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I knew that not any string of English words gets a probability, but I was naïve enough to think that all statements that are either true or false get one.
Well, I think this one is actually correct. But, as I said in the previous comment, the statement "Today is Monday" doesn't actually have a coherent truth value throughout the probability experiment. It's not either True or False. It's either True or True and False at the same time!
I was hoping they this sequence of posts which kept saying “don’t worry about anthropics, just be careful with the basics and you’ll get the right answer” would show how to answer all possible variations of these “sleep study” questions… instead it turns out that it answers half the questions (the half that ask about the coin) while the other half is shown to be hopeless… and the reason why it’s hopeless really does seem to have an anthropics flavor to it.
We can answer every coherently formulated question. Everything that is formally defined has an answer Being careful with the basics allows to understand which question is coherent and which is not. This is the same principle as with every probability theory problem.
Consider Sleeping-Beauty experiment without memory loss. There, the event Monday xor Tuesday also can't be said to always happen. And likewise "Today is Monday" also doesn't have a stable truth value throughout the whole experiment.
Once again, we can't express Beauty's uncertainty between the two days using probability theory. We are just not paying attention to it because by the conditions of the experiment, the Beauty is never in such state of uncertainty. If she remembers a previous awakening then it's Tuesday, if she doesn't - then it's Monday.
All the pieces of the issue are already present. The addition of memory loss just makes it's obvious that there is the problem with our intuition.
"You should anticipate having both experiences" sounds sort of paradoxical or magical, but I think this stems from a verbal confusion.
You can easily clear this confusion if you rephrase it as "You should anticipate having any of these experiences". Then it's immediately clear that we are talking about two separate screens. And it's also clear that our curriocity isn't actually satisfied. That the question "which one of these two will actually be the case" is still very much on the table.
Rob-y feels exactly as though he was just Rob-x, and Rob-z also feels exactly as though he was just Rob-x
Yes, this is obvious. Still as soon as we got Rob-y and Rob-z they are not "metaphysically the same person". When Rob-y says "I" he is reffering to Rob-y, not Rob-z and vice versa. More specifically Rob-y is refering to some causal curve through time ans Rob-z is refering to another causal curve through time. These two curves are the same to some point, but then they are not.
In case the bet is offered on every awakening: do you mean if she gives conflicting answers on Monday and Tuesday that the bet nevertheless is regarded as accepted?
Yes I do.
Of course, if the experiment is run as stated she wouldn't be able to give conflicting answers, so the point is moot. But having a strict algorithm for resolving such theoretical cases is a good thing anyway.
My initial idea was, that if for example only her Monday answer counts and Beauty knows that, she could reason that when her answer counts it is Monday, arriving at the conclusion that it is reasonable to act as if it was Monday on every awakening, thus grounding her answer on P(H/Monday)=1/2. Same logic holds for rule „last awakening counts“ and „random awakening counts“.
Yes, I got it. As a matter of fact this is unlawful. Probability estimate is about the evidence you receive not about what "counts" for a betting scheme. If the Beauty receives the same evidence when her awakening counts and when it doesn't count she can't update her probability estimate. If in order to arrive to the correct answer she needs to behave as if every day is Monday it means that there is something wrong with her model.
Thankfully for thirdism, she does not have to do it. She can just assign zero utility to Tuesday awakening and get the correct betting odds.
Anyway, all this is quite tangental to the question of utility instability. Which is about the Beauty making a bet on Sunday and then reflecting on it during the experiment, even if no bets are proposed. According to thirdism probability of the coin being Heads changes on awakening, so, in order for Beauty not to regret about making an optimal bet on Sunday, her utility has to change as well. Therefore utility instability.
There are indeed ways to obfuscate the utility instability under thirdism by different betting schemes where it's less obvious, as the probability relevant to betting isn't P(Heads|Awake) = 1/3 but one of thoses you meantion which equal 1/2.
