Re: no coherent “stable” truth value: indeed. But still… if she wonders out loud “what day is it?” at the very moment she says that, it has an answer. An experimenter who overhears her knows the answer. It seems to me that you “resolve” this tension is that the two of them are technically asking a different question, even though they are using the same words. But still… how surprised should she be if she were to learn that today is Monday? It seems that taking your stance to its conclusion, the answer would be “zero surprise: she knew for sure she would wake up on Monday so no need to be surprised it happened”
And even if she were to learn that the coin landed tails, so she knows that this is just one of a total of two awakenings, she should have zero surprise upon learning the day of the week, since she now knows both awakenings must happen. Which seems to violate conservation of expected evidence, except you already said that the there’s no coherent probabilities here for that particular question, so that’s fine too.
This makes sense, but I’m not used to it. For instance, I’m used to these questions having the same answer:
Yet here, the second question is fine (assuming tornadoes are rare enough that we can ignore the chance of two on consecutive days) while the first makes no sense because we can’t even define “today”
It makes sense but it’s very disorienting, like incompleteness theorem level of disorientation or even more
Ah, so I’ve reinvented the Lewis model. And I suppose that means I’ve inherited its problem where being told that today is Monday makes me think the coin is most likely heads. Oops. And I was just about to claim that there are no contradictions. Sigh.
Okay, I’m starting to understand your claim. To assign a number to P(today is Monday) we basically have two choices. We could just Make Stuff Up and say that it’s 53% or whatever. Or we could at least attempt to do Actual Math. And if our attempt at actual math is coherent enough, then there’s an implicit probability model lurking there, which we can then try to reverse engineer, similar to how you found the Lewis model lurking just beneath the surface of my attempt at math. And once the model is in hand, we can start deriving consequences from it, and Io and behold, before long we have a contradiction, like the Lewis model claiming we can predict the result of a coin flip that hasn’t even happened yet just because we know today is Monday.
And I see now why I personally find the Lewis model so tempting… I was trying to find “small” perturbations of the experiment where “today is Monday” clearly has a well defined probability. But I kept trying use Rare Events to do it, and these change the problem even if the Rare Event is not Observed. (Like, “supposing that my house gets hit by a tornado tomorrow, what is the probability that today is Monday” is fine. Come to think of it, that doesn’t follow Lewis model. Whatever, it’s still fine.)
As for why I find this uncomfortable: 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. And in particular 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.
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.
What seems clear is that the sample space is not the usual sleeping beauty sample space; it has to run metaphorically “skew” to it somehow.
If the question were “did the coin land on heads” then it’s clear that this is question is of the form “what world am I in?”. Namely, “am I in a world where the coin landed on heads, or not?”
Likewise if we ask “does a Tuesday awakening happen?”… that maps easily to question about the coin, so it’s safe.
But there should be a way to ask about today as well, I think. Let’s try something naive first and see where it breaks. 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
I tried to formalize the three cases you list in the previous comment. The first one was indeed easy. The second one looks “obvious” from symmetry considerations but actually formalizing seems harder than expected. I don’t know how to do it. I don’t yet see why the second should be possible while the third is impossible.
I hope it’s okay if I chime in (or butt in). I’ve been vaguely trying to follow along with this series, albeit without trying too hard to think through whether I agree or disagree with the math. This is the first time that what you’ve written has caused to go “what?!?”
First of all, that can’t possibly be right. Second of all, it goes against everything you’ve been saying for the entire series. Or maybe I’m misunderstanding what you meant. Let me try rephrasing.
(One meta note on this whole series that makes it hard for me to follow sometimes: you use abbreviations like “Monday” as shorthand for “a Monday awakening happens” and expect people to mentally keep track that this is definitely not shorthand for “today is Monday” … I can barely keep track of whether heads means one awakening or two… maybe should have labeled the two sides of the coin ONE and TWO instead is heads and tails)
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?
In fact, let’s try a somewhat ad-hoc and mostly unprincipled way to formalize this. Let’s say there’s a one percent chance per day that her friend forgets what day it is and decides to call her to ask. (One percent sounds like a lot but her friend is pretty weird) Then there’s a 2% chance of it happening if there are two awakenings, and one percent if there’s only one awakening. If there are two awakenings then Monday and Tuesday are equally likely; if there’s only one awakening then it’s definitely Monday. Thus, given that her friend is on the phone, today is more likely to be Monday than Tuesday.
Okay, maybe that’s cheating… I sneaked in a Rare Event. Suppose we make it more common? Suppose her friend forgets what day it is 10% off the time. The logic still goes through: given that her friend is calling, today is more likely to be Monday than Tuesday.
Okay, 10% is still too rare. Let’s try 100%. This seems a bit confusing now. From her friends perspective, Monday is just as good as Tuesday for coming down with amnesia. But from sleeping beauty’s perspective, GIVEN THAT the experiment is not over yet, today is more likely to be Monday than Tuesday. This is true even though she might be woken up both days.
Or is everything I just wrote nonsensical?
I think this is much easier to analyze if you think about your plans before the experiment starts, like on Sunday. In fact, let’s pretend we are going to write down a game plan on Sunday, and we will simply consult that plan wherever we wake up and do what it says. This sidesteps the whole half vs third debate, since both sides agree about how things look better the experiment begins.
Furthermore, let’s say we’re going to participate in this experiment 100 times, just so I don’t have to deal with annoying fractions. Now, consider the following tentative game plan: agree to the bet if and only if the room is red. Let’s see what happens. Out of 100 experiments, 50 will result in just one awakening. In 25 of them you will refuse the bet (costing you zero dollars), and in the other twenty five you will accept, which costs you $7500. So far so good. (Actually pretty bad, since we just lose lots of money.) The other 50 will result in two awakenings. Here, we don’t need to worry about probabilities anymore. It is guaranteed we will see a red room once and a blue room once. Thus, we will agree to the bet once, and thus the bet will be in effect. So we will win $200 fifty times, for a total of $10k. Once we subtract what we lost when the coin landed for one awakening, our net profit is $2500, or an average of $25 dollars per experiment.
Here’s how I think of what the list is. Sleeping Beauty writes a diary entry each day she wakes up. (“Nice weather today. I wonder how the coin landed.”). She would like to add today’s date, but can’t due to amnesia. After the experiment ends, she goes back to annotate each diary entry with what day it was written, and also the coin flip result, which she also now knows.
The experiment is lots of fun, so she signs up for it many times. The Python list corresponds to the dates she wrote in her dairy.
I think that that’s what he meant: more aluminum in the brain is worse than less. What he was trying to say in that sentence is this: high levels in the blood may not mean high levels in the brain unless the blood level stays high for a long time.
Thanks :) the recalibration may take a while… my intuition is still fighting ;)