Since SB doesn't remember Monday, she cannot feel the difference but the structure of the experiment KNOWS the difference.So if she is asked twice, Monday and Tuesday, that only happens with tails outcome. Even without memory, her credence may shift, but because the setup itself is informative. — Kizzy

It's also part of the protocol that although SB knows that she is being awakened a second time on Tuesday if and only if the coin landed tails, on each occasion where she is being awakened, she isn't informed of the day of the week. As specified in the OP (and in the Wikipedia article), she doesn't know if her awakening is happening on Monday or Tuesday (though she does know, or rather can infer, that it's twice as likely to be occurring on Monday). Hence, the relevant information available to her for establishing her credence is the same on each occasion of awakening.

I think I understand and see my mistake, I was confused from the start as I misunderstood the problem because of the way its laid out and my brain,[thing to note208am] and since I was and am assuming SB is using prior knowledge plus new information Tuesday, without amnesia Sunday to Monday. I just find it to be seemingly difficult to not use some sort of self locating knowledge to assign degree of belief in something, that something being a coin heads up with or without amnesia Sunday and the chance for belief to update come Tuesday.

Since I didn't account for the amnesia from Sunday to Monday, only Monday to Tuesday. My pov, though in error for mistaking PROTO, understood the OG problem (see wiki for Sleeping Beauty Paradox*) to mean after she goes to sleep Sunday, the experiment does not start until Monday, where she goes into the experiment asleep and awakened [inevitable], and asked her credence, put to sleep and either asked again Tuesday with amnesia or sleeps through it. awakening Wednesday and experiment is officially over. .

I do not put it passed me, that even though I just very well may have completely misinterpreted the problem before my own eyes, to be standing corrected, as I have to in order to carry on...

However, now knowing Sunday to Monday she is given amnesia, something about knowing the experiment duration and rules tells me self locating knowledge must come from somewhere, from something in time and our relation to it is relevant to the possible knowing, yes being awake is evidence of some thing but what else gives?

Thanks for your contributions to this thread. And also to your AI exploration experience and work that you've shared here on the forum as well. Of interest...

BUT even though, I was and still am off, in this way:

What else besides knowing what she signed up for, prior know ledge and new intel [ i am awake now] does not alter her credence Monday, only verifies indirect info about SB maybe? If she was given amnesia Sunday and put to bed, awakened Monday then what awakening is she forgetting? *see quote from wiki below, note bold words that are throwing me.....

Anyways, either way -

OH never mind, OF course if she knew it was Monday she wouldn't say 1/3, but what if she was off...and Tuesday comes around and it changes to 0? the chance to change or update belief still exists if tails and asked twice. On Monday she does not know for certain if heads or tails only gives her degree of belief in heads, knowing nothing Wednesday when experiment ends, tomorrow she will be awakened or sleep through the day, she can still guess reasonably participating, I think? I don't know, perhaps I am in over my head here...again!

* per Wikipedia, "On Sunday she will be put to sleep. Once or twice, during the experiment, Sleeping Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening." https://en.wikipedia.org/wiki/Sleeping_Beauty_problem

OH never mind, OF course if she knew it was Monday she wouldn't say 1/3, but what if she was off...and Tuesday comes around and it changes to 0? the chance to change or update belief still exists if tails and asked twice. On Monday she does not know for certain if heads or tails only gives her degree of belief in heads, knowing nothing Wednesday when experiment ends, tomorrow she will be awakened or sleep through the day, she can still guess reasonably participating, I think? I don't know, perhaps I am in over my head here...again! — Kizzy

You'll more easily wrap your head around the problem if you don't overcomplicate things (even though it will remain a tough problem). The purpose of the drug merely is to make it impossible for Sleeping Beauty on any occasion of awakening to know if this occasion was a first or a second one in the experiment (which she could otherwise deduce if she had a memory of the previous one or the lack thereof). This makes all three possibilities—Monday&Heads, Monday&Tails and Tuesday&Tails—indistinguishable from her subjective perspective although she knows at all times that over the course of the experiment all three of those situations could be experienced by her (without knowing which one it is whenever she's experiencing one of them). You can now place yourself in her shoes and start pondering what the chances are that the coin landed tails.

