> However, if the coin came up tails, I must be in the lab room, while if it came up heads, there’s only a 50% chance I’m in the lab room now, so if I am in the lab room, I should think there’s a 2/3 chance that the coin came up tails.

Beauty can't lawfully reason about the problem, while treating awakenings as individual outcomes. "This awakening" is not "random awakening" - awakenings in the experiment do not happen at random, they have order: Tails&Monday is always followed by Tails&Tuesday. Neither you can say that "this awakening" is "any awakening" because for first and second awakenings probability that the coin is Heads is different.

To reason correctly about the problem you need to talk about events that happen in this experiment, not in illdefined "this awakening". Aplying the same principle here we get:

P(Lab) = 1; P(Heads|Lab) = P(Tails|Lab) =1/2 - regardless of the outcome of the toss, in every experiment you will be awakened in the lab, so finding yourself in the lab in this experiment doesn't tell you anything about the state of the coin.

P(Darkness) = 1 P(Heads|Darkness) = P(Tails|Darkness) =1/2 - regardless of the outcome of the toss in every experiment you find yourself with closed eyes thinking about anthropics and it tells you nothing about the state of the coin.

P(Home) = 1/2; P(Heads|Home) = 1 - you find yourself home after a memory loss in every experiment only when the coin is Heads so you update in favor of it.

//Beauty can't lawfully reason about the problem, while treating awakenings as individual outcomes. "This awakening" is not "random awakening" - awakenings in the experiment do not happen at random, they have order: Tails&Monday is always followed by Tails&Tuesday. Neither you can say that "this awakening" is "any awakening" because for first and second awakenings probability that the coin is Heads is different.//

It doesn't matter if this awakening is random in some cosmic sense. The point is that beauty has no evidence about which day it is, so she should be indifferent between the possible outcomes.

You're assuming that the evidence is "I awake in the lab at least once." And if that were so then there'd be no update. But that violates conservation of evidence. You know, after you open your eyes, you'll be somewhere--not updating in favor of tails is heads I win tails we're even reasoning (see my response to Richard).

> she should be indifferent between the possible outcomes

And there are exactly two of them Heads and Tails. Which we can rename to Heads&Monday and Tails&Monday&Tuesday.

Awakening that she experiences are not possible outcomes, for the reasons I've explained.

> You're assuming that the evidence is "I awake in the lab at least once."

Yes. This is exactly what she observes. She would be able to observe "I awake in the lab the second time", but sadly amnesia prevents her to do it.

> But that violates conservation of evidence.

On the contrary, what does violate conservation of expected evidence is always updating in favor of Tails when you wake up - this is a completely predictable update, which you do every time.

> not updating in favor of tails is heads I win tails we're even reasoning

Suppose, that Jesus Christ descends from Heaven to Earth tomorrow. That would be a quite serious evidence in favor of Christianity, to say the least. On the other hand, if it doesn't happen tomorrow... well, that's not much evidence in against Christianity, personally, I wouldn't be surprised in the slightest. Does this contradict conservation of expected evidence in your opinion?

I don't think we're getting anywhere, so let's just chat when we do the YouTube debate. Let me just say two things, and feel free to respond:

"On the contrary, what does violate conservation of expected evidence is always updating in favor of Tails when you wake up - this is a completely predictable update, which you do every time."

When you wake up, you'll have misleading information, so you should expect to update. If you expect to *forget* certain information that you have, it can be rational to predictably update.

This does violate conservation. If you think that upon opening your eyes, there's a 1/4 chance your credence in heads will spike to 1, and 3/4 it will remain 50/50, then you expect to update to 2/3.

// On the other hand, if it doesn't happen tomorrow... well, that's not much evidence in against Christianity, personally, I wouldn't be surprised in the slightest. Does this contradict conservation of expected evidence in your opinion?//

No. However, it would violate it if your credence didn't go down at all if he didn't ascend from earth. But because the prior in that is so low overall, and the odds of it are so low conditional on Christianity, it occuring would confirm Christianity but it not occurring would be barely any evidence.

I [am a thirder, but] have flirted with halfing and can clearly see the attraction. As such, I don’t quite see how the conclusions you reach in your version of the experiment transfer over to the original version?

If I understand your version, you are giving the subject an equal number of observations (wake-ups) regardless of the outcome of the coin flip? That seems to materially change the experiment (whether or not the subject’s eyes are open). To me at least, the thing that makes the problem strain my intuition is exactly the mismatch between the uneven number of samplings/observations and the 50/50 coin flip. Even it out, and there’s nothing more to solve. I guess that’s maybe the point… To create a scenario that’s easier to parse, but I think something more is lost along the way.

[EDIT: I deleted the rest of this comment, as I had my mind challenged / changed enough by Ape in the coat, below, that I didn’t want to leave it up.]

