# Ape in the coat

Karma: 761
• The second one looks “obvious” from symmetry considerations but actually formalizing seems harder than expected.

Exactly! I’m glad that you actually engaged with the problem.

The first step is to realize that here “today” can’t mean “Monday xor Tuesday” because such event never happens. On every iteration of experiment both Monday and Tuesday are realized. So we can’t say that the participant knows that they are awakened on Monday xor Tuesday.

Can we say that participant knows that they are awakened on Monday or Tuesday? Sure. As a matter of fact:

P(Monday or Tuesday) = 1

This works, here probability that the coin is Heads in this iteration of the experiment happens to be the same as what our intuition is telling us P(Heads|Today) is supposed to be, however we still can’t define “Today is Monday”:

P(Monday|Monday or Tuesday) = P(Monday) = 1

Which doesn’t fit our intuition.

How can this be? How can we have a seeminglly well-defined probability for “Today the coin is Heads” but not for “Today is Monday”? Either “Today” is well-defined or it’s not, right? Take some time to think about it.

What do we actually mean when we say that on an awakening the participant supposed to believe that the coin is Heads with 50% probability? Is it really about this day in particular? Or is it about something else?

The answer is: we actually mean, that on any day of the experiment be it Monday or Tuesday the participant is supposed to believe that the coin is Heads with 50% probability. We can not formally specify “Today” in this problem but there is a clever, almost cheating way to specify “Anyday” without breaking anything.

This is not easy. It requires a way to define P(A|B), when P(B) itself is undefined which is unconventional. But, moreover, it requires symmetry. P(Heads|Monday) has to be equal to P(Heads|Tuesday) only then we have a coherent P(Heads|Anyday).

• First of all, that can’t possibly be right.

I understand that it all may be somewhat counterintuitive. I’ll try to answer whatever questions you have. If you think you have some way to formally define what “Today” means in Sleeping Beauty—feel free to try.

Second of all, it goes against everything you’ve been saying for the entire series.

Seems very much in accordance with what I’ve been saying.

Throughout the series I keep repeating the point that all we need to solve anthropics is to follow probability theory where it leads and then there will be no paradoxes. This is exactly what I’m doing here. There is no formal way to define “Today is Monday” in Sleeping Beauty and so I simply accept this, as the math tells me to, and then the “paradox” immediately resolves.

Suppose someone who has never heard of the experiment happens to call sleeping beauty on her cell phone during the experiment and ask her “hey, my watch died and now I don’t know what day it is; could you tell me whether today is Monday or Tuesday?” (This is probably a breach of protocol and they should have confiscated her phone until the end, but let’s ignore that.).

Are you saying that she has no good way to reason mathematically about that question? Suppose they told her “I’ll pay you a hundred bucks if it turns out you’re right, and it costs you nothing to be wrong, please just give me your best guess”. Are you saying there’s no way for her to make a good guess? If you’re not saying that, then since probabilities are more basic than utilities, shouldn’t she also have a credence?

Good question. First of all, as we are talking about betting I recommend you read the next post, where I explore it in more details, especially if you are not fluent in expected utility calculations.

Secondly, we can’t ignore the breach of the protocol. You see, if anything breaks the symmetry between awakening, the experiment changes in a substantial manner. See Rare Event Sleeping Beauty, where probability that the coin is Heads can actually be 13.

But we can make a similar situation without breaking the symmetry. Suppose that on every awakening a researcher comes to the room and proposes the Beauty to bet on which day it currently is. At which odds should the Beauty take the bet?

This is essentially the same betting scheme as ice-cream stand, which I deal with in the end of the previous comment.

• Sampling is not the way randomness is usually modelled in mathematics, partly because mathematics is deterministic and so you can’t model randomness in this way

As a matter of fact, it is modeled this way. To define probability function you need a sample space, from which exactly one outcome is “sampled” in every iteration of probability experiment.

But yes, the math is deterministic, so it’s not “true randomness” but a pseudo-randomness, so just like with every software library it’s hidden-variables model rather than Truly Stochastic model.

And this is why, I have troubles with the idea of “true randomness” being philosophically coherent. If there is no mathematical way to describe it, in which way can we say that it’s coherent?

Like, the point of many-worlds theory in practice isn’t to postulate that we should go further away from quantum mechanics by assuming that everything is secretly deterministic.

The point is to describe quantum mechanics as it is. If quantum mechanics is deterministic we want to describe it as deterministic. If quantum mechanics is not deterministic we do not want to descrive quantum mechanic as deterministic. The fact that many-words interpretation describes quantum mechanics is deterministic can be considered “going further from quantum mechanics” only if it’s, in fact, not deterministic, which is not known to be the case. QM just has a vibe of “randomness” and “indeterminism” around it, due to historic reasons, but actually whether it deterministic or not is an open question.

• You are already aware of this but, for the benefits of other readers all mention it anyway.

In this post I demonstrate that the narrative of betting arguments validating thirdism is generally wrong and is just a result of the fact that the first and therefore most popular ha;fer model is wrong.

Both thirders and halfers, following the correct model, make the same bets in Sleeping Beauty, though for different reasons. The disagreement is about how to factorize the product of probability of event and utility of event.

