concise from a mathematical point of view. I see that Gdr reverted those changes, which is good for laymen but means that we now have no links to appropriate probability theory pages. Perhaps a solution is for someone with more knowledge of probability than I to add the appropriate links to pages on probability theory terms to the current version, so that people wishing to dig deeper can do so?
Also, Pjacobi had a good point that the whole paradox arises from the lack of a clear formulation of the question, so I added his sentence to that effect near the top of the article. --BernardSumption 09:40, 26 Jul 2004 (UTC)
Somebody seems to have addressed a philisophical problem without even knowing it. Although guessing tails will make her win twice as much on average, she will remember the reward just the same either way. Does getting the reward an extra time really matter if she just forgets it? Daniel 02:49, 10 Feb 2005 (UTC)
In the current version, I have some difficulty in seeing the motivation for interpreting "credence" as a fraction out of the number of awakenings. (There seems to be a typo in the current version. "1/2" == "Tuesday"?) The final section that indicates that guessing "tails" will gain her more correct answers actually seems to motivate that interpretation better, although in somewhat different terms. I have not read the usenet thread. (I am assuming that 95% of it is simply different posters refusing to accept other posters' interpretations of an ambiguously worded question.) Is there perhaps a better way to explain/present that interpretation? Perhaps "what are the odds ..."? -- Wmarkham
Also, the article credits Adam Elga with the problem, but one of the linked pages states "The Sleeping Beauty problem, and the older Paradox of the Absentminded Driver, were first posed by Piccione and Rubinstein, in 1994." Is the Piccione and Rubinstein problem the same, but older? Is it possible that Elga should be credited with simply a more current wording? -- Wmarkham
I believe much of this article, and most of the posts on the talk page, are misguided. Our goal here should not be to solve the paradox or argue about which point is right -- Wikipeda isn't about original research -- but to state the current and historical (sure, it's only a couple of years old, but still) views. As it is now, there is no consensus in the philosophical literature. There are dozens of papers on the paradox, and they split fairly evenly on the answers. Moreover, the naive arguments offered in article as it is at present are certainly not all arguments, and indeed Elga doesn't use either of them (although he does mention them) to argue for his view in the original paper where the problem was presented. Another thing is that the inclusion of a link to Terry Horgan's paper is rather arbitrary, as it is obscure and rather uninteresting. If any paper should be linked to, it should be Elga's, or perhaps Lewis' response to Elga. Regarding who should be credited with inventing the problem, to answer Wmarkham's query: the Sleeping Beauty problem is indeed present in Piccione and Rubinstein's article, but they do not devote much attention to it, or give it a name. The Paradox of the Absentminded Driver, which the real focus of their paper, is not the same paradox. To make things even more complicated, Elga acknowledges in his original article that Stalnaker is the person to give SB its name, and Stalnaker had in turn read about "similar problems" in unpublished work by Zuboff. So there seems to be several independent discoveries of the paradox. Nonetheless, most credit Elga, as he was in any regard the first person to bring it to the forefront and recognize it as an important problem in its own right. Now, before someone asks why I don't do all these changes myself: hopefully, someday when I have an hour or two to spare I will. Right now I don't. -- Miai
Very sober analysis. Go ahead! Be bold in editing this page. --Pjacobi
[edit] Re-intuit, halfers!
I'm going to toss a coin. If it lands tails, I will write the name of a popular female singer in the next paragraph. If it lands heads, I will write the name of a less-popular-than-he-used-to-be male politician. Okay, here goes!
Now, remember the question: what is your credence now for the proposition that BRITNEY SPEARS my coin landed heads? Are you really going to tell me it's 50%? I'm not talking about gambling here; I really only want your opinion as to whether it landed heads or tails. Not whether it might have landed heads or tails when I flipped, and not whether a coin normally lands heads or tails. Your credence now.
The intuitive response is the correct one: zero percent. There might have been a 50% chance at one point, but not given the information you have. Similarly, if you flipped a coin and saw heads, you wouldn't analyse the probability as being 50% any more. From your perspective, the probability would have changed to 100%.
This is conditional probability: the probability of some event A, assuming event B (as defined in Wikipedia). Event B in this case is my naming Britney, or in my second example, your witnessing a monarch or president's picture facing up. In the Sleeping Beauty example, SB is assuming an event as well: her being woken up.