No. My point was that the *sense* of a paradox comes from the tension between the 50/50 of the coin flip vs. the three awakenings. If you create a 25/25/25/25 scenario (even if it is reduced to thirds as soon as beauty opens her eyes) it becomes a subtly different problem. You can still take different perspectives on it, but the way 4x25 neatly breaks down along 50/50 changes how it feels.

But in the last hours I’ve changed my thinking somewhat, and am losing sight of what I meant.

Basically, here’s how I see it now.

When Beauty wakes up, her chances are

Heads: 50%

Tails Monday: 25%

Tails Tuesday: 25%

In that sense I’m now a halfer.

However, run the experiment 100 times. Ask her each time she wakes up (~150 times), and give her a $10 prize every time she guesses correctly. In that case, her best strategy will be to guess tails each time. Because with that strategy she gets $20 every time the coin comes up tails (since she gets two guesses), but only “loses” $10 of the $1500 pot each it comes up heads. In that sense, there’s still something to the thirder position. Not about the probability, really, but the payout.

It really comes down to why we’re asking Beauty to think about the coin flip at all.

Well, yes, that explains why it doesn't work in your version of the problem, but not in the original.

And now that I have re-read your last comment, I need to change my reply: Yes, I think thirders should be halfers in your scenario, once they open their eyes, and find themselves in the lab. Because they are given more information than in the original. (It's basically the Monty Hall problem, removing one previously viable option halfway.)

As suggested above, I'm still a thirder in the sense that Beauty will, in fact, be right more often if she guesses tails. However, if I ran a casino where I had to pay out a prize for guessing correctly, and I wanted to stay in business, I would split the total pot more like 25/25/50 than 33/33/33. Wouldn't you?

Do you mean halfers should be thirders in the scenario?

// would split the total pot more like 25/25/50 than 33/33/33. Wouldn't you?//

Well prior to opening her eyes she should regard all 4 options as equal. But after she opens her eyes, she eliminates one of them, and so goes back to two thirds. I then later argue that this is like the original sleeping beauty problem.

And yes, I agree with you up until you say that it’s like the original Sleeping Beauty problem. I disagree about that, because in your version Beauty gets crucial, though not conclusive, information that takes a lot off the table.

But all of this – at least my claims about the problem – should be easy enough to test by simply flipping a coin enough times, and simulating the sleep and amnesia parts. I may, but honestly probably won’t.

> "However, if the coin came up tails, I must be in the lab room, while if it came up heads, there’s only a 50% chance I’m in the lab room now, so if I am in the lab room, I should think there’s a 2/3 chance that the coin came up tails."

Before opening your eyes, you should presumably have 25% credence in each of the four outcomes (Heads-Day1-lab, Heads-Day2-bed, Tails-Day1-lab, Tails-Day2-lab). If you're in the lab, that rules out the second of the four listed possibilities. But why assume that its probability must be redistributed equally amongst the remaining three? Wouldn't it seem more natural for a Halfer to assign (conditional on being in the lab): 50% Heads-day1, 25%-Tails-day1, 25%-Tails-day2?

Perhaps the issue is that Halfers seem to be precisely those who are inclined to reject the sort of anthropic reasoning you rely on here. So it shouldn't be surprising that inconsistencies result from combining Halfing with anthropic reasoning of that kind.

Well first as just a mathematical point, if there are some number of options, and one gets knocked out, you split the share of probability space it adopted among the remaining options.

Unless you do this in this case, you get a violation of the law of conservation of evidence, according to which you shouldn't expect your credence in some proposition to rise, on average, after you consider some evidence. The halfer with their eyes closed reasons that there's a 75% chance that their credence won't change in the coin having come up tails and a 25% chance that they'll get evidence that confirms heads. Thus, on average, they should expect their credence in heads after the coin flip to be 75%--but surely it's irrational to have a credence that you expect to rise literally when you open your eyes. This is heads I win tails we're even reasoning, to quote a turn of phrase that David Manley likes.

It's easy to see that waking up in the lab updates them in the direction of tails. P(them being in the lab)|tails=1, while P(Them being in the lab)|heads=.5. So updating on being in the lab, their credence should be 1/3.

I always leaned halfer, but your argument has shifted me to undecided (in a "the arguments sure seem to favor thirders, but I can't shake the suspicion that there must be a hidden flaw in those arguments!" kind of way).

> However, if the coin came up tails, I must be in the lab room, while if it came up heads, there’s only a 50% chance I’m in the lab room now, so if I am in the lab room, I should think there’s a 2/3 chance that the coin came up tails.

This is wrong.

