Upvoted because this is a really clean and well-argued presentation of the position and it is a position I agree with. But weren't basically all of these points made in that original discussion? I recall making a couple of them myself. I think this is the relevant sub-thread if people are interesting in seeing the arguments that were made on both sides.
One other consideration in favor of focusing on the payoff scheme is that in the literature I've seen hypothetical wagers are the only way of defining subjective probabilities to begin with. No one knows exactly what 'I believe P(X)=.1' means except that a gambler with such a belief would only take a bet on X if the payout greater than 10:1 and would only offer the bet if the payout was less than 10:1. I don't know of another way to make sense of probabilities for one-off events. If someone has one please share.
Anyway, this was the payoff scheme I liked. It seems less awkward than some of yours that result in a 1/2 answer and I think it ensures we don't violate conservation of information nicely.
I tend to agree with you, but apparently E. T. Jaynes does not:
It has always seemed objectionable to some, including this writer, to base probability theory on such vulgar things as betting, expectation of profit, etc. We think that the principles of logic ought to be on a higher plane.
He then goes on to point out (rather illogically, I think) that if probability estimates actually do depend on how the wager is structured, then
a well posed question would have more than one "right" answer, with nothing to choose between them. This, in our view, is another aspect of the superiority of [the approach of PT:TLOS based on Cox's theorem] which stresses logical consistency instead and, just for that reason, is more likely to have a lasting place in probability theory.
Quotes from the final paragraph of Chapter 13 of PT:TLOS
We have an apparently ambiguous word: "probability". That we accept it's ambiguous (the thesis of this post), means that there are situations where either of its multiple possibly contradictory meanings is intended. This doesn't negate a possibility of there being a concept that can be applied to most situations, doesn't itself suffer from ambiguity, and is usually referred to using the same word.
There is no apparent contradiction here: the disagreement you point out results from equivocating between the ambiguous word "probability" and (presumably) the more specific concept of probability that Jaynes refers to.
The sleeping beauty problem is not really a paradox of probability theory. The question is what Beauty's beliefs "should" be. As far as probability theory is concerned, there is a 1/2 chance that the coin came up heads, and conditioned on Beauty being interviewed the probability is still 1/2. I think it is reasonable to say that probability theory should be based on a strict logical formalism independent of feedback, but that the solution to this paradox involves recognizing the role of feedback in determining what Beauty "should" believe.
As far as probability theory is concerned, there is a 1/2 chance that the coin came up heads, and conditioned on Beauty being interviewed the probability is still 1/2.
Here we see an illustration of OP's argument that stating things in terms of probability can be ambiguous. But this ambiguity could be cleared up by instead saying either,
"As far as probability theory is concerned, there is a 1/2 chance that the coin came up heads, and conditioned on Beauty being interviewed at least once during the week, the probability is still 1/2."
or,
"As far as probability theory is concerned, there is a 1/2 chance that the coin came up heads; but conditioned on the observation that today is either Monday or Tuesday, and Beauty is being interviewed, the probability is 2/3."
Or, a commentator could be equally ambiguous using the language of wagers by saying,
"Beauty will be asked whether the coin came up heads or tails, and given $1 if she is correct."
Using the language of probability does not make one suddenly unable to speak clearly; and using the language of wagers doesn't guarantee precision.
Maybe the language of probability makes it easier to shoot yourself in the foot, but that doesn't mean there's something nonsensical about it.
From my own experience of awaking after days of artificial coma I find it unlikely to believe that the person will go through a good Bayesian reasoning.
It's very hard to form the belief that it's Thursday when the last day you remember is a Sunday. It goes against fundamental principles that evolution taught us over millions of years. It's one of those cases where our availability heuristics is really bad.
PS: I know this is only a thought experiment.
The ancestral environment didn't contain a lot of artificial comas, but to be fair it didn't contain many named week days either.
The problem with the Sleeping Beauty problem, more specifically, is that the intuitive translation of it into rules like P(woken up on Tuesday and Interviewed | experimenter got tails) = 1 violates normalization on the "number of observers," or the "probability she is interviewed at all."
Which is what you'd expect, since she can get interviewed more times than she remembers. But still, it violates normalization.
The 1/3, 2/3 solution corresponds to stating the problem intuitively and computing to the end, ignoring the normalization-breaking.