And if we investigate a bit deeper, halfer way to do it makes more sense, because its utilities do not shift back and forth during the same iteration of the experiment.

• 10 Apr 2024 8:38 UTC
−1 points
−6

I’m a bit surprised that you think this way, considering that you’ve basically solved the problem yourself in this comment.

P(Heads & Monday) = P(Tails & Monday) = 12

P(Tails & Monday) = P(Tails&Tuesday) = 12

Because Tails&Monday and Tails&Tuesday are the exact same event.

The mistake that everyone seem to be making is thinking that Monday/​Tuesday mean “This awakening is happening during Monday/​Tuesday”. But such events are ill-defined in the Sleeping Beauty setting. On Tails both Monday and Tuesday awakenings are supposed to happen in the same iteration of probability experiment and the Beauty is fully aware of that, so she can’t treat them as individual mutual exclusive outcomes.

You can only lawfully talk about “In this iteration of probability experiment Monday/​Tuesday awakening happens”.

In this post I explain it in more details.

• Meta: the notion of writing probability 101 wasn’t addressed to you specifically. It was a release of my accumulated frustration of not-particularly productive arguments with several different people which again and again led to the realizations that the crux of disagreement lies in the most basics, from which you are only one person.

You are confusing to talk to, with your manner to rise seemingly unrelated points and then immediately drop them. And yet you didn’t deserve the full emotional blow that you apparently received and I’m sorry about it.

Writing a probability 101 seems to me as a constructive solution to such situations, anyway. It would provide opportunity to resolve this kinds of disagreements as soon as they arise, instead of having to backtrack to them from a very specific topic. I may still add it to my todo list.

Ah yes, clearly, the problem is that I don’t understand basic probability theory. (I’m a bit sad that this conversation happened to take place with my pseudonymous account.) In my previous comment, I explicitily prepared to preempt your confusion about seeing the English word ‘experiment’ with my paragraph (the part of it that you, for some reason, did not quote), and specifically linking a wiki which only contains the mathematical part of ‘probability’, and not philosophical interpretations that are paired with it commonly, but alas, it didn’t matter.

i figured that either you don’t know what “probability experiment” is or you are being confusing on purpose. I prefer to err in the direction of good faith, so the former was my initial hypothesis.

Now, considering that you admit that you you were perfectly aware of what I was talking about, to the point where you specifically tried to cherry pick around it, the latter became more likely. Please don’t do it anymore. Communication is hard as it is. If you know what a well established thing is, but believe it’s wrong—just say so.

Nevertheless, from this exchange, I believe, I now understand that you think that “probability experiment” isn’t a mathematical concept, but a philosophical one. I could just accept this for the sake of the argument, and we would be in a situation where we have a philosophical consensus about an issue, to a point where it’s a part of standard probability theory course that is taught to students, and you are trying to argue against it, which would put quite some burden of proof on your shoulders.

But, as a matter of fact, I don’t see anything preventing us from formally defining “probability experiment”. We already have a probability space. Now we just need a variable going from 1 to infinity for the iteration of probability experiment, and a function which takes sample space and the value of this variable as an input and returns one outcome that is realized in this particular iteration.

I said that I can translate the math of probability spaces to first order logic, and I explicitly said that our conversation can NOT be translated to first order logic as proof that it is not about math

Sorry, I misunderstood you.

Also a reminder that you you still haven’t addressed this:

If a mathematical probabilistic model fits some real world process—then the outcomes it produces has to have the same statistical properties as the outcomes of real world process.

If we agree on this philosophical statement, then we reduced the disagreement to a mathematical question, which I’ve already resolved in the post. If you disagree, then bring up some kind of philosophical argument which we will be able to explore.

Anyway, are you claiming that it’s impossible to formalize what “today” in “today the coin is Heads” means even in No-Coin-Toss problem? Why are you so certain that people have to have credence in this statement then? Would you then be proven wrong if I indeed formally specify what “Today” means?

Because, as I said, it’s quite easy.

Today = Monday xor Tuesday

P(Today) = P(Monday xor Tuesday) = 1

Likewise we can talk about “Today is Monday”:

P(Monday|Today) = P(Monday|Monday xor Tuesday) = P(Monday) = 12

Now, do you see, why this method doesn’t work for Two Awakenings Either Way and Sleeping Beauty problems?

If you are not ready to accept that people have various levels of belief in the statement “Today is Monday” at all times, then I don’t think this conversation can go anywhere, to be honest. This is an extremely basic fact about reality.

In reality people may have all kind of confused beliefs and ill-defined concepts in their heads. But the question of Sleeping Beauty problem is about what the ideal rational agent is supposed to believe. When I say “Beauty does not have such credence” I mean, that an ideal rational agent ought not to. That probability of such event is ill-defined.

As you may’ve noticed I’ve successfully explained the difference in real life beliefs about optimal actions in the ice-cream stand scenario, without using such ill-defined probabilities.

The tragedy of the whole situation is that people keep thinking that.

Everything is “about philosophy” until you find a better way to formalize it. Here we have a better way to formalize the issue, which you keep ignoring. Let me spell it for you once more:

If a mathematical probabilistic model fits some real world process—then the outcomes it produces has to have the same statistical properties as the outcomes of real world process.