As I've explained here

https://www.lesswrong.com/posts/gwfgFwrrYnDpcF4JP/the-solution-to-sleeping-beauty

Beauty can't lawfully reason about the problem, while treating awakenings as individual outcomes. "This awakening" is not "random awakening" - awakenings in the experiment do not happen at random, they have order: Tails&Monday is always followed by Tails&Tuesday. Neither you can say that "this awakening" is "any awakening" because for first and second awakenings probability that the coin is Heads is different.

To reason correctly about the problem you need to talk about events that happen in this experiment, not in illdefined "this awakening". Aplying the same principle here we get:

P(Lab) = 1; P(Heads|Lab) = P(Tails|Lab) =1/2 - regardless of the outcome of the toss, in every experiment you will be awakened in the lab, so finding yourself in the lab in this experiment doesn't tell you anything about the state of the coin.

P(Darkness) = 1 P(Heads|Darkness) = P(Tails|Darkness) =1/2 - regardless of the outcome of the toss in every experiment you find yourself with closed eyes thinking about anthropics and it tells you nothing about the state of the coin.

P(Home) = 1/2; P(Heads|Home) = 1 - you find yourself home after a memory loss in every experiment only when the coin is Heads so you update in favor of it.

And so everything adds up to normality.

//Beauty can't lawfully reason about the problem, while treating awakenings as individual outcomes. "This awakening" is not "random awakening" - awakenings in the experiment do not happen at random, they have order: Tails&Monday is always followed by Tails&Tuesday. Neither you can say that "this awakening" is "any awakening" because for first and second awakenings probability that the coin is Heads is different.//

It doesn't matter if this awakening is random in some cosmic sense. The point is that beauty has no evidence about which day it is, so she should be indifferent between the possible outcomes.

You're assuming that the evidence is "I awake in the lab at least once." And if that were so then there'd be no update. But that violates conservation of evidence. You know, after you open your eyes, you'll be somewhere--not updating in favor of tails is heads I win tails we're even reasoning (see my response to Richard).

> she should be indifferent between the possible outcomes

And there are exactly two of them Heads and Tails. Which we can rename to Heads&Monday and Tails&Monday&Tuesday.

Awakening that she experiences are not possible outcomes, for the reasons I've explained.

> You're assuming that the evidence is "I awake in the lab at least once."

Yes. This is exactly what she observes. She would be able to observe "I awake in the lab the second time", but sadly amnesia prevents her to do it.

> But that violates conservation of evidence.

On the contrary, what does violate conservation of expected evidence is always updating in favor of Tails when you wake up - this is a completely predictable update, which you do every time.

> not updating in favor of tails is heads I win tails we're even reasoning

Suppose, that Jesus Christ descends from Heaven to Earth tomorrow. That would be a quite serious evidence in favor of Christianity, to say the least. On the other hand, if it doesn't happen tomorrow... well, that's not much evidence in against Christianity, personally, I wouldn't be surprised in the slightest. Does this contradict conservation of expected evidence in your opinion?

I don't think we're getting anywhere, so let's just chat when we do the YouTube debate. Let me just say two things, and feel free to respond:

"On the contrary, what does violate conservation of expected evidence is always updating in favor of Tails when you wake up - this is a completely predictable update, which you do every time."

When you wake up, you'll have misleading information, so you should expect to update. If you expect to *forget* certain information that you have, it can be rational to predictably update.

This does violate conservation. If you think that upon opening your eyes, there's a 1/4 chance your credence in heads will spike to 1, and 3/4 it will remain 50/50, then you expect to update to 2/3.

// On the other hand, if it doesn't happen tomorrow... well, that's not much evidence in against Christianity, personally, I wouldn't be surprised in the slightest. Does this contradict conservation of expected evidence in your opinion?//

No. However, it would violate it if your credence didn't go down at all if he didn't ascend from earth. But because the prior in that is so low overall, and the odds of it are so low conditional on Christianity, it occuring would confirm Christianity but it not occurring would be barely any evidence.

Thanks for this (and the LW posts). It challenged/changed my thinking on this.

You are very wellcome!

edited Apr 1I [am a thirder, but] have flirted with halfing and can clearly see the attraction. As such, I don’t quite see how the conclusions you reach in your version of the experiment transfer over to the original version?

If I understand your version, you are giving the subject an equal number of observations (wake-ups) regardless of the outcome of the coin flip? That seems to materially change the experiment (whether or not the subject’s eyes are open). To me at least, the thing that makes the problem strain my intuition is exactly the mismatch between the uneven number of samplings/observations and the 50/50 coin flip. Even it out, and there’s nothing more to solve. I guess that’s maybe the point… To create a scenario that’s easier to parse, but I think something more is lost along the way.

[EDIT: I deleted the rest of this comment, as I had my mind challenged / changed enough by Ape in the coat, below, that I didn’t want to leave it up.]

