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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 1/3.

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.

You would violate conservation of expected evidence if 

P(Monday) + P(Tuesday) = 1 

However this is not the case because P(Monday) = 1 and P(Tuesday) = 1/2

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) = 1/2

P(Tails & Monday) = P(Tails&Tuesday) = 1/2

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

P(Heads|Today) = P(Heads|Monday xor Tuesday) = P(Heads) = 1/3

Likewise we can talk about "Today is Monday":

P(Monday|Today) = P(Monday|Monday xor Tuesday) = P(Monday) = 1/2

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.

This whole conversation isn't about math. It is about philosophy.

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) = 1/2, 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!"

It's like you didn't even read my posts or my comments.

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?

Anytime Brandon updates he predictably updates in the direction of HEADS

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) = 1/2 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.

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