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.
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.
SnapDragon
7d ago
I think I'm Sleeping Beauty'd out, but thanks for your comments. I honestly don't think the problem's all that existentially weird - compared to many thought experiments, this one could at least take place in our physical universe.
SnapDragon
7d ago
Either way, I think you're basically right that it should by 2/3, but I don't think it's a paradox or even particularly interesting when properly formulated.
Absolutely! This is what I'm trying to get across. Unfortunately, Wikipedia does NOT present the problem this way: "an easy probability question that some people misinterpret."
SnapDragon
8d ago
I hope you knew what you were getting into bringing up Sleeping Beauty, haha.
Somewhat. I've gotten into arguments about this on astralcodexten before, and it honestly wasn't too bad. The way I try to sleep easy at night is by telling myself that 99% of people here are probably sensible, and it's only the 1% I end up having to argue with, who think that weird philosophical arguments can let you ignore the results of an easy-to-replicate experiment. (I'm not including you in this, to be clear.)
Put more simply, it's not fair to imply that there is a mathematically "correct" interpretation of probability.
Well, I understand what you're trying to say, but there IS a mathematically correct theory of probability, if you just stick with axioms and theorems. (Uh, without getting into the weeds of the Axiom of Choice, which shows up pretty quickly because probability is intricately tied with measure theory.) As your link says, there's a "standard" set of axioms that are pretty uncontroversial. However, you're right that there can be some tricky philosophical questions about how the real world maps to it. For instance, while the Doomsday Argument is wrong (you can't tell the future with anthropic arguments), there are other anthropic arguments that DO seem like they work and have some rather weird implications. I'd love to have a real discussion about those sometime instead of this minutia.
Regardless, the issue here is that this isn't a complex real-world problem, it's a simple experiment with clear results. And, like Monty Hall, it's one that you can even do yourself with slight modifications. As the experiment is repeated, 2/3 of the times she's asked, Sleeping Beauty will see tails. If she believes she'll see any other results, she's wrong. You can't philosobabble your way into changing this fact, any more than you could talk a coin into flipping Heads 100 times in a row. I absolutely do not agree that there is a reasonable way of defining a "trial" or "sample space" that somehow makes the halfer case make sense. You can see people in this thread trying, and it takes some real mental gymnastics.
When people bring up the Monty Hall problem, do you go around telling THEM that probability is philosophically complex and gosh, how can they really know they should switch with 2/3 confidence? No? Then why is Sleeping Beauty different?
SnapDragon
8d ago
Hmm, there may be some misunderstanding about the term "belief" here (or "credence" from Wikipedia, or "confidence", all of which can kind of be used interchangeably)? You don't "believe" that the coin was tails (or heads). After awakening, what you believe is that there's a 2/3 chance that it was tails. Which, as you said, matches with your observations if you repeat the experiment 100 times, indicating that your belief is well-calibrated.
Wouldn't you have the same issue with "belief" without the whole experiment setup, if I just flipped a coin behind my back? Isn't it reasonable to say that you "believe" the coin has a 50-50 chance of being heads, if you can't see it?
Rationalists like to make probabilistic predictions for events all the time (which I sure hope reflects what they "believe"). If you read astralcodexten, he'll often post predictions with probabilities attached, and he considers his beliefs well-matched with the real world not by getting everything right, but by getting 9/10 of the 90% predictions right, 8/10 of the 80% predictions right, etc.
SnapDragon
8d ago
Are you estimating observable X or observable Y? Just state this outright.
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?
I notice that you have now dropped any talk of "number of answers", which would have had, uh, implications here.
"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.
Stated without any justification.
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.
SnapDragon
8d ago
BTW, you can also just have people play the iterated version. After a few iterations your state of knowledge approaches Sleeping Beauty's, only without that tricky-to-arrange memory erasure.
SnapDragon
8d ago
I-frames can be ok in moderation, but yeah, they tend to look very "gamey". I think Cuphead (when equipping a particular ability) did it pretty well by making you disappear and teleport in a puff of smoke.
Since I'm complaining already, I hate games that replace visible damage to enemies with health bars. Stop being lazy, display damage that isn't more than a coat of red paint till the enemy suddenly goes from 100% combat capable to keeling over.
In fairness, that's REALLY hard to do. I don't play a lot of AAA games, but Helldivers 2 actually seems to do this really well, with enemies showing damage or even getting limbs blown off but still (less effectively) coming at you.
SnapDragon
8d ago
You can see both phrasings in the Wikipedia article. No mathematician would get a different answer to either of them. I suppose if you define "ambiguous" as "somebody ignorant could misread this", then ... sure? That's not a useful definition of "ambiguous" though. The solution there is to correct the misreading, which I hope someday will finally - finally! - percolate through the rationalist community, at the very least.
