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Roko comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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Not quite. The question of what do we mean by probability in this case is valid, but probability shouldn't be just about bets. Probability is bound to a specific model of the situation, with sample space, probability measure, and events. The concept of "probability" doesn't just mean "the password you use to win bets to your satisfaction". Of course this depends on your ontological assumptions, but usually we are safe with a "possible worlds" model.
It is for making decisions, specifically for expressing preference under the expected utility axioms and where uniform distribution is suggested by indifference to moral value of a set of outcomes and absence of prior knowledge about the outcomes. Preference is usually expressed about sets of possible worlds, and I don't see how you can construct a natural sample space out of possible worlds for the answer of 2/3.
Of course that's the obvious answer, but it also has some problems that don't seem easily redeemable. The sample space has to reflect the outcome of one's actions in the world on which preference is defined, which usually means the set of possible worlds. "Experience-moments" are not carved the right way (not mutually exclusive, can't update on observations, etc.)
By "can't update" I refer to the problem with marking Thursday "impossible", since you'll encounter Thursday later.
It's not a problem with the model of ontology and preference, it's merely specifics of what kinds of observation events are expected.
If the goal is to identify an event corresponding to observations in the form of a set of possible worlds, and there are different-looking observations that could correspond to the same event (e.g. observed at different time in the same possible world), their difference is pure logical uncertainty. They differ, but only in the same sense as 2+2 and (7-5)*(9-7) differ, where you need but to compute denotation: the agent running on the described model doesn't care about the difference, indeed wants to factor it out.
I humbly apologize for my inability to read (may the Values of Less Wrong be merciful).
A bet where she can immediately win, be paid, and consumer her winnings seems to me far more directly connected to the probability of "what state am I in" than a bet where whether the bet is consummated and the bet paid depends on what else happens in other situations that may exist later. It seems crazy to treat both of those as equally valid bets about what state she is in at the moment.
This has nothing to do with semantics. If smart people are saying "2+2=5" and I point out it's 4, would you say "what matters is why you want to know what 2+2 is"?
The question here is very well defined. There is only one right answer. The fact that even very smart people come up with the wrong answer has all kinds of implications about the type of errors we might make on a regular basis (and lead to bad theories, decisions, etc).
So the difficult question here is which probability space to set up, not how to compute conditional probabilities given that probability space.
(Posted as an antidote to misinterpretation of your comment I committed a moment before.)
If you mean something else by probability than "at what odds would you be indifferent to accepting a bet on this proposition" then you need to explain what you mean. You are just coming across as confused. You've already acknowledged that sleeping beauty would be wrong to turn down a 50:50 bet on tails. What proposition is being bet on when you would be correct to be indifferent at 50:50 odds?
There is a mismatch between the betting question and the original question about probability.
At an awakening, she has no more information about heads or tails than she had originally, but we're forcing her to bet twice under tails. So, even if her credence for heads was a half, she still wouldn't make the bet.
Suppose I am going to flip a coin and I tell you you win $1 if heads and lose $2 if tails. You could calculate that the p(H) would have to be 2/3 in order for this to be a fair bet (have 0 expectation). That doesn't imply that the p(H) is actually 2/3. It's a different question. This is a really important point, a point that I think has caused much confusion.
You seem to agree she should take a 50:50 bet on tails. What would be the form of the bet where she should be indifferent to 50:50 odds? If you can answer this question and explain why you think it is a more relevant probability then you may be able to resolve the confusion.
Roko has already given an example of such a bet: where she only gets one pay out in the tails case. Is this what you are claiming is the more relevant probability? If so, why is this probability more relevant in your estimation?
Yes, one pay out is the relevant case. The reason is because we are asking about her credence at an awakening.
How does the former follow from the latter, exactly? I seem to need that spelled out.
The interviewer asks about her credence 'right now' (at an awakening). If we want to set up a betting problem based around that decision, why would it involve placing bets on possibly two different days?
If, at an awakening, Beauty really believes that it's tails with credence 0.67, then she would gladly take a single bet of win $1 if tails and lose $1.50 if heads. If she wouldn't take that bet, why should we believe that her credence for heads at an awakening is 1/3?