The 1/2, 1/2 solution corresponds to restoring normalization either of the two most obvious ways and not thinking too hard about the unintuitive results.
I agree that translating the problem into a slightly different framework can help you resolve it, but I don't think it's necessary to generalize as much as you did or bring in decision theory. All we need to do is embrace frequentist statistics :D
That is to say, what you do is calculate expected frequencies rather than probabilities. Manipulating these lets you "hide" the normalization-breaking in a mathematically acceptable way, since an expected frequency of 1.5 doesn't have the same problems a probability of 1.5 does.
Definitely an odd problem, though.
I agree with the conclusions to most of the ways you frame the Sleeping Beaty problem, but not this important one.
Each interview consists of Sleeping Beauty being told whether the coin landed on heads or tails, followed by one question, “How surprised are you to hear that?” Should Sleeping Beauty be more surprised to learn that the coin landed on heads than that it landed on tails?
I would say no; this seems like a case where the simple probability-theoretic reasoning applies.
I disagree -- I think Monday's Sleeping Beauty should be surprised to learn that the coin landed on heads. I think my logic would be clear to you*, but your logic is opaque to me. This makes me suspect I'm missing something, and if I am, it's something important. Perhaps we're using different definitions of "surprised"?
*see the credence and log-scoring framing
ETA I'd appreciate an explanation as to what surprise has to do with likelihood. If I smell bacon when I wake up tomorrow, I'll be quite surprised to learn that George Foreman is grilling me bacon burgers. There's a high likelihood that I will smell bacon when I wake up tomorrow given George Foreman is grilling me bacon burgers, but that high likelihood would not temper my surprise.
Downvoted, both because I disagree with the main point, and because I don't see any good reason to kick this dead horse again. ("You still don't agree with me" is not a good reason.)
OP is really claiming, without a fair and balanced presentation of evidence, that people are less-able to make ambiguous statements about payoffs, than to make ambiguous statements about probability. OP draws the conclusion from this that probability is a meaningless and useless concept. But there is at best a quantitative, not a qualitative, distinction here between the precision of talking in terms of payoffs, and talking in terms of probability.
The fact that something is difficult to talk about unambiguously does not mean it should be tabooed.
OP is really claiming, without a fair and balanced presentation of evidence, that people are less-able to make ambiguous statements about payoffs, than to make ambiguous statements about probability.
I was going more for the point that some ambiguous questions about probabilities are disguised less-ambiguous questions about payoffs, not claiming that people aren't able to make ambiguous statements about payoffs or that reframing in terms of payoffs can resolve ambiguity in any probability problem.
OP draws the conclusion from this that probability is a meaningless and useless concept.
No I don't, and I'm surprised if the post gave the impression that I did. Thanks for the feedback and vote explanation though.
(ETA: I did not downvote you.)
Does this have some connection to the unbiased/maximal likelihood estimator dichotomy?
I don't really have a clear picture, so this should be treated only as a vague intution that someone could hopefully formalize, but I feel that somehow the maximal likelihood estimator would be in the 1/3 camp, since it optimizes just for the payoff - and, on the other hand, unbiased estimator would be in the 1/2 camp, since it optimizes for accuracy. Then, the whole problem comes down to the well-known issue that in some cases, MLE are not unbiased (e.g. classic problem with variance estimator).
“Now, it does seem to me that following the usual rules of probability theory ... tells Sleeping Beauty to assign .5 credence to the proposition that the coin landed on heads.”
I think that your 1/3 answer is entirely consistent with the usual rules of probability theory, at least once a highly intuitive caveat is added: Given no new information, the odds do not change unless some old information has been lost. Usually, we implicitly assume that information is not lost so there's no need for the caveat, but amnesia inducing drugs radically violate the assumption.
When sleeping beauty falls asleep on Sunday, she knows the day of the week, and that she's been interviewed exactly zero times. When she wakes up, she doesn't know the day or the number of times she's been interviewed. This loss of information opens the door to the odds changing.
An easier way to think about the same problem in Bayesian terms is to compare her belief during an interviews to her belief afterward. On Wednesday, she wakes up and thinks that if there's an interview, P(heads) = 1/3. But after realizing that there is no interview, she gains the information that it's Wednesday. From this, she infers that her experiences are no longer being skewed toward worlds where the result is tails by a factor of 2, so she updates her belief to P(heads) = 1/2.
