Dissolving Sleeping Beauty
The first thing I recommend to people interested in anthropics and the Sleeping Beauty Problem is, If a tree falls on Sleeping Beauty by ata. The whole article is excellent, I recommend reading in full, but the basic argument is simple:
There are different bets we can make regarding the coin. For example, we can pay out Sleeping Beauty each time they are woken up or only once.
If we pay them out each time they are woken up and then convert this betting structure into an equivalent probability, we obtain the thirder solution, whilst if we only pay them out once, we obtain the halver solution.
Therefore, the problem is underspecified. In order to be able to solve the problem, we need to explain which bet structure we are using.
Asking “Which is the tree bet structure?” is the same kind of confusion as asking, “If a tree fell in a forest, would it make a sound?”. Well, what exactly do you mean by “sound”? Answer that, and you’ll have the answer to your question.
I found this extremely insightful, but at the same time, I can understand why some people might find it unsatisfactory.
After all, when we ask, “What is the probably of the coin landing heads?”, we don’t have a particular bet structure in mind. And, I don’t just mean that we didn’t have an explicit bet structure in mind, even when asked to clarify I don’t think many people will immediately say, “Oh, I meant that bet structure!”.
What we have in the back of our minds isn’t some bet structure, but a whole lot of associations about what we mean and even though we can look for the bet structure that best matches these intuitions, the explanation given in “If a Tree Fell on Sleeping Beauty” won’t be satisfactory unless we bring these into the story. In this case, I suspect that there’s actually a single intution that does most of the work, so I’ll just focus on that.
The “Objective” Perspective
I suspect that most halvers have a strong intution that we’re trying to measure something objective about the coin. If we say, “Well, the probability of a coin landing on heads isn’t something intrinsic to it, but depends on the environment we put it in”, they’ll still try to make it as objective as possible by specifying the environment as precisely as they can and insisting on the most reliable-seeming way of measuring it.
In particular, to the extent that a particular observer affects the observation, something has gone wrong. Sleeping beauty gets woken up twice? “So what?”, they’ll say, ’That’s a distortion, not something we’re trying to remember? Obviously our experimental protocol should adjust to remove this distortion”. From this perspective, it’s as though we had a malicious experimentor who recorded tails twice every time it occurs. It’s outside of the protocol, so we halve in order to make it as if they followed it correctly. Or we, can imagine someone measuring the width of a door. Asking, “Well, what is the width of the door if we accidentally record it in inches rather than centimeters?”. It seems like a pointless question.
The “Subjective” Perspective
Some people have the intuition that we’re trying to measure something subjective. The appeal of this perspective is that it helps you in whatever situation you are in. They might consider those who want more objectivity to be unpragmatic. Their only question is, “What should I expect?”. Once you tell them that if you keep running the experiment, then they’ll be told that the coin came up head approximately a third of the times, that’s the matter is settled. If every time they’re asked about the coin they act on the basis of the one third probability—including making bets—then the result will be what they expected.
“What does it mean to say that the probability is really half?”, they may ask, “Pragmatically, I observe it a third of the time and if it doesn’t make a difference to the observations then it isn’t real or, even if is real, it’s completely irrelevant! Maybe I have been woken up twice, but I can’t know that, so I can’t adjust for it! Remember that probability is relative to the observer. Given a normal coin in normal circumstances, I’d assign p=0.5 to the coin being heads, but once I see the coin, my probability becomes p=1. Nothing about the coin has changed, just my epistemic position”.
The “Objective” Response
Since I’m trying to make both sides sound reasonable, I think I need to respond to provide an “objective” response to the “subjective” position. The “objective” position can be clarified as follows: that the claim is only that given a particular epistemic position regarding the coin toss, we can imagine an “objective” experiment corresponding to that which fits our experimental assumptions and not that our epistemic position is independent of our agent. That is, our knowledge of ourselves as an agent is only relevant in so far as it allows us to correct for the distortions we bring and is not part of what we are measuring.