Do you think this analysis works for the fact that a well-calibrated Beauty answers "1/3"? Do you think there's a problem with our methods of judging calibration?
What do you think the word "credence" means? I am thinking that perhaps that is the cause of your problems.
I'm treating credence for heads as her confidence in heads, as expressed as a number between 0 and 1 (inclusive), given everything she knows at the time. I see it as the same things as a posterior probability.
I don't think disagreement is due to different uses of the word credence. It appears to me that we are all talking about the same thing.
Yes. For example, let's take a clearer mathematical statement, "3 is prime". It seems that's true whatever people say. However, if you come across some mathematicians who are having a discussion that assumes 3 is not prime, then you should think you're missing some context rather than that they are bad at math.
I chose this example because I once constructed an integer-like system based on half-steps (the successor function adds .5). The system has a notion of primality, and 3 is not prime.
What you should say when asked "What is 2+2?" is a separate question from what is 2+2. 2+2 is 4, but you should probably say something else if the situation calls to that. The circumstances that could force you to say something in response to a given question are unrelated to what the answer to that question really is. The truth of the answer to a question is implicit in the question, not in the question-answering situation, unless the question is about the question-answering situation.
If you want a standard system where 3 is not prime consider Z[omega] where omega^3=1 and omega is not 1. That is, the set of numbers formed by taking all sums, differences, and products of 1 and omega.
Given that Beauty is being asked the question, the probability that heads had come up is 1/3. This doesn't mean the probability of heads itself is 1/3. So I think this is a confusion about what the question is asking. Is the question asking what is the probability of heads, or what is the probability of heads given an awakening?
Bayes theorem:
Where is the probability of heads? Actually we already assumed in the calculation above that p(heads) = 0.5. For a general biased coin, the calculation is slightly more complex:
I'm leaving this comment because I think the equations help explain how the probability-of-heads and the probability-of-heads-given-awakening are inter-related but, obviously -- I know you know this already -- not the same thing.
To clarify, since the probability-of-heads and the probability-of-heads-given-single-awakening-event are different things, it is indeed a matter of semantics: if Beauty is asked about the probability of heads per event ... what is the event? Is the event the flip of the coin (p=1/2) or an awakening (p=1/3)? In the post narrative, this remains unclear.
Which event is meant would become clear if it was a wager (and, generally, if anything whatsoever rested on the question). For example: if she is paid per coin flip for being correct (event=coin flip) then she should bet heads to be correct 1 out of 2 times; if she is paid per awakening for being correct (event=awakening) then she should bet tails to be correct 2 out of 3 times.
Actually .. arguing with myself now .. Beauty wasn't asked about a probability, she was asked if she thought heads had been flipped, in the past. So this is clear after all -- did she think heads was flipped, or not?
Viewing it this way, I see the isomorphism with the class of anthropic arguments that ask if you can deduce something about the longevity of humans given that you are an early human. (Being a human in a certain century is like awakening on a certain day.) I suppose then my solution should be the same. Waking up is not evidence either way that heads or tails was flipped. Since her subjective experience is the same however the coin is flipped (she wakes up) she cannot update upon awakening that it is more likely that tails was flipped. Not even if flipping tails means she wakes up 10 billion times more than if heads was flipped.
However, I will think longer if there are any significant differences between the two problems. Thoughts?
This is incorrect.
Given that Beauty is being asked the question, the probability that heads had come up is 1/2.
This is bayes' theorem:
p(H)=1/2
p(awakened|H)=p(awakened|T)=1
P(H|awakened)=p(awakened|H)P(H)/(p(awakened|H)p(H)+p(awakened|T)p(T))
which equals 1/2
By "awakened" here you mean "awakened at all". I think you've shown already that the probability that heads was flipped given that she was awakened at all is 1/2, since in both cases she's awakened at all and the probability of heads is 1/2. I think your dispute is with people who don't think "I was awakened at all" is all that Beauty knows when she wakes up.
Beauty also knows how many times she it likely to have been woken up when the coin lands heads - and the same for tails. She knew that from the start of the experiment.