After having done a lot of research on the Sleeping Beauty Problem as it was the topic of my bachelor's thesis (philosophy), I came to the conclusion that anthropic reasoning is wrong in the Sleeping Beauty Problem. I will explain my argument (shortly) below:
The principle that Elga uses in his first paper to validate his argument for 1/3 is an anthropic principle he calls the Principle of Indifference:
"Equal probabilities should be assigned to any collection of indistinguishable, mutually exclusive and exhaustive events."
The Principle of Indifference is in fact a more restricted version of the Self-Indication Assumption:
"All other things equal, an observer should reason as if they are randomly selected from the set of all possible observers."
Both principles are to be accepted a priori as they can not be attributed to empirical considerations. They are therefore vulnerable to counterarguments...
The counterargument:
Suppose that the original experiment is modified a little:
If the outcome of the coin flip is Heads, they wake Beauty up at exactly 8:00. If the outcome of the first coin flip is Tails, the reasearchers flip another coin. If it lands Heads they wake Beauty at 7:00, if Tails at 9:00. That means that when Beauty wakes up she can be in one of 5 situations:
Heads and Monday 8:00
Tails and Monday 7:00
Tails and Monday 9:00
Tails and Tuesday 7:00
Tails and Tuesday 9:00
Again, these situations are mutually exclusive, indistinguishable and exhaustive. Hence thirders are forced to conclude that P(Heads) = 1/5.
Thirders might object that the total surface area under the probability curve in the Tails-world would still have to equal 2/3, as Beauty is awakened twice as many times in the Tails-world as in the Heads-world. They are then forced to explain why temporal uncertainty regarding an awakening (Monday or Tuesday) is different from temporal uncertainty regarding the time (7:00 or 9:00 o’clock). Both classify as temporal uncertainties within the same possible world, what could possibly set them apart?
An explanation could be that Beauty is only is asked for her credence in Heads during an awakening event, regardless of the time, and that such an event occurs twice in the Tails-world. That is, out of the 4 possible observer-moments in the Tails-world there are only two in which she is interviewed. That means that simply the fact that she is asked the same question twice is reason enough for thirders to distribute their credence, and it is no longer about the number of observer moments. So if she would be asked the same question a million times then her credence in Heads would drop to 1/1000001!
We can magnify the absurdity of this reasoning by imagining a modified version of the Sleeping Beauty Problem in which a coin is tossed that always lands on Tails. Again, she is awakened one million times and given an amnesia-inducing potion after each awakening. Thirder logic would lead to Beauty’s credence in Tails being 1/1000000, as there are one million observer-moments where she is asked for her credence within the only possible world; the Tails-world. To recapitulate: Beauty is certain that she lives in a world where a coin lands Tails, but due to the fact that she knows that she will answer the same question a million times her answer is 1/1000000. This would be tantamount to saying that Mt. Everest is only 1m high when knowing it will be asked 8848 times! It is very hard to see how amnesia could have such an effect on rationality.
Conclusion:
The thirder argument is false. The fact that there are multiple possible observer-moments within a possible world does not justify dividing your credences equally among these observer-moments, as this leads to absurd consequences. The anthropic reasoning exhibited by the Principle of Indifference and the Self-Indication Assumption cannot be applied to the Sleeping Beauty Problem and I seriously doubt if it can be applied to other cases...
Heads and Monday 8:00
Tails and Monday 7:00
Tails and Monday 9:00
Tails and Tuesday 7:00
Tails and Tuesday 9:00
Again, these situations are mutually exclusive, indistinguishable and exhaustive. Hence thirders are forced to conclude that P(Heads) = 1/5.
I don't think that's the same experiment. Suppose that each time sleeping beauty is woken up, she is offered the opportunity to wager where the first coin ended up being heads. If she wins the wager she gets $1.00. Clearly her expected winnings from betting tails is still $1.00 and her expected utility from betting heads is still $0.50
Both classify as temporal uncertainties within the same possible world, what could possibly set them apart?
This strikes me as a rather Dark Artsy/debater type strategy. I thought that LessWrong had a social norm against that sort of thing.