Next we consider the notion that being doubled is irrelevant because we can’t observe it. We can respond that even if we can’t observe it, we can imagine an external observer outside the room who can. A good analogy is if we imagine the following experiment: Sleeping Beauty is asked to measure the height of an object, then put to sleep. When they are woken up, they have no memory of the first room apart from the figure they produced. They are given another object which is twice as tall and another ruler, but with the ruler having the marks twice as far apart. They are then asked whether the two objects were the same height.
Someone supporting the “subjective” view might say that for them the two objects are the same height for the subject, as the subject has no way of knowing that the height is different. As far, as I am concerned, this is a pure word game and the fact that an external observer (say someone like us who has the situation described to them) can see what happened means that the heights are different, even if we were to say Sleeping Beauty lacks the language to say what happened (which I don’t think is true).
My Position
As I’ve already indicated, my position that that a certain strong version of the “subjective” perspective—that presents the “objective” perspective as meaningless—is implausible. However, I’m not trying to dismiss the softer version which merely uses claim that it is more convenient to use “probability” to refer to the more subjective version. I happen to think both sides have a point and a reasonable case for why it makes more sense to define probability their way. And, I’m in agreement with Ata that the choice we ultimately make is merely a matter of convention. So, this ends up being about how draw the map and not what the territory is like.
Interesting take. Though I would thought halfers are the more “subjective” camp while thirders are the more “objective” camp. Main reason being halfers would have to endorse perspective disagreement: that two fully communicating agents can rightfully give different probabilities to the same question. While thirders would think they have to agree. (e.g. let someone else say Alex sleep in the same room as Beauty. He is going to be awakened on one random morning only. If Alex wakes up together with Beauty, he would reduce the probability of Heads to 1⁄3, whereas seeing Alex awake or asleep would not change Beauty’s probability. The two can communicate yet no one would change their answer) Halfers would have a hard time to justify this disagreement unless recognizing something irreducible/incommunicable about perspectives, about being Beauty herself at that moment. In my opinion that is more “subjective” than counting the relative frequency by number of awakenings as the thirders did.
Thanks, this comment is useful and if I read it before I’d written the post, then I would have been able to write a better post :-).
I agree that from initial appearances the halfer position appears more “subjective” and the thirder more “objective”, but I would also say surface level appearances can be deceiving.
Before I respond I’d like to ask you to clarify how Alex is woken up. If the coin is heads is he always woken up on Monday? And if the coin is tails, does he have a 50% chance of being woken up on Monday and a 50% chance of being woken up on Tuesday?
Or do you mean that whenever Sleeping beauty is woken up, he has a 1⁄3 chance of waking as well, with a chance that on some iterations that Alex is never woken up?
Or does it work some other way?
Sorry I could have give a better description of the disagreement. Alex is going to be waken up only once, whether it is Monday or Tuesday is random, independent of the coin toss. So when he wake up he will for sure find Beauty awake given Tails, and expect to find beauty awake or asleep with equal chances given Heads. For him seeing Beauty awake will reduce the probability of Heads to 1⁄3. Yet when Beauty wakes up she shall expect to find Alex asleep or awake with equal chance regardless of the coin result. They can communicate however they like. Yet none of them will change their answer.
I believe Katja Grace also pointed out the problem of disagreement (with a more complicated example imo) in her post. John Pittard also discussed this in his paper: When Beauties Disagree:Why Halfers Should Affirm Robust Perspectivalism. Though I personally don’t agree with their explanations.
Okay, well given that, I agree that if Alex is woken up with beauty he should change his probability of heads to 1⁄3. Of course, this isn’t P(Heads) anymore, but P(Heads|Beauty awake).
So I guess your issue then is that beauty also says that P(Heads)=1/2 whilst P(Heads|Beauty awake) = 1⁄3, when we might expect them to be the same thing. After all, beauty’s value of P(Heads) is based on all information beauty possesses and beauty knows that they are awake.
Well, I guess if we adopt what I’ve called the subjectivist intutions, then the above argument follows. But if we’re following the more objectivist intuitions then probability is only defined when there is no double counting going on; or if double counting is going on, but we add in an adjustment factor to undo it.
