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Believe me, Tanya does not think she just "missed" the ambiguous phrasing of the problem. What the problem is asking is quite clear - you will not get a different answer from different mathematicians based on their reading of it. The defense that it's "ambiguous" is how people try to retrofit the fact that their bad intuition of "what probability is" - which you've done a pretty good job of describing - somehow gets the wrong answer.
Um, yes? The field of probability arose because Pascal was trying to analyze gambling, where you want to be correct more often in an unpredictable situation. If you're in a situation where you will observe heads 1/3 of the time, either you say the probability is 1/3, or you're wrong. If I roll a die and you keep betting 50-50 odds on whether it's a 6, you don't get a pity refund because you were at least correct once, and we shouldn't say that's "less valuable" than the other five times...
Nothing in the problem says that only the last waking counts. But yes, if you add something to the problem that was never there, then the answer changes too.
Actually, the key insight of the Monty Hall problem is that the host knows which door the prize is behind. Ironically, unlike Sleeping Beauty, the usual way the Monty Hall problem is stated is actually ambiguous, because it's usually left implicit that the host could never open the prize door accidentally.
Indeed, in the "ignorant host" case, it's actually analogous to the Sleeping Beauty problem. Out of the 6 equal-probability possibilities (your choice of door) x (host's choice of door), seeing no prize behind the host's door gives you information that restricts you to four of the possibilities. You should only switch in two of them, so the odds are indeed 50/50.
Similarly, in the Sleeping Beauty problem, there are 4 equal-probability possibilities (Monday/Tuesday) x (heads/tails), and you waking up gives you information that restricts you to three of them.
This is asking a subtly different question. Here, you're asking, "When woken, you will be told, I am going to create an observable by showing you the result of the coin flip. What do you think an appropriate probability for that observable is?"
That is, you have taken one random variable, X, describing the nature of the coin flip, itself, and applied a transformation to get a different observable, Y, describing the random variable that you may see when awoken. This Y has X in it, but it also has the day and whether you're awake in it.
It is not clear to me that the original problem statement clearly identifies which observable we're asking about or betting on.
If the problem statement unambiguously stated, "What is your probability for Y, the coin I am about to show you?" then indeed, you should be a thirder. Forms of the question like what are listed in the Wiki presentation of the 'canonical form', "What is your credence now for the proposition that the coin landed heads?" are far more linguistically ambiguous as to whether we are asking about X or Y. "Landed" is past-tense, which to me indicates that it's simply asking about the thing that happened in the past, which is observable X, rather than the thing that is about to happen in the future, which is observable Y. There's nothing meaningful in there about payoffs or number of answers or anything.
Next, I'd like to join criticism of both the "number of answers" explanation and:
I think these are both flawed explanations, and I'll use one example alternative to explain.
Suppose you go to a casino. They say that either they have already flipped a coin or will flip a coin after you place a bet (I don't think it matters; you can't see it either way until after you bet). If the coin is heads, your bet will be simply resolved, but if the coin is tails, your bet will be taken as two identical bets. One can obviously compute the probabilities, the utilities, and calculate a correct wager, which would be the thirder wager. But in this case, everyone understands that they are not actually wagering directly on X, the direct probability of the coin flip. Nor are they making multiple separate "answers"; they are giving one answer, pre-computed at the beginning and simply queried in a static fashion. Likewise in the Sleeping Beauty problem; one is giving a single pre-computed answer that is just queried a different number of times depending.
It is also clear from this that there is no additional information from waking up or anything happening in the casino. You had all of the information needed at the initial time, about the Sleeping Beauty experimental set-up or about the structure of the casino's wager, when you pre-computed your one answer that would later be queried.
You just have to be very clear as to whether you're asking about X or Y, or what the actual structure of the casino game is for you to compute a utility. One you have that, it is, indeed, obvious. But I think your current explanations about number of answers or additional information from waking are flawed and that the 'canonical' language is more ambiguous.
This is the core thing you're getting wrong. You can learn things about past events that change your probability estimates!
If I roll a die and then tell you it was even, and then ask "what's the probability I rolled a 2?" - or, to use the unnaturally elaborate phrasing from the Wikipedia article, "what is your credence now for the proposition that I rolled a 2?" - do you answer 1/6? If your answer is "yes", then you're just abusing language to make describing math harder. It doesn't change the underlying math, it only means you're ignoring the one useful and relevant question that captures the current state of your knowledge.
Maybe you're the kind of guy who answers "if I have 2 apples and I take your 2 apples, how many do I have?" with "2 apples, because those others are still mine."
Your casino example is correct, but there's no analogue there to the scenario Sleeping Beauty finds herself in. If you'd like to fix it, imagine that you're one of two possible bettors (who can't see each other), and if the coin flip is heads then only one bettor (chosen at random) will be asked to bet. If it's tails, both will be. Now, when you're asked to bet, you're in Sleeping Beauty's situation, with the same partial knowledge of a past event.
Are you estimating observable X or observable Y? Just state this outright.
Are you learning something about observable X? Or are you simply providing a proper estimator for observable Y? I notice that you have now dropped any talk of "number of answers", which would have had, uh, implications here.
Obviously, there are ways to gain information about an observable. In this case, we can clearly state that we are talking about P(X|I), where I is the information from you telling me. Be serious. Tell me if you think we're saying something about X or Y.
No one has told you anything, no information has been acquired, when your pre-computed policy is queried. Where are you getting the information from? It's coming entirely from the pre-defined problem set-up, which went into your pre-computation, just like in my casino example.