(I'm glad you're enjoying my AI experiment reports!)

SB knows that Monday waking is guaranteed, no matter what the outcome of the coin toss, if so how can she eliminate the sleeping day and update the probabilities or her credence to 1/3 — Kizzy

SB does not know if a waking day is a Monday. Only that it is a waking day. She can eliminate the sleeping day because she knows this is a waking day.

Compare two versions:

• Three days where she is wakened and interviewed, and a fourth where she sleeps.
• Three days where she is wakened and interviewed, and a fourth where she is wakened and taken to DisneyWorld.

On a waking day in the second version, she clearly can eliminate the DisneyWorld day and the probability of Heads. Why is that different? Whether on not she would be awake on that "fourth day" is irrelevant. The important fact is not being able to observe it when it happens, it is being able to observe that it is not happening when it does not.

disagreements arise regarding the meaning of Sleeping Beauty's "credence" about the coin toss result when she awakens, and also about the nature of the information she gains (if any) when she is awakened and interviewed. — Pierre-Normand

Well, isn't this exactly that I tried to say about this being about information?

Should Sleeping Beauty express a 1/2 credence, when she is being awakened, that the coin landed heads? Should it be 1/3, or something else? — Pierre-Normand

Isn't the only the she can say simply that she's participating in the experiment... and she cannot know if its monday or tuesday. Information has an effect on the probability (as in the Monty Hall). Without the information, the probability cannot be accurately defined by her when waking up.

Since SB doesn't remember Monday, she cannot feel the difference but the structure of the experiment KNOWS the difference.So if she is asked twice, Monday and Tuesday, that only happens with tails outcome. Even without memory, her credence may shift, but because the setup itself is informative. — Kizzy

You are one of four volunteers gathered on Sunday Night. You see the combinations "Monday and Heads," "Monday and Tails," "Tuesday and Heads," and "Tuesday and Tails" written on four different note cards. They are turned over, shuffled, and distributed between you, but you can't look. You are told that after you go to sleep, a single fair coin will be flipped. Then, on Monday and again on Tuesday, three of you will be wakened asked some questions. The one who is left out will be the one whose card says the actual coin flip result, and the current day. Afterwards, you will be put back to sleep with amnesia.

Some time later, you find yourself awake and sitting in a room where you can see two of the other three volunteers on TV monitors (you are instructed to not try to communicate through them). One is labeled "Monitor A," and the other "Monitor B." You, and these other two, have their card face-down card on the table in front of them.

• Not knowing what your card says, you are asked for your credence that the coin result written on you card is the actual coin result. AND, the same question about your credence for it matching A's card, and B's card.
• Once you all have provided an answer (unseen by the others, of course), you are told to look at your card, without revealing it, and answer the same questions.

I say the answers in #1 cannot be anything but 1/3. You have the exact same information about each, and they have to add up to 1. If you disagree, please explain how it is possible. Note that the "structure of the experiment KNOWS" that there is a day, an a coin face, that apply. The importangt part is that yolu don't know these.

I say the answers in #2 can't change. Knowing the specific names applied to you "sleep day" does not change their existence what the "structure of the experiment KNOWS," in any way. You seem to think it can; that what the "structure of the experiment KNOWS" changes for you.

But the same applies to A and B. If it changes the same way, your answer for them changes the same way and eveybody's is 1/2. This is a paradox.

And if it changes in a different way for A and B, allowing you to say 1/4 for them, how did it change differently?

The Halfer's run-centered measure just is a way to measure the space of probabilities by partitioning the events that Sleeping Beauty's credence — Pierre-Normand

<Sigh.> I can repeat this as often as you ignore it.

The experiment, when viewed from the outside, consists of two possible runs. The experiment that SB sees is one day, from one run, and to her that one day is independent of whichever run she is in. Since she cannot know which run she is in, that is not information that is useful to her. Inside the experiment, an outcome consists of one "day" only. The only point that is significant to SB is that she can tell that an interview day is not a sleeping day. This constitutes "new information" in probability.