The thing to notice above is that P(Heads) and P(Heads|Beauty awake) are constructed over different sample spaces. The first is over just {{Heads},{Tails}}, while the later is over {{Heads, Mon},{Heads, Tues}, {Tails, Mon}, {Tails, Tues}} (Note: {Heads, Tues} never actually occurs).
And that’s why from the objectivist perspective they result in different probabilities—because they are asking different questions! In the first, knowledge of beauty’s awakeness state is taken to be relevant only in so far as it helps us to remove the distortions of beauty sometimes being asleep. While in the second, we’re trying to intentionally ignore the cases when beauty is asleep.
I think halfers would say from Beauty’s perspective P(Heads) and P(Heads|I am awake now) means the same thing. Any reasoning done by Beauty is based on the fact that “I am awake now”. Which is the basis for the argument of no new information: Beauty knew on Sunday she would find herself awake during the experiment. If halfers stand by this argument, (which I think is correct) then P(Heads) and P(Heads|I am awake) ought to both be half.
I didn’t use P(Heads|Beauty awake) because that kind of implies an outsider (e.g. Alex’s) perspective. I find that important. I think the disagreement is because for Alex, as long as he finds Beauty awake, he cannot tell which awakening it is. The complimentary event of seeing Beauty awake for him is to find beauty asleep. Whereas from Beauty’s perspective she focuses on this specific awakening. The complementary event from her perspective is finding Alex asleep in this awakening. That does not correspond to Alex finding Beauty asleep due to the memory wipe and possible double awakening of Beauty. (I don’t know if this explanation is easy to understand. It’s hard to express in anthropic paradoxes involving different moments. (I explained the same disagreement using a cloning thought experiment)[https://www.sleepingbeautyproblem.com/part-4-perspective-disagreement/]. Which I think is clearer)
In the end I do agree they are answering different questions. But still think halfers in some sense are more subjective.
Yes, at first glance it seems natural to assume this, but I see rejecting that claim as the only consistent way of developing the halfer view.
There’s a sense in which that’s true, which is that the fact that Beauty is awake is available for beauty to use in whatever calculations they perform. However, this isn’t the say as claiming they have to use it a particular way.
P(Heads|I am awake now) doesn’t just mean “I am awake now” is available in beauty’s databank, but also indicates that instead of sampling coin flips we’re sampling times when beauty is awake. So that’s the suble slight of hand; the invisible shift in assumptions.
The halfer position poorly developed is more subjective. Although maybe most halfers develop it that way, I don’t know.
This is what I think, thus our disagreement: Why should indexicals such as now, today or in some other anthropic problems I or myself, be considered as a sample at all? In my opinion they should be regarded as fundamentals/irreducibles. Their meanings are primitively understood from a given perspective. And the perspective disagreement is caused by this irreducible nature. I tried to explain my idea in this post. And that’s why I don’t think we should throwout the “no new information” argument too early.
My idea is by no means popular. And I am not trying to persuade you in accepting it. Just want to show the possibility that some halfer argument can be regarded as more “subjective based”. Whether they are poorly developed or not, I don’t know. Maybe time will tell.
I’m confused. I don’t think I understand what it means to regard them as irreducibles instead of a sample?
Take now as an example. I suggest it refers to the moment most immediate to the perception and subjective experience. A primitively understood concept that cannot be further reduced. So “I’m awake now” is a just a logical truth, i.e. no new information.
While traditionally people like to treat now as a randomly sampled moment during the experiment. As you did in the previous reply stating by using “I am awake now” indicates we are sampling times. People disagree about the exact sampling method (SSA or SIA). But I am suggesting now should not be regarded as a random sample at all.
We agree that the only objective chance in this setup is (a) the coin and (b) the choice of Alex’s waking day, right? So that’s a sample space of 4 outcomes, each equally probable a priori.
Alex and Beauty can communicate only when they are both awake. For both of them, this eliminates the outcome {Heads,AlexWakesOnTuesday}, leading both to the conclusion that Pr(Heads|both awake) = 13. No disagreement.