The way to define the scheme specifically for P(Heads|Awake), is this: you get asked to bet on every awakening. One agreement is sufficient, and only one agreement counts. No random selecting takes place.
This way the Beauty doesn't get any extra evidence when she is asked to bet, therefore she can't update her credence for the coin being Heads based on the sole fact of being asked to bet, the way you propose.
This makes me uncomfortable. From the perspective of sleeping beauty, who just woke up, the statement “today is Monday” is either true or false (she just doesn’t know which one). Yet you claim she can’t meaningfully assign it a probability. This feels wrong, and yet, if I try to claim that the probability is, say, 2/3, then you will ask me “in what sample space?” and I don’t know the answer.
Where does the feeling of wrongness come from? Were you under impression that probability theory promised us to always assign some measure to any statement in natural language? It just so happens that most of the time we can construct an appropriate probability space. But the actual rule is about whether or not we can construct a probability space, not whether or not something is a statement in natural language.
Is it really so surprising that a person who is experiencing amnesia and the repetetion of the same experience, while being fully aware of the procedure can't meaningfully assign credence to "this is the first time I have this experience"? Don't you think there has to be some kind of problems with Beauty's knowledge state? The situation whre due to memory erasure the Beauty loses the ability to coherently reason about some statements makes much more sense than the alternative proposed by thirdism - according to which the Beauty becomes more confident in the state of the coin than she would've been if she didn't have her memory erased.
Another intuition pump is that “today is Monday” is not actually True xor False from the perspective of the Beauty. From her perspective it's True xor (True and False). This is because on Tails, the Beauty isn't reasoning just for some one awakening - she is reasoning for both of them at the same time. When she awakens the first time the statement "today is Monday" is True, and when she awakens the second time the same statement is False. So the statement "today is Monday" doesn't have stable truth value throughout the whole iteration of probability experiment. Suppose that Beauty really does not want to make false statements. Deciding to say outloud "Today is Monday", leads to making a false statement in 100% of iterations of experiemnt when the coin is Tails.
P(today is Monday | heads) = 100% is fine. (Or is that tails? I keep forgetting.) P(today is Monday | tails) = 50% is fine too. (Or maybe it’s not? Maybe this is where I’m going working? Needs a bit of work but I suspect I could formalize that one if I had to.) But if those are both fine, we should be able to combine them, like so: heads and tails are mutually exclusive and one of them must happen, so: P(today is Monday) = P(heads) • P(today is Monday | heads) + P(tails) • P(today is Monday | tails) = 0.5 + .25 = 0.75 Okay, I was expecting to get 2/3 here. Odd. More to the point, this felt like cheating and I can’t put my finger on why. maybe need to think more later
Here you are describing Lewis's model which is appropriate for Single Awakening Problem. There the Beauty is awakened on Monday if the coin is Heads, and if the coin is Tails, she is awakened either on Monday or on Tuesday (not both). It's easy to see that 75% of awakening in such experiment indeed happen on Monday.
It's very good that you notice this feeling of cheating. This is a very important virtue. This is what helped me construct the correct model and solve the problem in the first place - I couldn't accept any other - they all were somewhat off.
I think, you feel this way, because you've started solving the problem from the wrong end, started arguing with math, instead of accepting it. You noticed that you can't define "Today is Monday" normally so you just assumed as an axiom that you can.
But as soon as you assume that "Today is Monday" is a coherent event with a stable truth value throughout the experiment, you inevitably start talking about a different problem, where it's indeed the case. Where there is only one awakening in any iteration of probability experiment and so you can formally construct a sample space where "Today is Monday" is an elementary mutually exclusive outcome. There is no way around it. Either you model the problem as it is, and then "Today is Monday" is not a coherent event, or you assume that it is coherent and then you are modelling some other problem.
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
P(Heads|Monday or Tuesday) = P(Heads) = 1/2
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 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.
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Universal guide to magic via anthropics:
Of course it is false. What are the reasons to even suspect that it might be true?
This actually sounds about right. What's paradoxical here?