Stated without any justification.
I will say that this is not analogous with the same justification you gave for mine.
Observable Y. Satisfied? It should be obvious that, when you're asking Sleeping Beauty for a probability estimate, it's about her current state of knowledge. Which has updated (excluding the Tuesday/heads case) by awaking. We don't normally go around asking people "hey, for no reason, forget what you know now, what was your probability estimate on last Thursday that it would rain last Friday?" What's the practical use of that?
"number of answers" was @kky's language, not mine. Anyway, are you trying to accuse me of playing language games here? I'm not. This isn't a clever trick question, and this certainly isn't a political question with both sides to it. There's a right answer (which is why the Wikipedia article is so frustrating). If I'm accidentally using unclear language, then it's my failure and I will try to do better. But it doesn't make your nitpicking valid. After all, if you were really honest about your criticisms, you could easily just rephrase the problem in a way that YOU think is clearly asking about your "observable Y". EDIT: Sorry, upon rereading I see you did do that. Your statement of the problem is fine too.
Uh... I need to spell out the obvious? There's nobody in your scenario that has 2/3 confidence that the coin flip was tails. Whereas, in mine, there is. Monday/Tuesday are analogous to bettor 1/bettor 2. If you're throwing out terms like "random variable" but you need me to walk you through this, then I'm sadly starting to suspect you're just trolling me.
Yes, thanks.
...about observable Y, yes.
One which you embraced, saying that this was core to the field of probability:
This was a significant component of why I entered this conversation in the first place.
This is simply asserting your conclusion. There is no justification here. There is absolutely someone who has a bet that has 2/3 confidence concerning the stated evaluation criteria. This is a pre-computed single decision and potentially queried multiple times, given all of the information prior to the event happening.
Let's make this simple. You say here:
Then just do this. You claimed that this was as simple as P(X|I), as though someone told you that they rolled an even number. Now, you're telling me that you're estimating P(Y). Use the axioms and theorems to get from one to the other. Hopefully your next comment will "stick with" them.
I'm confident from my background and career that I will be able to evaluate your formal proof. Just start from, "There is a binary random variable X," and proceed formally.
EDIT: Consolidating this other bit here:
Monty Hall has zero problem showing how exactly information changes over time. Your policy is clearly closed-loop feedback, rather than pre-computed static (done so in a way solely for the purpose of a stated utility criterion, as in the casino example). There is no ambiguity concerning what quantity you are providing an estimator for.
EDIT EDIT: Let me put it another way. I think a person is completely justified in saying, "My credence that the coin originally came up (X) tails is 1/2, and because of that and my knowledge of the experimental setup, my probability estimate for what I will see if you show me the coin now (Y) is 2/3. In fact, if my credence that the coin originally came up (X) tails was 2/3, then because I know the experimental setup, my probability estimate for what I will see if you show me the coin now (Y) would be 4/5 (I believe)."
I have no issues with this math. My only issue is that I really, honestly cannot wrap my mind around a mindset that doesn't treat Y as the obvious thing the question's about. Anyway, thanks for the debate, and let's try to leave it on as much of a consensus as we're going to get. I expect, like Tanya, I'm doomed to be perpetually pushing this boulder up this hill, so I might as well make the best of it.
Sorry to belabor this, because I think we've made progress and are maybe not on the same page, perhaps somewhere in the same chapter... but...
I think it's because people... sorry to say, like yourself... say things like...
and present it as though someone told you that they rolled an even number, which would be a case in which you are genuinely gaining information about the past event.
And I think that's probably the core of the philosophical debate and why people try to connect this problem to anthropics. Many people genuinely think that there is something here that "updates" (or "changes" or something) their belief about a past event. This is a genuinely tricky question, and I'm not completely confident of my own perspective. I clearly lean toward just saying that they're separate mathematical objects, and you're not saying anything about changing your estimate of X when you make an estimate of Y. But tons of people want it to say something about changing their estimate of X and they present it with language that clearly indicates that they're trying to say something about changing their estimate of X.
I think that if you mostly agree with my presentation that you can simply cleave them apart and say something separate about X and Y, and that your estimate for Y doesn't necessarily have some temporally-bound back-implications for beliefs about X, then you're actually taking a particular philosophical position... one that I think a lot of thirders would disagree with. One that many of them (like yourself, frankly) would start off vehemently denying and claiming that it's just obvious mathematics that you're saying something about X.
Bleh. I don't think we're even in the same book. I find this mostly incoherent, particularly your description of my views (and how you think they've changed ... unless that was just bait to draw me back in, which, if so, I'm a sucker). So-called "thirders" don't take any philosophical position outside of "if something happens 2/3s of the time we say it has probability 2/3".
Yes, that is indeed what's happening. It has to be what's happening. It is impossible for the conditional probability distribution on X (which we're perhaps-sloppily calling Y even though it's technically just a different distribution on the same variable) to change without you having learned information. They're two ways of saying the same thing.
So your sticking point is you don't see how waking could be information. That's what the results show, but it conflicts with your non-formal description of what's going on (that since you know you're going to wake at least once, you learn nothing from it). Would you at least agree with me that you're gaining information in the lollipop example here? i.e., your position is that among the ways of eliminating Tuesday/heads from the probability space: "waking up", "not getting a lollipop", or hell, just "being told it's not Tuesday/heads", the first is meaningfully different from the other two?
Does that also mean that you'd consider the probabilistic version (where the experimenters flip a second coin privately to determine whether to "simulate" Monday or Tuesday - no amnesia drugs required) uncontroversial?
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