What you are describing, 4 equal probable priori in terms of Monday/Tuesday is thirder’s logic. Then Beauty doesn’t even have to check on Alex, by realizing I’m awake beauty would eliminate (Heads, Tuesday) already. Seeing Alex changes nothing. Thirderism will not lead to any perspective disagreement.
One the other hand Halfers suggest finding “I am awake today” is to be expected (no new information) due to the experiment setup of one guaranteed awakening. Combining the status of Alex they will say the 4 equal probable priori are (Heads, Alex is awake today), (Tails, Alex is awake today),(Heads, Alex is asleep today), (Tails, Alex is asleep today). Finding Alex awake eliminate the last two. The probability of Heads remains unchanged at 1⁄2. Again seeing Alex changes nothing.
If you are suggesting Halfers should change the probability of Heads to 1⁄3 after seeing Alex awake then I don’t see how that can be coherent. Just adding an independent component (Alex’s random awakening) should not give Beauty additional information about the coin toss. Similar idea has been brought by Michael Titelbaum’s Technicolor Beauty and Radford Neal’s FNC. Both suggest Thirderism (or almost Thirderism in the case of FNC) is correct instead of Halfers.
No, am I not Thirding. Thirders consider Beauty’s wake-ups on Monday and Tuesday to be “events”. That’s not what I’m doing here.
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We’ll need to be a bit more specific about your Alex & Beauty set-up: I hope you don’t mind if we specify how Alex’s waking day is chosen. Let’s say it’s chosen to be Monday if a fair die roll turns out Odd, or Tuesday if Even.
Now, you must agree that there are precisely two independent sources of objective chance in your experiment: the coin and the die roll. There are four equiprobable outcomes, of course:
Pr(Heads,Odd) = 14
Pr(Heads,Even) = 14
Pr(Tails,Odd) = 14
Pr(Tails,Even) = 14
When Beauty sees Alex awake, she must discount the outcome (Heads,Even) from the sample space, leaving Pr(H,O) = Pr(T,O) = Pr(T,E) = 13. I really hope that everyone should agree that those are the correct posterior objective chances.
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This analysis says nothing about Beauty’s subjective credence for Heads, nor does it attempt to justify any philosophical issues arising. We can have that philosophical debate later. But let’s ground the discussion in something that’s not open to interpretation by at least agreeing on the objective chances.
I think using dice roll to decide Alex’s awakening is an excellent idea.
Back to the topic at hand, what I have been arguing is that the correct halfer argument should not be based on objective chance. Thus my disagreement with Chris so far. I think it should instead be based on the irreducible nature of perspectives, that it roots in the subjective. That’s why I brought up the perspective disagreement in the first place.
What you have argued: that after seeing Alex awake Beauty should eliminate one of the 4 events, reflects the fact that Alex has met Beauty in an awakening. But it does not use the whole information Beauty has: she specifically knows Alex has met Beauty in this awakening. And the specification of this awakening is based on her perspective, as it is the one which she has subjective experience of.
You said you are not thirding, only considering the objective chance. I believe you. But I was saying by considering the objective chance this way one has to endorse Thirderism. For example, if Beauty checks on Alex and see him asleep instead, by the same logic she must discount (Heads, Odd), leaving P(H,E) = P(T,O) = P(T,E) = 1⁄3. So it doesn’t matter whether Beauty finds Alex awake or asleep, she will say P(H)=1/3 regardless. Then Beauty should’ve regard 1⁄3 as the correct answer even before looking at Alex anyway. (This argument is due to Michael Titelbaum ’s Technicolor Beauty)
So do I think the objective chance you presented are correct? Yes, I do. Just as Beauty should think Alex’s answer of 1⁄3 is correct. Do I think Beauty should give the same answer of 1/3? No, I don’t. Because at the end of the day I am a halfer, and I believe perspectives are fundamental, the disagreement between Alex and Beauty is valid.
That’s good, I just wanted to establish that baseline. I think it’s healthy for everyone to confirm their understanding of the objective chances before moving on to debating philosophical issues of credence. Thank you.