An explanation could be that Beauty is only is asked for her credence in Heads during an awakening event, regardless of the time, and that such an event occurs twice in the Tails-world. That is, out of the 4 possible observer-moments in the Tails-world there are only two in which she is interviewed. That means that simply the fact that she is asked the same question twice is reason enough for thirders to distribute their credence, and it is no longer about the number of observer moments. So if she would be asked the same question a million times then her credence in Heads would drop to 1/1000001!
I don't understand your incredulity. Suppose that in each one of those million events, she is offered the wager. Clearly wagering tails is the winning move by far.
I don't think you fully understand my argument. It is not about being offered a wager or not, because that certainly would alter the experiment and make it very easy to decide whether halfer or thirder reasoning is the way to go.
Instead, it is about the fundamental principle the thirder's argument is based on; the anthropic principle Elga calls his Principle of Indifference. It is the key element used to justify Beauty's credence drop from 1/2 to 1/3 on waking up. This credence drop is in serious need of justification because Beauty learns nothing new when she wakes. She only learns 'Today is Monday or Tuesday' which she knew she would learn beforehand. That is, she receives no knew information on which she can conditionalise. Therefore thirders resort to anthropic reasoning, which goes like this: "I am in one of three awakenings now, which all look the same to me. Therefore I should didvide my credence equally over them."
My counterargument tries to show the fallacy of this reasoning by creating two other possible awakenings within the Tails-world. Hence there are then 4 possible awakenings within the Tails-world and thirders adhering to the Principle of Indifference should divide there credence equally over them. If they don't, then that means that it is not about Beauty's number of observer-moments within a possible world, but about the number of times Beauty is asked the same question.
Like you pointed out, Beauty is still awakened twice if Tails and once if Heads. Therefore she is indeed vulnerable to being Dutch Booked. The problem with the wager you proposed is that it is repeated twice if Tails and once if Heads, which makes it unfair. Suppose someone offered you a bet that paid 10$ if a coin comes up Heads and cost you 1$ if the coin comes up Tails. The catch is; if the coin comes up Tails the bet is repeated 100x times. Clearly you do not accept this bet, as the real bet is one where you stand to lose 100$ instead of 1$. However, this changes nothing about your belief that the objective chance of a coin to land Heads is 1/2. Beauty will not accept any bets that are repeated if lost. Dutch Book arguments in the Sleeping Beauty Problem are inconclusive since they are imaginable for both thirders and halfers. Hence they do not provide any deeper insights into the halfer and thrider arguments.
PS I'm sorry if I came on too strong; it was my first post here at LessWrong and I'm still reading my way through all the articles.
these situations are mutually exclusive, indistinguishable and exhaustive.
No, they aren't. "Indistinguishable" in that definition does not mean "can't tell them apart." It means that the cases arise through equivalent processes. That's why the PoI applies to things like dice, whether or not what is printed on each side is visually distinguishable from other sides.
To make your cases equivalent, so that the PoI applies to them, you need to flip the second coin after the first lands on Heads also. But you wake SB at 8:00 regardless of the second coin's result. You now have have six cases that the PoI applies to, counting the "8:00 Monday" case twice, and each has probability 1/6.
On the other hand, if you’re Dr. Evil and you’re in your moon base preparing to fire your giant laser at Washington, DC when you get a phone call from Austin “Omega” Powers
So, does this mean ata is going to write an Austin Powers: Superrational Man of Mysterious Answers fanfic?
I like the approach, but I feel you have not given justice to the 1/2 position. Some of Nick Bostrom's work on observer moments is very impressive in this regard; however I've been able to show that they imply that Sleeping Beauty can be money pumped.
I would personally take that to show that the 1/2 position is just wrong, but it also implies that some who do hold it are not clinging to a position that can be 'dissolved' in the way you are saying - they have a position that would result in them taking different odds on the same bets as 1/3's or some 1/2's.
Wei has a UDT approach that is similar to this; I can't seem to find it, though.
I've been able to show that they imply that Sleeping Beauty can be money pumped.
This is only true once you have described Beauty's reward scheme: the whole point of this post is that the probability to use depends on that.
Each interview consists of one question, “What is your credence now for the proposition that our coin landed heads?”, and the answer given will be scored according to a logarithmic scoring rule, with the aggregate result corresponding to the number of utilons (converted to dollars, let’s say) she will be penalized after the experiment.