Wait so in the scenario where beauty goes home the next day, you think halfers should be thirders?

No. My point was that the *sense* of a paradox comes from the tension between the 50/50 of the coin flip vs. the three awakenings. If you create a 25/25/25/25 scenario (even if it is reduced to thirds as soon as beauty opens her eyes) it becomes a subtly different problem. You can still take different perspectives on it, but the way 4x25 neatly breaks down along 50/50 changes how it feels.

But in the last hours I’ve changed my thinking somewhat, and am losing sight of what I meant.

Basically, here’s how I see it now.

When Beauty wakes up, her chances are

Heads: 50%

Tails Monday: 25%

Tails Tuesday: 25%

In that sense I’m now a halfer.

However, run the experiment 100 times. Ask her each time she wakes up (~150 times), and give her a $10 prize every time she guesses correctly. In that case, her best strategy will be to guess tails each time. Because with that strategy she gets $20 every time the coin comes up tails (since she gets two guesses), but only “loses” $10 of the $1500 pot each it comes up heads. In that sense, there’s still something to the thirder position. Not about the probability, really, but the payout.

It really comes down to why we’re asking Beauty to think about the coin flip at all.

See my response to Richard for why that doesn't work.

Well, yes, that explains why it doesn't work in your version of the problem, but not in the original.

And now that I have re-read your last comment, I need to change my reply: Yes, I think thirders should be halfers in your scenario, once they open their eyes, and find themselves in the lab. Because they are given more information than in the original. (It's basically the Monty Hall problem, removing one previously viable option halfway.)

As suggested above, I'm still a thirder in the sense that Beauty will, in fact, be right more often if she guesses tails. However, if I ran a casino where I had to pay out a prize for guessing correctly, and I wanted to stay in business, I would split the total pot more like 25/25/50 than 33/33/33. Wouldn't you?

Do you mean halfers should be thirders in the scenario?

// would split the total pot more like 25/25/50 than 33/33/33. Wouldn't you?//

Well prior to opening her eyes she should regard all 4 options as equal. But after she opens her eyes, she eliminates one of them, and so goes back to two thirds. I then later argue that this is like the original sleeping beauty problem.

Yes, that’s what I meant. Sorry.

And yes, I agree with you up until you say that it’s like the original Sleeping Beauty problem. I disagree about that, because in your version Beauty gets crucial, though not conclusive, information that takes a lot off the table.

But all of this – at least my claims about the problem – should be easy enough to test by simply flipping a coin enough times, and simulating the sleep and amnesia parts. I may, but honestly probably won’t.

> "However, if the coin came up tails, I must be in the lab room, while if it came up heads, there’s only a 50% chance I’m in the lab room now, so if I am in the lab room, I should think there’s a 2/3 chance that the coin came up tails."

Before opening your eyes, you should presumably have 25% credence in each of the four outcomes (Heads-Day1-lab, Heads-Day2-bed, Tails-Day1-lab, Tails-Day2-lab). If you're in the lab, that rules out the second of the four listed possibilities. But why assume that its probability must be redistributed equally amongst the remaining three? Wouldn't it seem more natural for a Halfer to assign (conditional on being in the lab): 50% Heads-day1, 25%-Tails-day1, 25%-Tails-day2?

Perhaps the issue is that Halfers seem to be precisely those who are inclined to reject the sort of anthropic reasoning you rely on here. So it shouldn't be surprising that inconsistencies result from combining Halfing with anthropic reasoning of that kind.

Well first as just a mathematical point, if there are some number of options, and one gets knocked out, you split the share of probability space it adopted among the remaining options.

Unless you do this in this case, you get a violation of the law of conservation of evidence, according to which you shouldn't expect your credence in some proposition to rise, on average, after you consider some evidence. The halfer with their eyes closed reasons that there's a 75% chance that their credence won't change in the coin having come up tails and a 25% chance that they'll get evidence that confirms heads. Thus, on average, they should expect their credence in heads after the coin flip to be 75%--but surely it's irrational to have a credence that you expect to rise literally when you open your eyes. This is heads I win tails we're even reasoning, to quote a turn of phrase that David Manley likes.

It's easy to see that waking up in the lab updates them in the direction of tails. P(them being in the lab)|tails=1, while P(Them being in the lab)|heads=.5. So updating on being in the lab, their credence should be 1/3.

Thanks, yeah, the "heads I win tails we're even" objection does seem pretty devastating!

Richard are you a halfer or a thirder or undecided?

I always leaned halfer, but your argument has shifted me to undecided (in a "the arguments sure seem to favor thirders, but I can't shake the suspicion that there must be a hidden flaw in those arguments!" kind of way).