No. If I asked you to calculate the objective chances then I hope you (and everyone else here) would perform exactly the same calculation. Presented as a maths problem, as I did, it says nothing about my (or your, or anyone else’s) philosophical leanings.
Perhaps you meant that by equating the objective Pr(...) with Credence one has to be a Thirder?
I do like the example though. It confused me enough that I felt I had to post again (replying to myself … bad form). What follows is pure speculation.
I think there’s something here about scopes/closures.
Within the scope of a single wake-up, “Alex is awake at the same time as me” is admissible as an event: specifically the subset { (H,O), (T,O), (T,E) } of the sample space.
Within the scope of the whole experiment, “Alex is awake at the same time as me” is inadmissible as an event: it is not a well-defined subset of the sample space.
So for games whose payout decomposes additively over wake-ups, you may consider each wake-up in isolation and, if you do so, you must use all available events including Alex’s sleeping state, Thirder-style.
For games whose payout does not decompose additively over wake-ups, you must consider the experiment as a whole and, in doing so, find that you have no new information, Halfer-style.
Hey, don’t worry about replying to oneself. I do it quite often :)
I just realized the root cause of our debate after reading this:
You regard “objective chance” as the baseline, any philosophical leaning is built on top of that. But I think the philosophical aspect of the problem cuts deeper than that.I regard (postulates) perspective reasoning as fundamental and objective reasoning as derived. While in most people recognize (postulate) objective reasoning (i.e. perspective-independent reasoning) as fundamental and indexicals as additional information that need to be incorporated on top of objective reasoning. That’s why I call my approach Perspective Based Reasoning.
So when seeing “objective chance” I was not considering it as the baseline, but treating it as “thinking from a god’s eye view” (which give the same answer as Alex’s perspective). To me it is no more fundamental than thinking from Beauty’s first-person viewpoint. Just two different systems of reasoning. And the two perspective gives different answers, both are valid, thus the disagreement.
It is why when you ask what should Beauty think about the objective chance I keep stressing what should Beauty think from her perspective. You want to leave the philosophical leanings out of it for now, but to me using objective chance as the baseline already has a philosophical leaning.
It is also why I don’t think halfers should change the probability to 1⁄3 after seeing Alex awake. I think the best rebuttal is still the one I just discussed. By the same logic Beauty would change the probability to 1⁄3 after seeing Alex asleep as well. Since it is going to be 1⁄3 regardless of Alex’s status her probability should have been 1⁄3 right from the start, i.e. after waking up.
I don’t think we’re a million miles apart on this. We can both tolerate Halfing and Thirding, in your case because they’re both reasonable perspectives and in my case because it doesn’t matter anyway. Your characterisation of our disagreement seems about right: let’s come back to that in the comments of some future unsuspecting author’s post on Sleeping Beauty :)
I’ve added the Sleeping Beauty tag for you.
Since the rational agent in SBP has full knowledge of the experimental set-up, she has no need of any subjective credence: any bet she takes can be made solely on the facts of the experiment.
Paradoxes like SBP seem to evaporate if you distinguish two types of credence: (A) credence for the experiment that the agent is a part of, vs. (B) credence for the outcomes encountered within the experiment. Type (A) credence for the experimental protocol is all you need for decision-making. Type (B) credence is implied by the experimental protocol and adds nothing to the agent’s decision-making capabilities, so you’re free to define that however you wish.
See https://www.lesswrong.com/posts/8ecgHXWQBiuY97B2X/ for more detail.
From your description here, it sounds like we’re arguing essentially the same thing, but you think it’s different? How?
I think my argument is more stark: I’m saying that there are fundamentally two types of credence, (A) and (B) in the comment above, and only one of them is relevant to rational decision-making.
I agree with intuition being the thing that separates Halfers and Thirders, but I’m saying that they’re fighting over irrelevant Type (B) credence—“inner” credence in the OOBR post I linked to.
“And only one of them is relevant to rational decision-making”—okay, that’s a significant difference!