Here is a diagram I made of this situation, inspired by my adventures here.
The semantics are such that for example, because all the player nodes are inside a blue blob, p=q=r, unless you choose an answer by flipping coins or something.
Edit: fixed; thanks, RobinZ
This may be kinda cheating, but how's this for a solution: First, I check to see if my left shoe is untied. If it isn't, I say my credence is 50% and then untie it. If it is already untied, I say my credence is 0% because I know for certain that it is tuesday and the coin landed on tails. I won't remember coming up with this plan on tuesday, but I'll think of it again and then derive that I already implemented it. This does, of course, require that the interviewers don't realize what I'm doing and tie it while I'm asleep (and that it was tied to begin with). If I'm not mistaken, this method maximizes my expected score on the negative log test, at -1 bit regardless of the result.
Am I right to say that for classic probability theory the probability is the number of positive outcomes over the number of experiments? If that's correct, then the answer depends on what we consider to be an experiment. If it's the Sleeping Beauty's awakening, then it's two tails and one heads over three awakenings. If the whole procedure (or just a coin flip) is an experiment, then it's one heads and one tails over two procedures.
Each interview consists of one question, “What is your credence now for the proposition that our coin landed heads?”, and the answer given will be scored according to a logarithmic scoring rule, with the aggregate result corresponding to the number of utilons (converted to dollars, let’s say) she will be penalized after the experiment.
Here is a diagram I made of this situation, inspired by my adventures here.
The semantics are such that for example, because all the player nodes are inside a blue blob, p=q=r, unless you choose an answer by flipping coins or something.
Edit: fixed; thanks, RobinZ
...where's the flip corresponding to getting scored for the other outcome? Doesn't that make it 1-p=q=r?
Several months ago, we had an interesting discussion about the Sleeping Beauty problem, which runs as follows:
Sleeping Beauty volunteers to undergo the following experiment. On Sunday she is given a drug that sends her to sleep. A fair coin is then tossed just once in the course of the experiment to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday, given a second dose of the sleeping drug, and awakened and interviewed again on Tuesday. The experiment then ends on Tuesday, without flipping the coin again. The sleeping drug induces a mild amnesia, so that she cannot remember any previous awakenings during the course of the experiment (if any). During the experiment, she has no access to anything that would give a clue as to the day of the week. However, she knows all the details of the experiment.
Each interview consists of one question, “What is your credence now for the proposition that our coin landed heads?”
In the end, the fact that there were so many reasonable-sounding arguments for both sides, and so much disagreement about a simple-sounding problem among above-average rationalists, should have set off major alarm bells. Yet only a few people pointed this out; most commenters, including me, followed the silly strategy of trying to answer the question, and I did so even after I noticed that my intuition could see both answers as being right depending on which way I looked at it, which in retrospect would have been a perfect time to say “I notice that I am confused” and backtrack a bit…
And on reflection, considering my confusion rather than trying to consider the question on its own terms, it seems to me that the problem (as it’s normally stated) is completely a tree-falling-in-the-forest problem: a debate about the normatively “correct” degree of credence which only seemed like an issue because any conclusions about what Sleeping Beauty “should” believe weren’t paying their rent, were disconnected from any expectation of feedback from reality about how right they were.
It may seem either implausible or alarming that as fundamental a concept as probability can be the subject of such debates, but remember that the “If a tree falls in the forest…” argument only comes up because the understanding of “sound” as “vibrations in the air” and “auditory processing in a brain” coincide often enough that most people other than philosophers have better things to do than argue about which is more correct. Likewise, in situations that we actually encounter in real life where we must reason or act on incomplete information, long-run frequency is generally about the same as optimal decision-theoretic weighting. If you’re given the question “If you have a bag containing a white marble and two black marbles, and another bag containing two white marbles and a black marble, and you pick a bag at random and pick a marble out of it at random and it’s white, what’s the probability that you chose the second bag?” then you can just answer it as given, without worrying about specifying a payoff structure, because no matter how you reformulate it in terms of bets and payoffs, if your decision-theoretic reasoning talks about probabilities at all then there’s only going to be one sane probability you can put into it. You can assume that answers to non-esoteric probability problems will be able to pay their rent if they are called upon to do so, and so you can do plenty within pure probability theory long before you need your reasoning to generate any decisions.
