I kind of skimmed that article, but basically he's arguing his problem has no concrete answer, and he's absolutely right, because the question does not give enough information to answer it. That is not the same problem as the OP though.

Knowing that I'm not disagreeing with Nick Bostrum makes me even more confident that 1/3 is the answer to OP.

The question isn't very clear, but if you read the arguments for the 1/2 and 1/3 view, it becomes obvious that the question is supposed to be the same as in the OP.

PairTheBoard:

Same experiment, but over a million days. If heads, she only wakes up once. If tails, she wakes up a million times over a million days. Same amnesia, etc.

When awake, she can reason, "I am awake at this moment in time. I don't know what point in time it is, but I'm clearly at a point in time, and that much is relevant. Therefore tails is much more likely to have been flipped than heads."

What is the flaw in this reasoning?

How is this not analogous to the OP?




Similarly, take the dude from Memento (5 min memory). If coin lands heads, he will be woken up for 10 minutes. If it lands tails, he will be woken up for 10 hours. He is only woken up once in each case.

What's the probability of a fair coin? : 50/50

What's more likely to have been flipped for this experiment? : I'm awake, therefore tails.

Same logic again.



You can't just ignore time because "durdle, durdle, coins are 50/50". Time is a relevant piece of info here.

I think the problem people have with this is they think they have no new information, when they really do, although it was new information they knew they were going to have.

If she only woke up 1/2 the time for heads, and 100% for tails, people could reason, "She woke up, therefore it's not the 50% of heads where I'm not awaken." I assume most people wouldn't say 50/50 here.

In this situation, it is true she will be woken up 100% of the time. However, that's misleading, because she will be woken up AT LEAST ONCE 100% of the time. The fact that she exists in a universe where she is woken up is still relevant information, because SHE WILL NOT BE WOKEN UP WITH EQUAL FREQUENCY in each universe.

In this case, 100% is a very poor indicator of how often she will wake up. It is misleading.

Similarly, if she were dreaming in OP's experiment, she could reason, "I'm asleep, therefore it's more likely that the coin landed heads. I can't assign probabilities because I don't know how many dreams like this I will have, or how long I will be asleep for, but pH>pT"

Oh, just incase.

PTB: I saw earlier you responded, "I'm done with this thread until you've read posts from 2007"

I've read most of the last four pages of that thread, and didn't come across any new arguments. With every mathematical reasoning for the answer being 1/2, my response is the same. Your formula does not factor in all of the information we have. There is information you are flat out ignoring.

Another way to look at this. The experiment is in a blue room. If she wakes up in a red room she knows the experiment is over.

Take OP but add that 3% of MH 2% of MT and 1% of TT she is not woken up as expected. She is aware of these odds. If she misses her chance to awake on the final day, she awakens in a red room which indicates it's over.

So she awakens in a blue room and is asked her credence of coin having landed heads. Now the 50/50 side can't use their logic because we have concrete probabilities. Upon being interviewed she will clearly say it's <1/3rd to be heads.

Yet we've only changed the OP by a few percent. Clearly it's not going from 50% to under 1/3rd with such a minor change, therefore the 50/50 camp is wrong.

I don't think you're understanding all the arguments. For example, in my big post, I never appeal to the indifference principle to establish 1/2 (I do use it to make monday|tails = tuesday|tails, although this is not actually necessary, and it needs an even more annoying probability space to be rigorous). In you post 57 example, my method would slightly discount heads, but not to 1/3 or anything close.

what is wrong with the following reasoning

I wake after a fair toss of a coin and am told it is Monday. I am the asked "what credence..." Obviously I answer 50%.

However I am not privy to this information. All I know is that a coin was flipped and I am awake.

I reason thus. If this is Monday I have a 50% chance of getting the answer right if I respond "tails". If this is Tuesday I have no chance of getting the answer right if I respond "heads".

Since there is a non zero probability of this day being Tuesday and since the answer "tails" has a 50% chance of being right on Monday anyway, I must add the possibility of it being Tuesday to my equation and answer tails every time I am wakened.

I would like to see a response from the 50/50 camp to this.

Did you study my post #36? I'll copy it below for your convenvience. Notice that in all scenarios Beauty's "Awake Time" remains constant at 1 hour. When her Tails awake time is interupted by a period of unconsciousness the total pre interuption time and post interuption time always equals 1 hour. And of course her Heads awake time is always an uninterupted hour.

The original Sleeping Beauty problem should really have this same "equal awake time" stipulation because the arguments for 1/3 do not appeal to time awake but to the number of "checked credences". The cases I present in post #36 are consistent on this point so the arguments for 1/3 should apply to them just as well as they do in the Original Problem. But I don't think they work for my examples. And to maintain the 1/3 position I believe you must take a stand at some point in my sequence of scenarios and claim 1/3 begins to apply there. To do that you must explain why 1/3 applies where you say it does but not in the previous scenario. I don't think that can be credibly done.

--------------------
Post #36
Consider the following sequence of related experiments.