If we agree on this philosophical statement, then we reduced the disagreement to a mathematical question, which I’ve already resolved in the post. If you disagree, then bring up some kind of philosophical argument which we will be able to explore.

If you are a layman

I’m not. And frankly, it baffles me that you think that you need to explain that it’s possible to talk about math using natural language, to a person who has been doing it for multiple posts in a row.

mathematical objects itself have no concept of ‘experiment’ or ‘time’ or anything like those.

The more I post about anthropics the clearer it becomes that I should’ve started with posting about probability theory 101. My naive hopes that average LessWrong reader is well familiar with the basics and just confused about more complicated cases are crushed beyond salvation.

Can a probability space model a person’s beliefs at a certain point in time?

This question is vague in a similar manner to what I’ve seen from Lewis’s paper. Let’s specify it, so that we both understand what we are talking about

Did you mean to ask 1. or 2:

1. Can a probability space at all model some person’s belif in some circumstance at some specific point in time?

2. Can a probability space always model any person’s belief in any circumstances at any unspecified point in time?

The way I understand it, we agree on 1. but disagree on 2. There are definetely situations where you can correctly model uncertanity about time via probability theory. As a matter of fact, it’s most of the cases. You won’t be able to resolve our disagreement by pointing to such situations—we agree on them.

But you seem to have generalized that it means that probability theory always has to be able to do it. And I disagree. Probability space can model only aspects of reality that can be expressed in terms of it. If you want to express uncertanity between “today is Monday” or “today is Tuesday” you need a probability space for which Monday and Tuesday are mutually exclusive outcomes and it’s possible to design a specific setting—like the one in Sleeping Beauty—where they are not, where on the same trial both Monday and Tuesday are realized and the participant is well aware of it.

In particular, Beauty, when awoken, has a certain credence in the statement “Today is Monday.”

No she does not. And it’s easy to see if you actually try to formally specify what is meant here by “today” and what is meant by “today” in regular scenarios. Consider me calling your bluff about being ready to translate to first order logic at any moment.

Let’s make it three different situations:

1. No-Coin-Toss problem.

2. Two awakenings with memory loss, regardless of the outcome of the coin.

3. Regular Sleeping Beauty

Your goal is to formally define “today” using first order logic so that a person participating in such experiments could coherently talk about event “today the coin is Heads”.

My claim is: it’s very easy to do so in 1. It’s a harder, but still doable in 2. And it’s not possible to do so in 3, without contradicting the math of probability theory.

setting up an icecream stand which is only open on Monday in one direction from the lab, another in the opposite direction which is only open on Tuesday and making this fact known to subjects of an experiment who are then asked to give you icecream and observe where the go

This is not a question about simply probability/​credence. It also involves utilities and it’s implicitly assumed that the participant preferes to walk for less distance than more. Essentially you propose a betting scheme where:

P(Monday)U(Monday) = P(Tuesday)U(Tuesday)

According to my model P(Monday) = 1, P(Tuesday) = 12, so:

2U(Monday) = U(Tuesday), therefore odds are 2:1. As you see, it deals with such situations without any problem.

• What she is really surprised about however, is not that she has observed an unlikely event ({HHTHTHHT}), but that she has observed an unexpected pattern.

Why do you oppose these two things to each other? Talking about patterns is just another way to describe the same fact.

In this case, the coincidence of the sequence she had in mind and the sequence produced by the coin tosses constitutes a symmetry which our mind readily detects and classifies as such a pattern.

Well, yes. Or you can say that having a specific combination in mind allowed to observe event “this specific combination” instead of “any combination”. Once again this is just using different language to talk about the same thing.

One could also say that she has not just observed the event {HHTHTHHT} alone, but also the coincidence which can be regarded as an event, too. Both events, the actual coin toss sequence and the coincidence, are unlikely events and both become extremely unlikely with longer sequences.

Oh! Are you saying that she has observed the intersection of two rare events: “HHTHTHHT was produced by coin tossing” and “HHTHTHHT was the sequence that I came up with in my mind” both of which have probability 1/​2^8 so now she is surprised as if she observed an event with (1/​2^8)^2?

That’s not actually the case. If the person came up with some other combination and then it was realized on the coin tosses the surprise would be the same—there are 1/​2^8 degrees of dreedom here—for every possible combination of Heads and Tails with lenghth 8. So the probability of the observed event is still 1/​2^8.

• I meant to show you that if you don’t start out with “centered worlds don’t work”, you CAN make it work

The clever way isn’t that clever to be honest. It’s literally just: don’t assume that it does not work and try it.

I didn’t start believing that “centred worlds don’t work”. I suspect you got this impression mostly because you were reading the posts in the wrong order. I started from trying the existent models noticed when they behave weirdly if we assume that they are describing Sleeping Beauty and then noticed that they are actually talking about different problems—for which their behavior is completely normal.

And then, while trying to understand what is going on, I stumbled at the notion of centred possible worlds and their complete lack of mathematical justification and it opened my eyes. And then I was immediately able to construct the correct model, which completely resolves the paradox, adds up to normality and has no issues whatsoever.

But in hindsight, if I did start from the assumption that centred possible worlds do not work, - that would be the smart thing to do and I’d save me a lot of time.