But when you start getting into problems where there may be multiple copies of you and you don’t know how their responses will be aggregated — or, more generally, where you may or may not be scored on your probability estimate multiple times or may not be scored at all, or when you don’t know how it’s being scored, or when there may be other agents following reasoning correlated with but not necessarily identical to yours — then I think talking too much about “probability” directly will cause different people to be solving different problems, given the different ways they will implicitly imagine being scored on their answers so that the question of “What subjective probability should be assigned to x?” has any normatively correct answer. Here are a few ways that the Sleeping Beauty problem can be framed as a decision problem explicitly:
Each interview consists of Sleeping Beauty guessing whether the coin came up heads or tails, and being given a dollar if she was correct. After the experiment, she will keep all of her aggregate winnings.
In this case, intending to guess heads has an expected value of $.50 (because if the coin came up heads, she’ll get $1, and if it came up tails, she’ll get nothing), and intending to guess tails has an expected value of $1 (because if the coin came up heads, she’ll get nothing, and if it came up tails, she’ll get $2). So she should intend to guess tails.
Each interview consists of Sleeping Beauty guessing whether the coin came up heads or tails. After the experiment, she will be given a dollar if she was correct on Monday.
In this case, she should clearly be indifferent (which you can call “.5 credence” if you’d like, but it seems a bit unnecessary).
Each interview consists of Sleeping Beauty being told whether the coin landed on heads or tails, followed by one question, “How surprised are you to hear that?” Should Sleeping Beauty be more surprised to learn that the coin landed on heads than that it landed on tails?
I would say no; this seems like a case where the simple probability-theoretic reasoning applies. Before the experiment, Sleeping Beauty knows that a coin is going to be flipped, and she knows it’s a fair coin, and going to sleep and waking up isn’t going to change anything she knows about it, so she should not be even slightly surprised one way or the other. (I’m pretty sure that surprisingness has something to do with likelihood. I may write a separate post on that, but for now: after finding out whether the coin did come up heads or tails, the relevant question is not “What is the probability that the coin came up {heads,tails} given that I remember going to sleep on Sunday and waking up today?”, but “What is the probability that I’d remember going to sleep on Sunday and waking up today given that the coin came up {heads,tails}?”, in which case either outcome should be equally surprising, in which case neither outcome should be surprising at all.)
Each interview consists of one question, “What is the limit of the frequency of heads as the number of repetitions of this experiment goes to infinity?”
Here of course the right answer is “.5, and I hope that’s just a hypothetical…”
Each interview consists of one question, “What is your credence now for the proposition that our coin landed heads?”, and the answer given will be scored according to a logarithmic scoring rule, with the aggregate result corresponding to the number of utilons (converted to dollars, let’s say) she will be penalized after the experiment.
In this case it is optimal to bet 1/3 that the coin came up heads, 2/3 that it came up tails:
| Bet on heads: | 1/2 | 1/3 | ||
| Actual flip: | Heads | Tails | Heads | Tails |
| Monday: | -1 bit | -1 bit | -1.585 bits | -0.585 bits |
| Tuesday: | n/a | -1 bit | n/a | -0.585 bits |
| Total: | -1 bit | -2 bits | -1.585 bits | -1.17 bits |
| Expected: | -1.5 bits | -1.3775 bits | ||
(If you’re not used to the logarithmic scoring rule enough to trust that 1/3 is better than every other option too, you can check this by graphing y = (log2x + 2 log2(1 - x))/2, where x is the probability you assign to heads, and y is expected utility.)
So I hope it is self-evident that reframing seemingly-paradoxical probability problems as decision problems generally makes them trivial, or at least agreeably solvable and non-paradoxical. What may be more controversial is that I claim that this is satisfactory not as a circumvention but as a dissolution of the question “What probability should be assigned to x?”, when you have a clear enough idea of why you’re wondering about the “probability.” Can we really taboo concepts like “probability” and “plausibility” and “credence”? I should certainly hope so; judgments of probability had better be about something, and not just rituals of cognition that we use because it seems like we’re supposed to rather than because it wins.