OK, I see now why you are emphasizing being awoken at all. That is the relevant event, because that is exactly what she experiences and all that she has to base her decision upon.
(But keep in mind that people are just busy answering different questions, they're not necessarily incorrect for answering a different question.)
Why was this comment down-voted so low? (I rarely ask, but this time I can't guess.) Is it too basic math? If people are going to argue whether 1/3 or 1/2, I think it is useful to know their debating about two different probabilities: the probability of heads or the probability of heads given an awakening.
Re: "But if we specify that the money will be put into an account (and she will only be paid one winning) that she can spend after the experiment is over, which is next week, then she will find that 1/2 is the "right" answer"
That seems like a rather bizarre way to interpret: "What is your credence NOW for the proposition that our coin landed heads?" [emphasis added]
NOW. One bet.
Again, consider the scenario where at each awakening we offer a bet where she'd lose $1.50 if heads and win $1 if tails, and we tell her that we will only accept whichever bet she made on the final interview.
If her credence for heads on an awakening, on every awakening (she can't distinguish between awakenings), really was 1/3, she would agree to accept the bet. But we all know accepting the bet would be irrational. Thus, her credence for heads on an awakening is not 1/3.
So: you are debating what:
"What is your credence now for the proposition that our coin landed heads?"
...actually means. Personally, I think your position on that is indefensible.
This would make it clear exactly where the problem lies - if not for the fact that you also appear to be in a complete muddle about how many times Beauty awakens and is interviewed.
We both know what question is being asked. We both know how many times she awakens and is interviewed. I know what subjective probability is (I assume you do too). I showed you my math. I also explained why your ratio of expected frequencies does not correspond to the subjective probability that you think it does.
Does it not concern you even a little that the Wikipedia article you linked to quite clearly says you are wrong and explains why?
I started by reading the wikipedia page. At that point, the 1/3 solution made some sense to me, but I was bothered by the fact that you couldn't derive it from probability laws. I then read articles by Bostrom and Radford. I spent a lot of time working on the problem, etc. Eventually, I figured out precisely why the 1/3 solution is wrong.
Is Wikipedia a stronger authority than me here? Probably. But I know where the argument there fails, so it's not very convincing.
I think we are nearing the end here. Someone just wrote a whole post explaining why the correct answer is 1/3: http://lesswrong.com/lw/28u/conditioning_on_observers/
It's fascinating to me that you won't tell me which probability is wrong, p(H)=1/2, P(monday|H)=1
It's also interesting that you won't defend your answer (other than saying I'm wrong). You are in a situation where the number of trials depends on outcome, but are using an estimator that is valid for independent trials. Show me that yours converges to a probability. Standard theory doesn't hold here.
Probabilities are subjective. From Beauty's POV, if she has just awakened to face an interview, then p(H)=1/3. If she has learned that is Friday and the experiment is over, (but she has not yet been told which side the coin came down), then she updates on that info, and then p(H)=1/2. So, the value of p(H) depends on who is being asked - and on what information they have at the time.
It's the first one - P(H)=1/2 is wrong. Before going any further, we should adopt Jaynes' habit of always labelling the prior knowledge in our probabilities, because there are in fact two probabilities that we care about: P(H|the experiment ran), and P(H|Sleeping Beauty has just been woken). These are 1/2 and 1/3, respectively. The first of these probabilities is given in the problem statement, but the second is what is asked for, and what should be used for calculating expected value in any betting, because any bets made occur twice if the coin was tails.
How can these things be different, P(H|the experiment ran) and P(H|Sleeping Beauty has just been woken)?
Yes, a bet would occur twice if tails, if you set the problem up that way. But the question has to do with her credence at an awakening.
The 1/3 calculation is derived from treating the 3 counts as if they arose from independent draws of a mulitinomial distribution. They are not independent draws. There is 1 degree of freedom, not 2. Thus, the ratio that lead to the 1/3 value is not the probability that people seem to think it is. It's not clear that the ratio is a probability at all.
What's this about a multinomial distribution and degrees of freedom? I calculated P(H|W) as E(occurances of H&&W)/E(occurances of W) = (1/2)/(3/2) = 1/3.