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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?
Sure, there are ways to add actual gain of information that is relevant for X. I'd have to work through different precise formulations.
Anyway, this morning, after having written my last comment (and before reading yours, as it happens), I was feeling very confident in it. I figured (as I should figure) that I should actually check out the literature in the area a bit, and see what's there. Of course, I was also looking for whether anyone in the literature had proposed a similar solution... and if so, whether there was any responding literature saying that it was insufficient in some way.
I proceeded with a mix of Wikipedia cites and Google Scholar, but it turns out that Wikipedia actually sums up what I now think is a great representation of my view pretty well, with reference to Groisman's 2008 paper. It's in the Wiki section on Ambiguous-question position:
I hadn't quite hit on the right language, but I was getting there with random variables X and Y. I was pretty sure, and I'm still pretty sure, that if one actually spells out, in detail, formal definitions of X and Y, one can see that they do not relate in the form of a simple conditional probability that can be used to 'update' X. What I hadn't yet specified was that the main way that they differ is that they're describing different sample spaces.
One can make this analogy even more explicit by saying that if heads is flipped, a green ball that has written on it, "Monday, heads" is placed in the box. If tails is flipped, one red ball labeled "Monday, tails" and one red ball labeled "Tuesday, tails" are placed in the box.
I think very clearly here, one can say that when you pull a ball from the box, there is a 1/3 chance that you see a green ball. That is exactly the same as saying that there's a 1/3 chance that you see "heads" written on the ball. Similarly, there is a 2/3 chance that you see "tails" written on a red ball. "Seeing heads/tails on a ball" is random variable Y.
...but you cannot say that there was a 2/3 chance of tails having been flipped (random variable X). That's just a different sample set. You don't "gain information" about what was flipped by knowing that a ball has been drawn from the box (waking up). You had all the information you needed at the first moment, because you knew the experimental setup and how the depositing/withdrawing mechanism worked.
Balls are deposited according to the sample set {One Green, Two Red}, where they have some stuff written on them, but they're withdrawn according to the sample set {Green/Monday, Red/Monday, Red/Tuesday}.
This is also, I believe, the key intellectual step that justifies the naive thirder position against the naive halfer position in the first place - that because you have no information about which situation you're waking up in, you have to realize that the set of possibilities has three elements (over which, you take a typical uniform distribution), only one which has you seeing heads (green) and two which have you seeing tails (red).
To reiterate, yes, the correct betting strategy for what you observe will be 2/3 red/tails, but I don't believe that any property of conditional probability implies that your estimate should be that tails was flipped with probability 2/3. I think it would actually directly violate the laws of probability, for if you apply the laws of probability to the actual mechanics of the experiment and say that tails is flipped with probability 2/3, then you should observe tails with probability 4/5. This is actually a pretty straightforward calculation.
p = probability of flipping tails/depositing 2red
Un-normalized probability of observing tails/observing red: 2p
Un-normalized probability of observing heads/observing green: (1-p)
2p + (1-p) = p+1 is our normalization constant over the three possible balls
Normalized probability of observing tails/observing red: 2p/(p+1)
Normalized probability of observing heads/observing green: (1-p)/(p+1)
This is just the mechanics of the game. For p=1/2, we get 2/3 and 1/3. For p=2/3, we get 4/5 and 1/5.
This math should work for the extended versions of the game, too. If you wake up once (have one green ball) on heads, but wake up n times (have n green balls) on tails, then
p = probability of flipping tails/depositing n red
Un-normalized probability of observing tails/observing red: np
Un-normalized probability of observing heads/observing green: (1-p)
np + (1-p) = (n-1)p+1 is our normalization constant over the n+1 possible balls
Normalized probability of observing tails/observing red: np/((n-1)p+1)
Normalized probability of observing heads/observing green: (1-p)/((n-1)p+1)
I also took a little time on Google Scholar to check some of the papers that cited this Groisman paper. Many of them did the typical thing of just citing a paper because it came up in their own GS search, clearly not having read it (this has happened to my own work plenty, much to my chagrin). I already can't remember whether there was one or two papers that actually said something about what Groisman did and complained about it, but their complaint wasn't really comprehensible to me (maybe if I spent more than a morning on it, I could figure out whether I think it's valid or not). Perhaps you'll still disagree along some lines like that and be able to explain it better.
Maybe I could still imagine a critique, perhaps in terms of moving sums around (I.e., there are cases of multiple summation where you can/can't moving an inner sum out to an outer sum), but sums are often not that hard to move around. I'd definitely need to see a pretty detailed formal argument of where exactly a problem occurs. Otherwise, I'm pretty doubtful that any more informal argument is going to move me much.
It also comports with my casino game example. A static, non-feedback policy is just queried a different number of times, so it observes tails more often. I know that it'll observe tails more often (2p/(p+1) of the time), so that's how I should bet on what it observes. Perhaps to reach your preference for saying that whether you bet right matters the most, let's say this casino has you play two games simultaneously. In the first game, you're just betting on the outcome of a coin flip with probability p (maybe we even remove p=1/2 to remove possible degeneracies). In the second game, at the same time, you're betting on this modified game where your policy is queried twice if it's tails. They use the same coin and then evaluate both games, with your separate bets. If someone is not betting according to p and 2p/(p+1) in the two respective games, then I think you would declare that they are wrong. The difference between these two bets is simply that these two static policies have different observation/evaluation functions. The second policy doesn't somehow update mid-game and think that the properties of the coin flip have changed. If it did, your two policies would have weird and conflicting estimates for the properties of the coin flip. How would you even make your second set of bets?
...I guess finally, since I can't shut up, go back to computing policies for parallel Sleeping Beauty games. One is betting on a normal coin flip, while the other has this weird observation function. They use the same coin. Should those policies (people) have different estimates for the coin flip when they wake up... or just different estimates for what they will observe in their appropriately-blinded state?
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