Assuming her being woken as outlined in the article, the probability of heads is 1/3. I don't think this is merely a question of how you interpret the question. There are two perspectives: the coin tosser's and SB's. It's intuitive to always want to revert to the coin tosser's perspective, because 50-50 is intuitive. But it's also wrong - her subjective experience affects her perception of the probability.
Experience always affects probability. If two teams of equal ability are almost finished their game, and the score is 8-0, then most people would put the odds of the team with 8 winning at close to 100%. You don't say "there's a 50-50 chance of each team winning."
The article seems to favour the 'halfer' viewpoint. I disagree with the analogy (in which a second coin toss has no effect on the first). It's true that SB's waking had no effect on the toss, but it does have an effect on her analysis, which is what she's been asked for. Worrying about whether she has an effect on the toss is still moving back to the intuitive coin-tosser's perspective, not hers.
To highlight my point: how can anyone possibly think that the gambler's odds are different than the probability?24.64.223.203 08:21, 31 October 2005 (UTC)
[edit] ?
This makes absolutely no sense.
[edit] Is it a frivolous article?
I think that this is not actually frivolous (as per note on version 2006-03-05 01:57:51 of the article), or at any rate less frivolous than the Monty_Hall_problem. The arguments for both sides given at the external link seem quite serious, and it evidently confuses people.
The Monty Hall problem is of course not a logical paradox, but is one in the sense that is is counter-intuitive. I am actually less sure about this, as it seems to raise the question of what we mean by 'credence'.
PJTraill 14:06, 5 March 2006 (UTC)
[edit] What is credence?
I have adopted the term credence as it occurs in the original articles, but I was unfamiliar with it.
I get the impression from http://users.ox.ac.uk/~mert0130/teaching/lecture16.doc that it is a subjective estimate of probability, but that seems to eliminate any possibility of discussing the rightness of Beauty's answer.
I see in http://fitelson.org/probability.pdf a lot more discussion, making it plausible that there should be various answers.
It seems to me that there should be a Wikipedia article on Credence (probability theory), or at least that it should come in the the article on probability, but a search threw up nothing much.
PJTraill 15:14, 5 March 2006 (UTC)
Credence is not a subjective estimate of some objective probability, but is rather a measure of partial belief. It often results from an estimate of objective probabilities, but often doesn't. Even if there are no objective chances (for example, if the world is deterministic) then credences will still be important, because people don't know everything for certain. Similarly, even if the actual situation is chancy, one can have evidence that goes against this chance (especially after the fact, as mentioned elsewhere here), so credence is importantly different from objective chance too.
- Isn't credence just another word for subjective probability in this context? If that is the case (as is supported by the contributor above as well as in the first link PJTraill refers to) the simplest way to solve the dead crecence-link would be to let it point to the subjective probability page. INic 15:39, 20 September 2006 (UTC)
[edit] Duplicate
I see now that there is already an article at Sleeping beauty paradox. Sorry about this. That has been around since 2004-07-15 11:42:50 and gives more explanations, though I'm not sure if it covers all the same ground. If nobody else does, I'd better clear this up (make a redirect or scrap?), and redirect the links, though I'm a bit busy just now. My point about Credence (probability theory) remains. PJTraill 15:28, 5 March 2006 (UTC)
- Make a redirect. - Rainwarrior 04:19, 21 June 2006 (UTC)
- I just merged the articles and the talk pages. -- Reinyday, 17:21, 24 June 2006 (UTC)
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- Thanks PJTraill 00:55, 6 November 2006 (UTC)
[edit] Paragraph deleted
A small section titled "Credence" completely denies the 1/3 possibility thus negating the factual accuracy of the entire article explaining a paradox. you are twice as likely to be awaken if it was tails, so 1/3. The section titled "Variations" seems to be a rebuttle to the false section, without being bold enough to delete it.
[edit] Example
I was trying to come up with a simpler example (maybe with less sordid drug abuse) and while it's not completely the same, I think it makes the 1/3 answer clearer...
The Mad Coin Tosser has been tossing a single coin each day for several years although lately he has become tired of his daily ritual. Last month he decided that each time the coin lands heads he will toss the coin again the next day, but should the coin land tails, he will take a break the next day and toss the next coin the day after that.
What are the chances that the last coin he tossed was heads?
Since you don't know whether today is a coin tossing day or a break day, you are in the same position as the Sleeping Beauty. Indeed, the only real difference with the cruel SB experiment is whether it is run once or forever, and I'm pretty sure that doesn't change the odds. --88.113.96.178 12:34, 12 March 2007 (UTC)