Assume throughout that Beauty is not only given amnesia but also has no sense of time while awake. She has no sense of how long she's been awake since her last sleep. However, when awake within the experiment she will be aware that she is awake within the experiment.

All cases have the same settup where Beauty is told Sunday Night before she goes to sleep that a fair coin will be flipped after she goes to sleep.

Also assume the experimenters have technology which unobtrusively allows them to read Beauty's mind at any time and accurately tell them what credence she holds at that moment in the proposition that the coin fell heads. Beauty cannot tell when her mind is being read.

Beauty is told all this to begin with.


1.) Beauty is also told that in either case, Heads or Tails, she will be awakened in the morning and put back to sleep after an hour until the following Sunday at which time the experiment is over. What credence should Beauty have for Heads when she finds herself awake within the experiment?

Answer: 50%
Both cases H,T are identical.



2.) Same as 1) except Beauty is told her mind will be read once if Heads, and twice if Tails. If Tails her mind is read 1 minute after waking and again 31 minutes after her awakening. What credence should Beauty have for Heads when she finds herself awake within the experiment?

Answer: 50%
She reasons that the situation is identical to 1) where she clearly has credence 50%. They can read her mind as many times as they want, it doesn't change her credence that Heads was flipped. Her experience hasn't changed.



3.) Same as 2) except Beauty is told that if they read her mind and her credence is 50% they will take that as an indication she would be happy to bet on Heads and so they will place a $100 bet for her on Heads. What credence should Beauty have for Heads now when she finds herself awake within the experiment?

Answer: 50%
Whether they read her mind or place bets for her, nothing has changed from 1) to affect her credence the coin landed heads. Her experience remains identical to 1). However, when told of the bet she replies, "Well, if you are going to place two bets for me on heads when you know it's Tails I want at least 2-1 odds on the bet. Otherwise, play tricks with your own money."



4.) Same as 3) except Beauty is told that if Tails, 30 minutes after awakening she will be instantly put to sleep and reawakened 1 millisecond later with amnesia and no sense of how long she has been awake. To keep her total awake time the same when Tails, they will put her to sleep again 1 hour and 1 millisecond after her first awakening. What credence should Beauty have for Heads now when she finds herself awake within the experiment?

Answer: 50%
Beauty recognizes that while her experience is no longer identical to 1) it remains indistinguishable to her. If there were no mind reading nor bets she would certainly not change her credence because of a 1 millisecond loss of consciousness in the middle of the Tails awakening - all else being equal. And the mind reading and bets remain just as irrelevant as before.



5.) Same as 4) except Beauty is told it will be a 1 hour instead of 1 millisecond loss of consciousness in the middle of her Hour Long Tail Awake Time. What credence should Beauty now have for Heads when she finds herself awake within the experiment?

Answer: 50%
Beauty recognizes her experience will still be indistinguishable from 1). However they chop up her 1 hour Tails awake time with amnesia, having no sense of time her subjective experience of it remains the same. Chopping the time up has no bearing on which way the coin landed.



6.) Same as 4) and 5) except Beauty is told it will be a 24 hour loss of conciousness in the middle of her Hour Long Tail Awake Time. So half her Tail Awake Time will be Monday morning and the other half on Tuesday morning. What credence should Beauty now have for Heads when she finds herself awake within the experiment?

Answer: 50%
Like she reasoned in 5), they can chop up her Hour Tails Awake Time however they like, it has no bearing on either her subjective experience of it nor on whether the coin landed heads or tails.



2'.) Recall in 2) there was no interuption of consciousness, just the mind readings. In 2') it's same as 2) except the experimenters tell Beauty they're going to just ask her what her credence is for Heads rather than read her mind. To do this they give her a short term memory loss drug so that not only does she have no sense of time but when Tails and asked the second time she has no memory of having been asked the first time. What now Beauty's credence for heads?

answer: 50%
Beauty reasons that having her credence checked by asking rather than mind reading has no bearing on how the coin landed. She already knew in 2) that her credence would be getting checked twice when Tails. If the Tails double-checking didn't change her credence when by mind reading it shouldn't change it doing it the old fashioned way by asking Her.

If she changes her credence when asked because she thinks the act of being asked is twice as likely when Tails, then she must hold this alternate credence for the entire Hour Awake time whether she is being asked at the moment or not. So it would not be the "being asked" that would change her credence but knowing she was going to be getting double-asked when Tails. But she knew just as well in 2) that her credence was being double-checked when Tails by mind reading. If knowing the double check of her credence by mind reading didn't change her credence then knowing the double-check by asking shouldn't either.

So she maintains 50% whether her credence is checked by mind reading or by asking. Of course if they introduce betting in a 3') she will still demand at least 2-1 odds to bet on heads when asked her credence.



6'.) Same as 6) except they inform Beauty she will be asked her credence 1 minute after each awakening rather than reading her mind at those times. Her credence in heads?

Answer: 50%
Same argument as 2') together with argument in 6).



Notice 6') is the original Sleeping Beauty problem.
-----------------------------


PairTheBoard

I suppose an argument could be that in 50% of cases she still only wakes up once. That that day is Monday and that no matter how often she wakes up in the other 50% it will only add up to 50% of the time across infinite experiments.