In fact, "new information" is not defined in probability. The information that allows for probability updates is whatever eliminates outcomes that exist in the prior sample space, but are inconsistent with that information. Yes, this usually means a positive fact about the outcome, that does not apply to all, hence some call it "new information." But being "new" isn't what is important, it is the elimination. H&Tue is a member of the prior sample space. It is eliminated when she is awoken and interviewed.

And you can check this is several ways, all of which you ignore. One, you can change the sleeping day to one where she is awakened, but not interviewed. I'll stick with the example that you take her to DisneyWorld. Now one "day" in what you call the "Heads run" is eliminated when she is interviewed. Since the "Heads run" has a 50% probability, and she is can't be in all of the probability-weight of the "Heads Run,", her credence in Heads must be less than 50%/

But it cannot matter what happens on H&Tue. What affects SB's credence is that she knows that the current "day" is not H&Tue. Which she knows whatever happens on H&Tue.

Or you could address the Camp Sleeping Beauty version with more than just "it illustrates the thirder view." You could try to apply the "six day run" theory to Camp Sleeping Beauty. And you will not be able to do so consistently.

• Is SB's credence in each die roll the number of times today's activity appears in that row, divided by the number of times it appears in the 6x6 calendar. If you disagree, please say what it is based on the "six day run" theory.
• If one of the activities is replaced with "sleep through this day," does that change her credence in any way? HOW, and TO WHAT?

SB knows that Monday waking is guaranteed, no matter what the outcome of the coin toss, if so how can she eliminate the sleeping day and update the probabilities or her credence to 1/3 — Kizzy

Because her current knowledge and existence in the experiment is fully limited to one day. Knowing that she will always be awakened on Monday does not change that. See above.

I do think this related to the Monty Hall problem where information affects probabilities. Information does affect probabilities, you know. — ssu

This is called conditional probability.

It's easier indeed to understand the Monty Hall when there's a lot more doors

What that does is make it more intuitive. Since there is a 99.9999% chance Monty Hall picked that one door for the specific reason that it has the car, and a 0.0001% chance that he picked a goat door randomly, it makes sense top go with the 99.9999%. This is harder to wee then the numbers are 66.7% and 33.3%.

So yes, there is similarity in that the information that allows conditional probability to be used is hard to see. But the reasons are quite different. In Sleeping Beauty, it is because philosophers want to propose inconsistent ways to view information.

<Sigh.> I can repeat this as often as you ignore it.

The experiment, when viewed from the outside, consists of two possible runs. The experiment that SB sees is one day, from one run, and to her that one day is independent of whichever run she is in. — JeffJo

SB doesn't have the magical power to make the other awakenings, or their mutual causal relationships, drop out of existence on the occasion where she awakens. She still knows that the two potential T-awakenings live on the same timeline (and hence that when she's experiencing one of them, she will also go on to experience, or will have experienced, the other one in the same run).

Since she cannot know which run she is in, that is not information that is useful to her. Inside the experiment, an outcome consists of one "day" only. The only point that is significant to SB is that she can tell that an interview day is not a sleeping day. This constitutes "new information" in probability.

The fact the the information isn't new to her doesn't make the possibility of there being other potential awakenings in the same run irrelevant. She already has information about those possibilities (and long run frequencies) since she was told about them before the experience began. The Halfer stance, just like the Thirder (equally valid) stance, does not depend on her learning anything new when she awakens since it merely depends on her knowledge of the relative frequencies of H-runs to T-runs.

You're saying that when she awakens, she learns that an interview day is not a sleeping day. But she already knew that interview days never are sleeping days. She can't be asleep and awake at the same time. She knew before the experiments began that the awakenings she would potentially experience in the future would equally as often turn out (merely unbeknownst to her at the time) to have been T-Mon, T-Tue and H-Mon and hence that, when she experiences any of them, those three possibilities would be equally likely from her epistemic standpoint. The Halfer-credence isn't either based on anything new that she learns upon awakening but it is about a differently partitioned relative frequency of events.

To recap what I had said earlier: When SB, as a Thirder, says that the odds that the coin landed tails are 2/3, what she means is that that her current awakening episode just is one from a set of indistinguishable awakening epodes that, in the long run, will turn out to have been T-awakenings 2/3 of the time. When SB, as a Halfer, says that the odds that the coin landed tails are 1/2, what she means is that her current awakening episode is part of a set of indistinguishable runs that, in the long run, will turn out to have been T-runs one half of the time.