With my previous comment I meant to show you that if you don’t start out with “centered worlds don’t work”, you CAN make it work (very important: here, I haven’t yet said that this is how it works or how it ought to work, merely that it CAN work without some axiom of probability getting hurt).

Well, you didn’t. All this time you’ve just been insisting on a privileged treatment for them: “Can work until proven otherwise”. Now, that’s not how math works. If you come up with some new concept, be so kind to prove that they are coherent mathematical entities and what are their properties. I’m more than willing to listen to such attempts. The problem is—there are none. People just seem to think that saying “first person perspective” allows them to build sample space from non-mutually exclusive outcomes.

Still, I struggle to see what your objection is apart form your intuition that “NO! It can’t work!”

By definition of a sample space it can be constructed only from elementary outcomes which has to be mutually exclusive. Tails&Monday and Tails&Tuesday are not mutually exclusive—they happen to the same person in the same iteration of probability experiment during the same outcome of the coin toss. “Centredness” framework attempts to treat them as elementary outcomes, regardless. Therefore, it contradicts the definition of a sample space.

This is what statistical analysis clearly demonstrates. If a mathematical probabilistic model fits some real world process—then the outcomes it produces has to have the same statistical properties as the outcomes of real world process. All “centred” models produce outcomes with different properties, compared to what actually running Sleeping Beauty experiment would do. Therefore they do not correctly fit the Sleeping Beauty experiment.

I want to argue how it CAN work in another way with credences/​centeredness/​bayesianism.

If you want to understand how centered world/​credence/​bayesian epistemology works

Don’t mix bayesianism and credences with this “centredness” nonsense. Bayesianism is not in trouble—I’ve been appealing to Bayes theorem a lot throughout my posts and it’s been working just fine. Likewise, credence in the event is simply probability conditional on all the evidence—I’m exploring all manner of conditional probabilities in my model. Bayesianism and credences are not some “another way” It is the exact same way. It’s probability theory. “Centredness”—is not.

experiment isn’t a good word, because it might lock you into a third-person view

Your statistical analysis is of course also assumes the third-person

I don’t understand what you mean by “third-person view” here, and I suspect neither do you.

Statistical test is very much about Beauty’s perspective—only awakenings that she experiences are noted down, not all the states of the experiment. Heads&Tuesday isn’t added to the list, which would be the case if we were talking about third person perspective.

On the other hand, when you were talking about justifying an update on awakening, you are treating the situation from the observer perspective—someone who has non zero probability for Heads&Tuesday outcome and could realistically not observe the Beauty being awakened and, therefore, updates when sees her indeed awaken.

“Centred” models do not try to talk about Beauty’s perspective. They are treating different awakened states of the Beauty as if they are different people, existing independently of each other, therefore contradicting the conditions of the setting, according to which all the awakenings are happening to the same person. Unless, of course, there is some justification why treating Beauty’s awakened states this way is acceptable. The only thing resembling such justification, that I’ve encountered, is vaguely pointing towards the amnesia that the Beauty is experiencing, with which I deal in the section Effects of Amnesia. If there is something else—I’m open to consider it, but the initial burden of proof is on the “centredness” enthusiasts.

• I’ll start from adressing the actual crux of our disagreement

You often do this mistake in the text, but here it’s too important to not mention that “Awake” does not mean that “Beauty is awakened.”, it means that “Beauty is awake” (don’t forget that centeredness!) and, of course, Beauty is not awake if it is Tuesday and the coin is heads.

As I’ve written in this post, you can’t just said magical word “centredness” and think that you’ve solved the problem. If you wont a model that can have an event that changes its truth predicate with the passage of time during the same iteration of the probability experiment—you need to formally construct such model, rewriting all the probability theory from scratch, because our current probability theory doesn’t allow that.

In probability theory, one outcome of a sample space is realized per an iteration of experiment. And so for this iteration of experiment, every event which includes this outcome is considered True. All the “centred” models therefore, behave as if Sleeping Beauty consist of two outcomes of probability experiment. As if Monday and Tuesday happen at random and that to determine whether the Beauty has another awakening the coin is tossed anew. And because of it they contradict the conditions of the experiment, according to which Tails&Tuesday awakening always happen after Tails&Monday. Which is shown in Statistical Analysis section. It’s a model for random awakening not for current awakening that. Because current awakening is not random.

So no, I do not do this mistake in the text. This is the correct way to talk about Sleeping Beauty. Event “The Beauty is awaken in this experement” is properly defined. Event “The Beauty is awake on this particular day” is not, unless you find some new clever way to do it—feel free to try.

Consider the following problem: “Forgetful Brandon”

I must say, this problem is very unhelpful to this discussion. But sure, lets analyze it regardless.

I hope you agree that Brandon not actually doing the Bayesian calculation is irrelevant to the question.

I suppose? Such questions are usually about ideal rational agents, so yes, it shouldn’t matter, what a specific non-ideal agent does, but then why even add this extra complication to the question if it’s irrelevant?

Well, that’s his problem, honestly, I though we agreed that what he does is irrelevant to the question.

Also his behavior here is not as bad as what you want the Beauty to do—at least Brandon doesn’t update in favor of Heads on literally every iteration of experiment.

should we point out a failure of conservation of expected evidence?