Therefore, if she wakes up and says tails she'll still be wrong 50% of the time.

It's basically an appeal to the indifference principle to declare all days equally likely, although you already know the experiment isn't symmetric with respect to days of the week. You could, if you really wanted to, claim that "It is H1 or T1" "It is H1 or T2" "It is H1 or T3" ... should be equally likely, or "given tails, T1, T2, T3... should be equally likely", because the experiment is symmetric in those regards, although neither of these give you an answer directly.

This thread is full of reasons why you really should write out probability spaces and define everything properly.

I think 2/3'ers could argue here that she is really awaken 2 times when tails so now he has 3 possible states to be in and arguably 3 of them with the same probability.

I think this is where you're wrong. In the original problem she is aware she has just woken up. Since she wakes up twice when it is tails this is where the difference comes from. A fair representation of this problem would be if you (in your examples) allowed her to be awake for twice as long when it is tails (with the no awareness of time remaining intact). In that scenario I would still argue she should believe tails to be more likely.

Pair, I agree with everything you say (except the wording in #4 is a bit off) up until this point:

I'm assuming you are implying, every time she is asked a question in case 2' that she is given amnesia.

If this is the case, when she's awake and not being asked a question, she thinks 50/50.

Spoiler for irrelevant nitpickery:


When she's asked a question, she thinks 1/3.

When her mind is being read, she's given no information.
When she's being asked the question, she is.

Isn't this just blatantly level 1 obviously correct?

For the record, I think the wording in #4 could be interpreted in two very different ways. The way that makes sense for you to intend it, I agree with.

I reserve the right to disagree with it, but 2' seems much more to the point and far more worth discussing.

While we're at it, am I seriously the only person in ~30 years to realize that in the doomsday argument, the very first step modeling of n/N by U(0,1] doesn't work because n/N can only take rational values, and there's no such thing as U(0,1] on rationals between (0,1]? (among other reasons)

Is the DA something people take seriously? I could list quite a few flaws with it.

They pretty much all fall under this description: "DA makes ambitious assumptions when they would support the conclusion, and ignores empirical data when it would dispute the conclusion."

You are basically told "If we ignore everything else, and just look at this", and then are introduced to very specific assumptions right after you have been told to forget everything else.

It seems very clear to me that she should conclude that there is a 2/3 probability that tails is the correct answer.

There are two probabilistic questions that she faces on awakening. What day is it and how did the coin land?

Tuesday wakeups occur with 1/2 the probability of Mondays and all wakeups are Monday or Tuesday. Therefore the probability of Monday is 2/3 and Tuesday is 1/3. On Monday the coin is 50/50 heads or tails. Thus, Monday/Heads is 1/3 and Monday/Tails is 1/3. On Tuesday, the coin is 100% tails so Tuesday/Heads is 0 and Tuesday /Tails is 1/3.

All told then:

Monday/Heads = 1/3
Monday/Tails=1/3
Tuesday/Tails=1/3.

Since 2/3 of the possible events include a tails, she concludes it is 2/3 likely that the coin flipped tails.

I would hope not, but it got a freaking Nature article in 1993.

I think you erred when you included the bet in 4 and that made everything after that suspect. If she bets once on what amounts to the first awakening then all of the awakenings are no longer equivalent and that biases the experiment in your subsequent examples. I still believe 2/3 is the correct answer.

The experiment is not well designed to answer the question on coin toss probability because the result of the toss couples with the asking of the question so that the final answer is not 50/50.

The difference between the 2/3 and 50% camp is that the 50% camp hope they are right...

What is the response of the 50% camp to the questions?

What separates these next 2 experiments;

1) N=1 pill after tails

2) N=2 pills after tails

They probably view them as identical. How can that be?


The 50% camp is speechless about the fact that an unfair coin that gives 2/3 tails with no pill involved behaves exactly the same in response to the question how often a tail lead to this waking up. The difference starts appearing only when streaks are studied but the empirical ratio that defines probability is not deriving any correction from them. And yes frequency and probability are not the same thing depending on situations, but they are here because the real experiment is to take N trials of realized wake ups and see how often when your wake up its Monday or how often it was tails that took you there and this has got to be the definition of what is the probability tails was served out on Monday's flipping or what is the probability its Monday.

And of course nobody from the 2/3 camp is ever disputing the fact that this is a fair coin and that the 50% branching on Monday is the very reason you have 2/3 after waking up.

So i am really waiting for the 50% camp to tell me that they are confident that when they wake up its Monday at x% probability or Tuesday at 1-x (give me an x guys).

Would it help at all the discussion if we imagined that the experiment performed is a permanent flow one where every day of the life of one person the question is asked and of course its performed with pill or no pill depending on the head tails outcome.

Would that speed of flow of the experiment convince anyone that the probability 2/3 makes sense when waking up? The space we are drawing from is the days of our lives.

The experiment record keeping would basically look like that in some typical sequence...

10011000000100110011100100100100001000010011000000 11110000100