Just as you view it as irrelevant to your Thirder claim that T-mon and T-tue belong to the same run, which it indeed is, a Halfer views it as irrelevant to their claim that T-runs spawn more than one awakening episode, which it indeed is. The Halfer and yourself simply are talking about different things.

SB doesn't have the magical power to make the other awakenings, or their mutual causal relationships, drop out of existence on the occasion where she awakens. — Pierre-Normand

Exactly. That is the opposite side of the ability you claim she could have, to make one "other awakening" selectively pop into significance based on know;edge she does not possess. That is, to treat the "other Tails awakening" when the coin landed Tails differently than the "Heads awakening."

Again: the prior sample space comprises FOUR combinations of Coin+Day. In the prior, each is equally likely to apply at the moment the lab techs decide whether or not to awaken her. If they do, to entirety of her information about it is that it is one of the THREE combinations that correspond to an awakening. To her, there is no more, or less, of a connection to the "other" day in this two-day run that indicates, to her, whether it is Monday or Tuesday, if the coin landed Heads or Tails, or which "run" she is in. If you think otherwise, I'd be glad to hear why. An explicit reason why, on T&Mon, she could be more or less likely to think it is H&Mon. This requires knowledge of whether she is in a Heads or Tails run, not the knowledge that such runs are possibilities.

When SB, as a Halfer, says that the odds that the coin landed tails are 1/2, what she means is that her current awakening episode is part of a set of indistinguishable runs that, in the long run, will turn out to have been T-runs one half of the time. — Pierre-Normand

Indistinguishable? You contradict yourself here, because in the long run you do distinguish them.

But you use this argument to once again evade answering the direct questions I have asked several times. One of them is "If the sleeping day is changed to a non-interview waking day, what should her answer be on an interview waking day?" It can't be 1/2, because that would not allow here to have 100% credence in Heads in the non-interview waking day. So she must answer 1/3. But if she answers 1/3, what is different in her knowledge on an interview waking day in the original version?

And if you try to hand-wave a difference, how does in work in the Camp Sleeping Beauty version when each run can contain a different number of waking days?

But I've given up the silly notion that you will address these questions. Which probably means you can't.

Indistinguishable? You contradict yourself here, because in the long run you do distinguish them. — JeffJo

No. I just mean that when she awakens she isn't able to tell if she's in a T-run anymore than she can tell if she's in a T-Monday-awakening or any other possible awakening. That's why the best she can express is a rational credence. She distinguishes runs, and awakenings, and coin toss results, as distinct possibilities that are realized with frequencies determined by the experiment's protocol. If those possibilities were irrelevant, then her knowledge of the protocol that sets their long run frequencies would also be irrelevant. But it's clearly relevant to both Halfers and Thirders.

No. I just mean that when she awakens she isn't able to tell if she's in a T-run anymore than she can tell if she's in a T-Monday-awakening or any other possible awakening. — Pierre-Normand

And I'm saying that this is the exact reason why she cannot base credence on what may, or may not, be the other part(s) of the "run" she is in. I'm saying that all she can base credence on is the one day she can see. And this is trivial to confirm, by addressing the questions you refuse to acknowledge.

And I'm saying that this is the exact reason why she cannot base credence on what may, or may not, be the other part(s) of the "run" she is in. I'm saying that all she can base credence on is the one day she can see. And this is trivial to confirm, by addressing the questions you refuse to acknowledge. — JeffJo

I've made quite a few points that you've never acknowledged, some of them in responses to questions of yours that I responded to more than once. But some of the objections you raise are so vague and bear so little relationship to what I've said that the best I can do in response to them is to try to reiterate my own view more clearly. You repeatedly claimed that I'm disallowed to make reference to any awakening opportunity Sleeping Beauty isn't currently experiencing. But how do you yourself arrive at a credence of 2/3 without making reference to the fact that there are three possible awakening opportunities in total and not just the single one that she is experiencing?