I mean, if we want to explain Brandon’s failure at rationality—we should. The reason why Brian’s behaviour is not rational is exactly that—he fails at conservation of expected evidence. There are two possible signals that he may receive: “Yay”, “No yay and getting ice cream”. These signals are differently correclated with the outcome of the coin toss. If he behaved rationally he updated on both of them in opposite direction, therefore following the conservation of expected evidence.

In principle, it’s possible to construct a better example where Brandon doesn’t update not because of his personal flaws in rationality, but due to the specifics of the experiment. For example, if he couldn’t be sure when exactly Adam is supposed to shout. Say, Adam intended to shout one minute after he saw the result of the coin toss, but Brandon doesn’t knows it, according to his information Adam shouts “Yay” in the interval of three minutes sicnce the coin was tossed. And so he is still waiting, unupdated aftre just one minute.

But then, it won’t be irrelevant to the question as you seem to want it for some reason.

I don’t see why you object to Sleeping Beauty not doing the calculation in case she is not awakened. (Which is the only objection you wrote under the “Freqency Argument” model)

I do not object to the fact that the Beauty doesn’t do calculation in case she is not awakened—she literally can’t do it due to the setting of the experiment.

I object to Beauty predictably updating in favor of Tails when she awakens in every iteration of the experiment which is a blatant contradiction of conservation of expected evidence. Updating model, as a whole descrives Observer Sleeping Beauty problem, where the observer can legitimately not see that the Beauty is awake and therefore update on awakening is lawful

Which is the only objection you wrote under the “Freqency Argument” model

See also Towards the Correct Model where I point to core mathematical flaw of Frequency Argument—ignoring the fact that it works only when P(Heads|Awake) = 12 which is wrong for Sleeping Beauty. And, of course, Updating Model fails the Statistical Analysis as every other “centred” model.

Uninformed Sleeping Beauty

When the Beauty doesn’t know the actual setting of the experiment she has a different model, fitting her uninformed state of knowledge, when she is told what is actually going on she discards it and starts using the correct model from this post.

• Again, that depends.

I think, I talk about something like you point to here:

If I forget what is the current day of the week in my regular life, well, it’s only natural to start from a 17 prior per day and work from there. I can do it because the causal process that leads to me forgetting such information can be roughly modeled as a low probability occurrence which can happen to me at any day.

It wouldn’t be the case, if I was guaranteed to also forget the current day of the week on the next 6 days as well, after I forgot it on the first one. This would be a different causal process, with different properties—causation between forgetting—and it has to be modeled differently. But we do not actually encounter such situations in everyday life, and so our intuition is caught completely flat footed by them.

• I’ve started at your latest post and recursively tried to find where you made a mistake

I think you’d benefit more if you read them in the right order starting from here.

Philosophers answer “Why not?” to the question of centered worlds because nothing breaks and we want to consider the questions of ‘when are we now?’ and ‘where are we now?’.

Sure, we want a lot of things. But apparently we can’t always have everything we want. To preserve the truth statements we need to follow the math wherever it leads and not push it where we would like it to go. And where the math goes—that what we should want.

Am I understanding you correctly that you reject P(today is Monday) as a valid probability in general (not just in sleeping beauty)?

This post refers several alternative problems where P(today is Monday) is a coherent probability, such as Single Awakening and No-Coin-Toss problems, which were introduced in the previous post. And here I explain the core principle: when there is only one day that is observed in the one run of the experiment you can coherently define what “today” means—the day from this iteration of the experiment. A random day. Monday xor Tuesday.

This is how wrong models try to treat Monday and Tuesday in Sleeping Beauty. As if they happen at random. But they do not. There is an order between them, and so they can’t be treated this way. Today can’t be Monday xor Tuesday, because on Tails both Monday and Tuesday do happen.

As a matter of fact, there is another situation where you can coherently talk about “today”, which I initially missed. “Today” can mean “any day”. So, for example, in Technicolor Sleeping beauty from the next post, you can have coherent expectation to see red with 50% and blue with 50% on the day of your awakening, because for every day it’s the same. But you still can’t talk about “probability that the coin is Heads today” because on Monday and Tuesday these probabilities are different.

So in practice, the limitation is only about Sleeping Beauty type problems where there are multiple awakenings with memory loss in between per one iteration of experiment, and no consistent probabilities for every awakening. But generally, I think it’s always helpful to understand what exactly you mean by “today” in any probability theory problem.

axiomatically deciding that 13 is the wrong probability for sleeping beauty

I do not decide anything axiomatically. But I notice that existent axioms of probability theory do not allow to have predictable update in favor of Tails in 100% of iterations of experiment, neither they allow a fair coin toss to have unconditional probability for Heads equal 13.

And then I notice that the justification that people came up with for such situations, about “new type of evidence” that a person receives is based on nothing but some philosopher wanting it to be this way. He didn’t come with any new math, didn’t prove any theorems. He simply didn’t immediately notice any contradictions in his reasoning. And when an example was broiught up, he simply doubled dowm/​ Suffice to say, its absolutely not how anything supposed to work.

if everything else seems to work, is it not much simpler to accept that 13 is the correct answer and then you don’t have to give up considering whether today is Monday?