But when I try to replace “probability” with what I mean by it, and when I mean it in any normative sense — not, like, out there in the territory, but just “normative” by whatever standard says that assigning a fair coin flip a probability of .5 heads tends to be a better idea than assigning it a probability of .353289791 heads — then I always find myself talking about optimal bets or average experimental outcomes. Can that really be all there is to probability as degree of belief? Can’t we enjoy, for its own sake, the experience of having maximally accurate beliefs given whatever information we already have, even in circumstances where we don’t get to test it any further? Well, yes and no; if your belief is really about anything, then you’ll be able to specify, at the very least, a ridiculous hypothetical experiment that would give you information about how correct you are, or a ridiculous hypothetical bet that would give you an incentive to optimally solve a more well-defined version of the problem. And if you’re working with a problem where it’s at all unclear how to do this, it is probably best to backtrack and ask what problem you’re trying to solve, why you’re asking the question in the first place. So when in doubt, ask for decisions rather than probabilities. In the end, the point (aside from signaling) of believing things is (1) to allow you to effectively optimize reality for the things you care about, and (2) to allow you to be surprised by some possible experiences and not others so you get feedback on how well you’re doing. If a belief does not do either of those things, I’d hesitate to call it a belief at all; yet that is what the original version of the Sleeping Beauty problem asks you to do.
Now, it does seem to me that following the usual rules of probability theory (the ones that tend to generate optimal bets in that strange land where intergalactic superintelligences aren’t regularly making copies of you and scientists aren’t knocking you out and erasing your memory) tells Sleeping Beauty to assign .5 credence to the proposition that the coin landed on heads. Before the experiment has started, Sleeping Beauty already knows what she’s going to experience — waking up and pondering probability — so if she doesn’t already believe with 2/3 probability that the coin will land on tails (which would be a strange thing to believe about a fair coin), then she can’t update to that after experiencing what she already knew she was going to experience. But in the original problem, when she is asked “What is your credence now for the proposition that our coin landed heads?”, a much better answer than “.5” is “Why do you want to know?”. If she knows how she’s being graded, then there’s an easy correct answer, which isn’t always .5; if not, she will have to do her best to guess what type of answer the experimenters are looking for; and if she’s not being graded at all, then she can say whatever the hell she wants (acceptable answers would include “0.0001,” “3/2,” and “purple”).
I’m not sure if there is more to it than that. Presumably the “should” in “What subjective probability should I assign x?” isn’t a moral “should,” but more of an “if-should” (as in “If you want x to happen, you should do y”), and if the question itself seems confusing, that probably means that under the circumstances, the implied “if” part is ambiguous and needs to be made explicit. Is there some underlying true essence of probability that I’m neglecting? I don’t know, but I am pretty sure that even if there were one, it wouldn’t necessarily be the thing we’d care about knowing in these types of problems anyway. You want to make optimal use of the information available to you, but it has to be optimal for something.
I think this principle should help to clarify other anthropic problems. For example, suppose Omega tells you that she just made an exact copy of you and everything around you, enough that the copy of you wouldn’t be able to tell the difference, at least for a while. Before you have a chance to gather more information, what probability should you assign to the proposition that you yourself are the copy? The answer is non-obvious, given that there already is a huge and potentially infinite number of copies of you, and it’s not clear how adding one more copy to the mix should affect your belief about how spread out you are over what worlds. On the other hand, if you’re Dr. Evil and you’re in your moon base preparing to fire your giant laser at Washington, DC when you get a phone call from Austin “Omega” Powers, and he tells you that he has made an exact replica of the moon base on exactly the spot at which the moon laser is aimed, complete with an identical copy of you (and an identical copy of your identical miniature clone) receiving the same phone call, and that its laser is trained on your original base on the moon, then the decision is a lot easier: hold off on firing your laser and gather more information or make other plans. Without talking about the “probability” that you are the original Dr. Evil or the copy or one of the potentially infinite Tegmark duplicates in other universes, we can simply look at the situation from the outside and see that if you do fire your laser then you’ll blow both of yourselves up, and that if you don’t fire your laser then you have some new competitors at worst and some new allies at best.
So: in problems where you are making one judgment that may be evaluated more or less than one time, and where you won’t have a chance to update between those evaluations (e.g. because your one judgment will be evaluated multiple times because there are multiple copies of you or your memory will be erased), just ask for decisions and leave probabilities out of it to whatever extent possible.
In a followup post, I will generalize this point somewhat and demonstrate that it helps solve some problems that remain confusing even when they specify a payoff structure.