SnapDragon
8d ago
Or maybe just keep the coin flip but use 1000 wakings instead? I do love expressing things this way, but I've found that (unlike Monty Hall) people will still continue to get the Sleeping Beauty problem wrong even afterwards. The issue here is that they know they should bet based on the 2/3 odds, they just think that the concept of "probability" they have in their heads is some ineffable philosophical concept that goes beyond measuring odds.
"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.
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.
SnapDragon
8d ago
Well, ok, but you chose that ambiguous phrasing. The Wikipedia article has two different statements of the problem, neither of which is unclear. You have to be very careful with your wording (as you were) to make it a misleading question that sounds like it's asking about a result but is actually, uh, about a "reality that continues to exist".
SnapDragon
8d ago
This is just not true. Waking up doesn't give you any information, because you already know that you will wake up. You are 100% expecting to wake up.
You are not expecting to wake up on Tuesday if the coin is heads. If it clears your confusion, imagine that instead you always wake up, but at 8:00 am a researcher will come in and give you a lollipop if (and only if) it's Tuesday and the coin was heads. Mathematically, it is exactly the same scenario, only without the "sleeping through the experiment" part that seems to be throwing you. At 7:59 am you have 50% confidence that the coin was tails. At 8:01 am you have either 66% confidence that the coin was tails, or 100% confidence that the coin was heads. You have been given partial information.
This is because 'probability' here is being used in two different ways - in the first, about our estimation about how the world actually is or was in the past, and in the second on a physical outcome in the future that can go different ways.
You're using the passive "is being used" here, but you're the one making this mistake. (Note that probabilities can differ, even for the same event, based on knowledge.) Sleeping Beauty is just asked "was the flip tails?" Not something silly like "do we live in a world where coin flips are fair?"
(BTW, your computer program/anthropic example is fine, and I've seen scripts to do it. Of course the answer you get is 2/3.)
SnapDragon
8d ago
I actually loved the Zero Escape series - except Zero Time Dilemma, sadly, which I bounced on because I really didn't care for the graphics and the nonlinear format. Sounds like I should go back to finish it, though.
SnapDragon
9d ago
Thanks for the recommendations. I've definitely heard good things about Sekiro.
While the rest of the game was neat, I didn't like most of the bosses in Nine Sols. The last boss, in particular, was ridiculously overtuned, harder than anything in HK/Silksong even in the two-phase version (and the "true ending" has an even harder third phase). I think the only reason people tolerated it was you could switch to the trivial "easy" difficulty at any time.
SnapDragon
9d ago
IMO right or wrong doesn't really come into it.
You could argue that this is because the world of HK is much closer to the "survive" side of survive/thrive. Morality is something you can spend more time agonizing over when the existence of your entire civilization isn't hanging by a thread.
SnapDragon
9d ago
If Sleeping Beauty is asked each time she awakens for a probability distribution over which side the coin landed on, and will be paid on Wednesday an amount of money proportional to the actual answer times the average probability she put on that answer across wakings, she should be a halfer to maximize payout.
I appreciate that you're trying to steelman the halfer position, but that's a really artificial construction. In fact, in this framing, the payout is 1/2 regardless of what she answers (as long as she's consistent). That's what happens when you try to sidestep the obvious way to bet (where even the Wikipedia article admits she should wager 1/3 on heads - and then somehow fails to definitively end the article there).
p.s. you might enjoy the technicolor sleeping beauty problem.
Nice, I think I'd encountered it before (I've unfortunately read a lot of "Ape in the coat"'s voluminous but misguided Sleeping Beauty posts), but I didn't specifically remember that one. Commit to betting only if the room is red. Then of the four equal-weight possibilities (Monday is red/blue) x (heads/tails), you win in red/tails and blue/tails, you lose in red/heads, and you don't bet in blue/heads. Expected payout per experiment is 1/4*(200+200-300) = 25.
He does seem to be wrong about "for reference, in regular Sleeping Beauty problem utility neutral betting odds for once per experiment bet are 1:1", because if you have any source of randomness yourself, you can actually get better odds (by ensuring that you'll "take the bet" more often when you have two chances at it). I see you actually posted a really nice analysis of the problem yourself in the link. It's fun that there's a distinction between an external source of randomness (where the results on Monday/Tuesday are dependent) and an internal source (where the results on Monday/Tuesday must be independent).
SnapDragon
9d ago
I believe, but can’t prove, that a lot of Elden Ring’s difficulty complaints come from DS3 veterans who refuse to do anything but dodge roll and light attack and don’t engage with the new mechanics.
Or, I guess, people like me who were new to FromSoftware games and were trying to bash our way through without looking up online explanations for all the mechanics the game doesn't explain.
SnapDragon
9d ago
Only partially - I genuinely think this is an example of a failure of Wikipedia as a repository of knowledge. And believe me, I'd like nothing more than for rationalists to grok Sleeping Beauty like they (mostly) grok Monty Hall.