If everything actually worked then the situation would be quite different. However, my previous post explores how every attempt to model the Sleeping Beauty problem, based on the framework of centred possible worlds fail one way or another.

You can also clearly see it in Statistical Analysis section of this post. I don’t see how this argument can be refuted, frankly. If you treat Tails&Monday and Tails&Tuesday as different elementary outcomes then you can’t possibly keep their correct order, and it’s in the definition of the experiment that on Tails, Monday awakening is always followed by the Tuesday awakening and that the Beauty is fully aware of it. Events that happen in sequence can’t be mutually exclusive and vice versa. I’m even formally proving it in the comments here.

And so, we can just accept that Tails&Monday and Tails&Tuesday are the same outcome of the probability space and suddenly everything adds up to normality. No paradox, no issues with statistical analysis, no suboptimal bets, no unjustified updates and no ungrounded philosophical handwaving. Seems like the best deal to me!

• Well done!

Halfer and thirder are about answer to the initial question of the Sleeping Beauty problem: What is the probability that the coin landed tails when you awake in the experiment?

• Yes, you are correct, thanks!

# [Question] Should you re­fuse this bet in Tech­ni­color Sleep­ing Beauty?

4 Apr 2024 8:55 UTC
14 points
• A probability experiment is a repeatable process

On every iteration we have exactly one outcome from a sample space that is realized. And every event from event space which has this outcome is also assumed to be realized. When I say “experiment” I mean a particular iteration of it yes, because one run of sleeping beauty experiment correspond to one iteration of the probability experiment. I hope it cleared the possible misunderstanding.

THE OBSERVATION IS NOT AN EVENT

Event is not an outcome, it’s a set of one or more outcomes, from the sample space, which itself has to belong to the event space.

What you mean by “observation” is a bit of a mystery. Try tabooing it—after all probability space consists of only sample space, event space and probability function, no need to invoke this extra category for no reason.

A common way to avoid rebuttal

It’s also a common way to avoid unnecessary tangents. Don’t worry we will be back to it as soon as we deal with the more interesting issue, though I suspect then you will be able to resolve your confusion yourself.

No, thats how you try to misinterpret my version to fit your incorrect model. You use the term for Elga’s one-coin version as well. Strawman arguments are another avoidance technique.

I don’t think that correcting your misunderstanding about my position can be called “strawmanning”. If anything it is unintentional strawmannig from your side, but don’t worry, no offence taken.

Yes, One-coin-version has the exact same issue, where sequential awakenings Tails&Monday, Tails Tuesday are often treated as disconnected mutually exclusive outcomes.

But anyway, it’s kind of pointless to talk about it at this point when you’ve already agreed to the the fact that the correct sample space for two coins version is {HT_HH, TT_TH, TH_TT, HH_HT}. We agree on the model, let’s see where it leads.

Huh? What does “connect these pairs” mean to pairs that I already connected?

It means that you’ve finally done the right thing of course! You’ve stopped talking about individual awakenings as if they are themselves mutually exclusive outcomes and realized that you should be talking about the pairs of sequential awakenings treating them as a single outcome of an experiment. Well done!

No, I am not.

But apparently you still don’t exactly undertand the full consequences of it. But that’s okay, you’ve already done the most difficult step, I think the rest will be easier.

I am saying that I was able to construct a valid, and useful, sample space

And indeed you did! Once again—good job! But let’s take a minute and understand what it means.

Suppose that in a particular instance of the experiment outcome TT_TH happened. What does it mean for the Beauty? It means that she is awakened the first time before the second coin was turned and then awakened the second time after the coin was turned. This outcome encompases both her awakenings.

Likewise, when outcome HT_HH happens, the Beauty is awakened before the coin turn and is not awakened after the coin turn. This outcome describes both her awakening astate and her sleeping state.

And so on with other two outcomes. Are we on the same page here?

If there was no amnesia the Beauty could easily distinguish between the outcomes where she awakes twice orr only once. But with amnesia she is none the wiser. In the moment of awakening they feel exactly the same for her.

The thing you need to properly acknowledge, is that in the probability space you’ve constructed P(Heads) doesn’t attempt to describe probability of first coin being Heads in this awakening. Once again—awakenings are not treated as outcomes themselves anymore. Now it describes probability that the coin is Heads in this iteration of experiment as a whole.

I understand, that this may be counterintuitive for you if you got accustomed to the heresy of centred possible words. This is fine—take your time. Play with the model a bit, see what kind of events you can express with it, how it relates to betting, make yourself accustomed to it. There is no rush.

I am describing how she knows that she is in either the first observation or the second.

You’ve described two pairs of mutually exclusive events.

{HT_HH, TT_TH, TH_TT}; {HH_HT} - Beauty is awakened before the coin turn; Beauty is not awakened before the coin turn

{HH_HT, TT_TH, TH_TT}; {HT_HH} - Beauty is awakened after the coin turn; Beauty is not awakened after the coin turn.

Feel free to validate that it’s indeed what these events are.

And you correctly notice that

and

Once again, I completely agree with you! This is a correct result that we can validate through a betting scheme. A Beauty that bets on Tails exclusively when she is awoken before the coin is turned is correct 66% of iterations of experiment. A Beauty that bets on Tails exclusively when she is awoken after the coin is turned is also correct in 66% of iterations of experiment. Once again, you do not have to trust me here, you are free to check this result yourself via a simulation.