What the SB problem amounts to is a Reductio ad absurdum against the principle of indifference being epistemically normative, a principle that in any case is epistemically inadmissible, psychologically implausible, and technically unnecessary when applying probability theory; a rational person refrains from assigning probabilities when ignorant about frequency information; accepting equal odds is not a representation of ignorance (e.g Bertrand's Paradox).

- It is commonly falsely argued by thirders, that halvers are suspect to a Dutch-book argument, by virtue of losing twice as much money if the coin lands tails, than they gain if the coin lands heads (since the dutch-book is defined as an awoken SB placing and losing two bets, each costing her $1 in the case of tails, one on monday and one on tuesday, versus her placing and winning only one bet rewarding her with $1 on Monday if the coin lands heads). But this dutch book argument is invalidated by the fact that it it equivalent to SB beingapriori willing to win $1 in the case of heads and losing $2 in the case of tails, i..e. SB knowingly accepting a Dutch Book with an expected loss of 0.5x1 - 0.5x2 = -$0.5 before the experiment begins, given her prior knowledge that P(H) = 0.5. So the Dutchbook argument is invalid and is actually an argument against the thirder position.

The (frankly unnecessary) lesson of SB is that meaningful probabilities express causal assumptions, and not feelings of indifference about outcomes.

You repeatedly claimed that I'm disallowed to make reference to any awakening opportunity Sleeping Beauty isn't currently experiencing. — Pierre-Normand

You can refer to any part of the experiment you want. Sleeping Beauty knows all of the parts (*), but has no means to relate her current awake period to any others. You are saying halfers base their answer on doing that. They can't.

But how do you yourself arrive at a credence of 2/3 without making reference to the fact that there are three possible awakening opportunities in total and not just the single one that she is experiencing?

Are you really that obtuse? As I indicated with the (*), she knows all of the parts. That's what establishes the prior sample space. All four possibilities, with equal probabilities. Since she is awake, she eliminates the one she sleeps through.

And as I have said, betting arguments don't work because you have to agree on how many bets are placed. But there is no logical fallacy in a direct probability analysis, as I have done.

There is nothing vague about my questions, unless you refuse to understand it.

• Compare two versions of the popular problem; one where she stays asleep on H+Tue, and one where she is awakened but taken to Disney World instead of being interviewed. In the halfer, two-runs model, does her credence in Heads change between these two versions? What is her credence in Heads when she goes to DisneyWorld?
• In my Camp Sleeping Beauty version, is her credence in die roll D (# times today's activity appears in row D)/(# times today's activity appears in table), as thirders would claim, or is it 1/6 as halfers would claim? How does the halfer's answer change if today's activity does not appear in all rows?

What the SB problem amounts to is a Reductio ad absurdum against the principle of indifference being epistemically normative, a principle that in any case is epistemically inadmissible, psychologically implausible, and technically unnecessary when applying probability theory; a rational person refrains from assigning probabilities when ignorant about frequency information; accepting equal odds is not a representation of ignorance (e.g Bertrand's Paradox). — sime

I don't see any questionable appeal to the principle of indifference being made in the standard Thirder arguments (though JeffJo may be making a redundant appeal to it, which isn't needed for his argument to go through, in my view.) Sleeping Beauty isn't ignorant about frequency information since the relevant information can be straightforwardly deduced from the experiment's protocol. SB doesn't infer that her current awakening state is a T-awakening with probability 1/3 because she doesn't know which one of three indistinguishable states it is that she currently is experiencing (two of which are T-awakenings). That would indeed be invalid. She rather infers it because she knows the relative long run frequency of such awakenings to the 2/3 by design.

The (frankly unnecessary) lesson of SB is that meaningful probabilities express causal assumptions, and not feelings of indifference about outcomes.

I don't think that is the salient lesson from the thought experiment but I agree with your claim.

Regarding the Dutch-book arguments, they represent specific payout structures. They highlight why it's rational for Halfers to be indifferent between betting on H or T when only one betting opportunity and payout is afforded to them in one run of the experiment. They also highlight why Thirders are not likewise indifferent between betting on H or T when one betting opportunity and payout is afforded to them on any awakening occasion.