SnapDragon
9d ago
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.
Do you count getting a correct answer twice "more valuable" than getting it once?
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...
If she is told that she's going to get cash ONLY if she correctly answers on the last waking, then it doesn't matter what she picks, her odds of a payday are equal.
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.
This problem strongly reminds me of the Monty Hall problem, where of course the key insight is that the ordering matters and that eliminating possibilities skews the odds off of 50%.
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.
SnapDragon
9d ago
You may like Dark Souls 2 more than Elden Ring, as IMO it satisfies both the "less mindless mashing of the roll button" and "more rewarding exploration" criteria. (It's my understanding that Dark Souls 2 is a controversial member of the series. But I haven't played any other Soulslike games. Some 4chan users feel that Dark Souls 3 and Elden Ring degenerated into mindless "rollslop" in comparison to Demon's Souls, Dark Souls, and Dark Souls 2.)
Ah, that's good to hear - and "rollslop" is a great word! I have enough FOMO that I probably will try dipping my toes into the genre a few more times, even after a few negative experiences.
SnapDragon
9d ago
Well, yes, this is what I mean when I say that some people don't understand what probability measures. If you pretend "schmrobability" is some weird mystical floaty value that somehow gets permanently attached to events like coin flips, then you get confused as to why the answer, as you can observe by trying forms of the experiment yourself, somehow becomes 1/3. Mathematicians say "ok, please fix your incorrect understanding of probability." Philosophers say "oh, look at this fascinating paradox I've discovered." Yeesh.
SnapDragon
9d ago
With Wikipedia, if I read an article on Abraham Lincoln, I am pretty confident the dates will be correct and the life and political events will be real and sourced. Sure, sometimes there are errors and there are occasional trolls and saboteurs (I once found an article on a species of water snake that said their chief diet was mermaids), and if you are a Confederate apologist you will probably be annoyed at the glazing, but you still won't find anything that would be contradicted by an actual biography.
So, yes, I'm sure most of us are aware that Wikipedia political articles are going to be as misleading as they can get away with, but let me just say that there are some completely non-political articles that are factually wrong, too. If you look up the Sleeping Beauty problem, the article states that there is "ongoing debate", which is ridiculous. For actual mathematicians, there's no debate; the answer is simple. The only reason there's a "debate" is because some people don't quite understand what probability measures. Imagine if the Flat Earth page said that there was "ongoing debate" on the validity of the theory...
And don't even get me started on the Doomsday argument, which is just as badly formed but has a bunch of advocates who are happy to maintain a 20-page article full of philosobabble to make it sound worthy of consideration.
I'm sure there are many other examples from fields where I'm not informed enough to smell the bullshit. Crowdsourcing knowledge has more failure modes than just the well-known political one.
First, it's "metroidvania", not "metrovania".
Hollow Knight and Silksong are both masterpieces, standing at the pinnacle of the metroidvania genre. But their lore (which is really unique and cool, and I like that you can deconstruct and analyze it to death) isn't really why I feel that way. It just comes down to gameplay - the controls are near-perfect and the challenges they throw in your way, particularly the bosses, are amazing and incredibly varied.
I actually kind of resent the Dark Souls comparison. I've barely played a real Dark Souls game, but I actively disliked Elden Ring (despite it, too, having incredible aesthetics and ridiculously deep lore). So many of the bosses felt exactly the same - oh, here's a screen-filling attack, I've memorized how many frames it takes so I can dodge-roll at the right time. Oh whoops, it was his fake-out attack instead, now I'm dead. (I guess I should have allocated my stats differently in their ridiculously-badly-explained leveling system so I could take two hits instead of one.) And I hate that other games considered soulslikes (Salt & Sanctuary, Nine Sols) have latched on to this style, too. You know, you can have a good, challenging game without making it ALL about i-frames!
Notably, there are no i-frames at all in Silksong (I believe the same was true in HK, but it's been a while). You are expected to move all around the screen to actively avoid boss attacks, not weirdly absorb an attack because you rolled with the correct timing. And the bosses are incredibly varied - from huge slow juggernauts, to ranged jerks firing at you while you leap around crumbling platforms, to nimble teleporting fighters who can parry and punish your attacks if you're overeager. For many of them, even on my 20th run back I still had a smile on my face, thinking about how I could do better next time. That's what 10 years of genius-level game craftsmanship can do.
The other thing that HK and Silksong do better than almost any other game is rewarding exploration. In most games, finding a secret wall will give you a small optional upgrade, and you do it because you want the 100% completion mark. In Silksong (even more than HK), finding a well-hidden secret might unlock a key quest item, or a hidden encounter, or even an entire new zone. It's kind of nuts, and it did mean I missed some big things by playing without a guide, but I loved it anyway.
If it weren't for Blue Prince, I think Silksong would easily have been my Game of the Year. (Disclaimer: I have not played Clair Obscur yet.)
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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