And from this you assumed that the Beauty can always reason that the awakening that she is experiencing either happened before the second coin was turned or after the second coin was turned and therefore P(Heads|(HT_HH, TT_TH, TH_TT), (HH_HT, TT_TH, TH_TT)) = 13.

But this is clearly wrong, which is very easy to see.

First of all

Which is 12, because it is probability of Heads conditional on the whole sample space, where exactly 12 of the outcomes are such that the first coin is Heads. But also we may appeal to a betting argument, a Beauty that simply bets on Tails every time is correct only in 50% of experiments. This is a well known result—that per experiment betting in Sleeping Beauty should be done at 1:1 odds. But you are, nevertheless, also free to validate it yourself if you wish.

With me so far?

Now you have an opportunity to find the mistake in your reasoning yourself. It’s an actually interesting result, with fascinating consequences, by the way. And I don’t think that many people properly understand it, based on the current level of discourse about Sleeping Beauty and anthropics as a whole. So, even though it’s going to be a bit embarrasing for you, you will also discover rare and curious new piece of knowledge as a compensation for it.

• With these definitions, we can see that the SB Problem is one random experiment with a single result.

Yes! I’m so glad you finally got it! And the fact that you simply needed to remind yourself of the foundations of probability theory validates my suspicion that it’s indeed the solution for the problem. You may want to reread the post and notice that this is exactly what I’ve been talking about the whole time.

Now, I ask you to hold in mind the fact that “SB Problem is one random experiment with a single result”. We are goin to use this realization later.

That result is observed twice (yes, it is; remaining asleep is an observation of a result that we never make use of, so awareness as it occurs is irrelevant

This is false, but not crucial. We can postpone this for later.

What you call “sequential events” are these two separate observations of the same result.

No, what I call sequential events are pairs HH and HT, TT and TH, corresponding to exact awakening, which can’t be treated as individual outcomes.

The sample space for the experiment is {HH1_HT2, HT1_HH2, TH1_TT2, TT1_TH2}.

On the other hand, as soon as you connect these pairs and got HH_HT, HT_HH, TT_TH and TH_TT, they totally can create a sample space, which is exactly what I told you in this comment. As soon as you’ve switched to this sound sample space we are in agreement.

Each outcome has probability 14. The first observation establishes the condition as {HT1_HH2, TH1_TT2, TT1_TH2} and its complement as {HH1_HT2}. Conditional probability says the probability of {HT1_HH2} is 13. The second observation establishes the condition as {HH1_HT2, TH1_TT2, TT1_TH2} and its complement as {HT1_HH2}. Conditional probability says the probability of {HH1_HT2} is 13.

You are describing a situation where the Beauty was told whether she is experiencing an awakening before the second coin was turned or not. If the Beauty awakens and learns that it’s the awakening before the coin was turned, she indeed can reason that she observed the event {HT1_HH2, TH1_TT2, TT1_TH2} and that the probability that the first coin is Heads is 13. This, mind you, is not sneaky thirder idea of probability, where P(Heads) can be 13 even though the coin is Heads in 12 of the experiments. This is actual probability that the coin is Heads in this experiment. Remember the thing I asked you to hold in mind, our mathematical model doesn’t attempt to describe the individual awakening anymore, as you may be used to, it describes the experiment as a whole. Let this thought sink through.

The Beauty which learned that she is awakened before the coin was turned, can bet on Tails and win with 66% chance per experiment. So she should agree for per experimental betting odds up to 1:2 - which isn’t usually a good idea in Sleeping Beauty when you do not have any extra information about the state of the coin.

Likewise, if she knows for certain that she is experiencing an awakening after the second coin was turned. The same logic applies. She can lawfully update in favor of Tails and win per experimental bets on Tails with 66% probability.

And the point is that it does not matter which observation corresponds to SB being awake, since the answer is 13 regardless.

So one might think. But strangely enough, this doesn’t work this way. If the Beauty awakens without learning whether she is experiencing the awakening before the coin turn or after, she can’t just reason that whatever awakening she is experiencing, the probability is 13 and win per experiment bets with 66% probability. She will be right only in 50% of experiments. As it turns out:

however

P(Heads|Before or After Coin Turn) = 12

How can this be the case?

I could walk you through the solution to this truly fascinating problem, but you’ve demonstrated much better ability to arrive the the correct(ish) answer when you are doing it on your own, than when I give you all the answers—so feel free to engage with this problem on your own. I believe you do have all the pieces of the puzzle now and the only reason you haven’t completed it yet is because you’ve seen the number “1/​3”, decided that it validates thirdism and refused to think further. But now you know that it’s not the case, so your motivated reasoning is less likely to be stopping you.

As a matter of fact, thirdism is completely oblivious to the fact that there can be situations in Sleeping Beauty where betting per experiment at up to 1:2 odds may be a good idea. So you are discovering some new grounds here!

• One of my ways of thinking about these sorts of issues is in terms of “fair bets”

Well, as you may see it’s also is not helpful. Halfers and thirders disagree on which bets they consider “fair” but still agree on which bets to make, whether they call them fair or not. The extra category of a “fair bet” just adds another semantic disagreement between halfers and thirders. Once we specify whether we are talking per experiment or per awakening bet and on which, odds both theories are supposed to agree.

I don’t actually know what the Thirder position is supposed to be from a standpoint from before the experiment, but I see no contradiction in assigning equal utilities per awakening from the before-experiment perspective as well.

Thirders tend to agree with halfers that P(Heads|Sunday) = P(Heads|Wednesday) = 12. Likewise, because they make the same bets as the halfers, they have to agree on utilities. So it means that thirders utilities go back and forth which is weird and confusing behavior.

A Halfer has to discount their utility based on how many of them there are, a Thirder doesn’t. It seems to me, on the contrary to your perspective, that Thirder utility is more stable

You mean how many awakenings? That if there was not two awakenings on tails, but, for instance, ten, halfers will have to think that U(Heads) has to be ten times as much as U(Tails) for a utility neutral per awakening bet?

Sure, but it’s a completely normal behavior. It’s fine to have different utility estimates for different problems and different payout schemes—such things always happen. Sleeping Beauty with ten awakenings on Tails is a different problem than Sleeping Beauty with only two so there is no reason to expect that utilities of the events has to be the same. The point is that as long as we specified the experiment and a betting scheme, then the utilities has to be stable.

And thirder utilities are modified during the experiment. They are not just specified by a betting scheme, they go back and forth based on the knowledge state of the participant—behave the way probabilities are supposed to behave. And that’s because they are partially probabilities—a result of incorrect factorization of E(X).

Speculation; have you actually asked Thirders and Halfers to solve the problem? (while making clear the reward structure?

I’m asking it right in the post, explicitly stating that the bet is per experiment and recommending to think about the question more. What did you yourself answer?

My initial state that thirders model confuses them about this per experiment bet is based on the fact that a pro-thirder paper which introduced the technicolor sleeping beauty problem totally fails to understand why halfers scoring rule updates in it. I may be putting to much weight on the views of Rachael Briggs in particular, but it apparently was peer reviewed and so on, so it seems to be decent evidence.

… and I in my hasty reading and response I misread the conditions of the experiment

Well, I guess that answers my question.

Thirders can adapt to different reward structures but need to actually notice what the reward structure is!

Probably, but I’ve yet to see one actually derive the correct answer on their own, not post hoc after it was already spoiled or after consulting the correct model. I suppose I should have asked the question beforehand, and then publish the answer, oh well. Maybe I can still do it and ask nicely not to look.

The criterion I mainly use to evaluate probability/​utility splits is typical reward structure

Well, if every other thirder reason like this, that would indeed explain the issue.

You can’t base the definition of probability on your intuitions about fairness. Or, rather, you can, but then you are risking contradicting the math. Probability is a mathematical concept with very specific properties. In my previous post I talk about it specifically and show that thirder probabilities for Sleeping Beauty are ill-defined.

• This is surprising to me. Are you up to a a more detailed discussion? What do you think about the statistical analysis and the debunk of centred possible worlds? I haven’t seen these points being raised or addressed before and they are definitely not about semantics. The fact that sequential events are not mutually exclusive can be formally proven. It’s not a matter of perspective at all! We could use the dialogues feature, if you’d like.

Probability is what you get as a result of some natural desiderata related to payoff structures.

This is a vague gesture to a similarity cluster and not an actual definition. Remove fancy words and you end up with “Probability has something to do with betting”. Yes it does. In this post I even specify exactly what it does. You don’t need to read E.T. Jayne’s to discover this revelation. The definition of expected utility is much more helpful.

When anthropics are involved, there are multiple ways to extend the desiderata, that produce different numbers that you should say, depending on what you get paid for/​what you care about, and accordingly different math.

There are always multiple ways to “extend the desiderata”. But more importantly, you don’t have to say different probability estimates depending on what you get paid for/​what you care about. This is the exact kind of nonsense that I’m calling out in this post. Probabilities are about what evidence you have. Utilities are about what you care about. You don’t need to use thirder probabilities for per awakening betting. Do you disagree with me here?

When there’s only a single copy of you, there’s only one kind of function, and everyone agrees on a function and then strictly defines it. When there are multiple copies of you, there are multiple possible ways you can be paid for having a number that represents something about the reality, and different generalisations of probability are possible.

How is it different from talking about probability of a specific person to observe an event and probability of any person from a group to observe an event? The fact that people from the group are exact copies doesn’t suddenly makes anthropics a separate magisteria.

Moreover, there are no independent copies in Sleeping Beauty. On Tails, there are two sequential time states. The fact that people are trying to make a sample space out of them directly contradicts its definition.

When we are talking just about betting, one can always come up with its own functions, it’s own way to separate expected utility of an event into “utility” and “probability”. But then their “utilities” will be constantly shifting due to receiving new evidence and “probabilities” will occasionally ignore new evidence, and shift for other reasons. And pointing at this kind of weird behavior is a completely reasonable reaction. Can a person still use such definitions consistently? Sure. But this is not a way to carve reality by its joints. And I’m not just talking about betting. I specifically wrote a whole post about fundamental mathematical reasons, before starting talking about it.