For example, if one were to ask the average person to express their credences regarding the outcome of a two horse race that they know absolutely nothing about, they will simply say "I don't know who will win" and refrain from assigning any odds, equal or otherwise. They will also tend to accept bets in which they have knowledge that the physical probabilities are 50/50 over bets that they are totally ignorant about. — sime

Two Envelopes seems to encourage abuse of the principle of indifference in exactly this way. Maybe it's just something like this: rationality requires treating "It's one or the other but I've literally no idea which" as an uninformative 50:50 prior only when there's the real possibility of acquiring new information upon which to update that prior. I'd rather just say, no, don't do that, "I don't know" doesn't mean "It's 50:50", but there are a great many usages in which the prior is quickly swamped by actual information, and the PoI is a harmless formality. --- In Two Envelopes, you know there will never be any new data, so that harmless prior metastasizes.

Y'all know the math much better than I, so maybe I'm speaking out of turn.

trying to move away from the idea that one's credence in the state H is entirely determined by the specification of the ways in which one can come to be in that state — Pierre-Normand

Even the word "state" feels too coarse for Sleeping Beauty, since it could denote the situation a robust well-defined subject finds themselves in, or it could denote the very identity of that subject. --- At least, that's how the two main camps look to me. One wonders, where am I? how did I get here? One wonders, what am I? what has made me into this?

As you say, it's all about individuation. Lacking a fixed point of individuation, you can push the lever however you like but you won't actually move anything.

The thirder's position is indeed a ratio of possible words, but there is scant evidence to support the idea that credences are accurately represented by taking ratios over possible worlds. — sime

Elsewhere Lewis is pretty careful about what he calls de se modality -- epistemic questions are not just about possible worlds but irreducibly about your epistemic counterpart's status in a given possible world.

That's in the neighborhood of what I've been musing about anyway.

Here's one more ridiculous comparison to clarify the difference between Where am I? and What am I? (if it's even relevant to Sleeping Beauty): the Chomsky-Foucault debate. Chomsky wants to know how we got here and what we can do about it, and all he needs is reason, plenty of careful analysis. Foucault, steeped in the hermeneutics of suspicion, doesn't believe there is a neutral faculty of reason which could deploy a battery of neutral concepts like "justice" as if they were not compromised, even tainted, by history and capital. Chomsky's concern is that we don't understand where we are and how we got here, Foucault's that we don't understand what we are and what we're doing.

you know there will never be any new data, so that harmless prior metastasizes. — Srap Tasmaner

:up:

Priors supposedly representing no prejudice turning out to have major influence on results has been a thing in recent years.

Even the word "state" feels too coarse for Sleeping Beauty, since it could denote the situation a robust well-defined subject finds themselves in, or it could denote the very identity of that subject. --- At least, that's how the two main camps look to me. One wonders, where am I? how did I get here? One wonders, what am I? what has made me into this? — Srap Tasmaner

Also mega agree on this, the principle of indifference has a bunch of knowledge built into it. It assumes you know all the possible configurations something could be in, in the first place. And not just that, you know how all the variables relate within your stipulated model.

Two Envelopes seems to encourage abuse of the principle of indifference in exactly this way. Maybe it's just something like this: rationality requires treating "It's one or the other but I've literally no idea which" as an uninformative 50:50 prior only when there's the real possibility of acquiring new information upon which to update that prior. I'd rather just say, no, don't do that, "I don't know" doesn't mean "It's 50:50", [...] — Srap Tasmaner

I think there are good grounds for Sleeping Beauty upon awakening to ascribe strictly equal probabilities P = 1/3 to the three cases: (1) Today is Monday and the coin landed tails, (2) Today is Monday and the coin landed heads and (3) Today is Tuesday and the coin landed tails, rather than simply claiming ignorance. This doesn't involve any abuse of the principle of indifference so far as I can see. Rather, Sleeping Beauty can deduce those three outcomes to have equal probabilities from the statement of the problem, or experimental setup. Suppose upon awakening she would always bet on the same one of the three outcomes chosen in advance. In that case, upon repeating the whole experiment run many times, she's be right roughly 1/3 of the time she awakens and make this bet. This would be the case regardless of which one of the three outcomes she chooses to always bet on. Halfers have an obvious rejoinder to this line of thinking but I just wanted to make this point for now.

The halver's position is based on the common-sense postulate that the physical properties and propensities of objects remain unchanged when their states are conditioned upon irrelevant information. Given acceptance of that postulate plus the premise of a fair coin, why shouldn't the halver insist that they are talking about reality, as opposed to their mere mental state?

The thirder's position is indeed a ratio of possible words, but there is scant evidence to support the idea that credences are accurately represented by taking ratios over possible worlds. — sime

Taking ratios over possible worlds is a vivid way to illustrate the issue at hand when it comes to Bayesian updating of credences in light of new information. Regarding credences, in well-defined problems like Sleeping Beauty, as Pradeep Mutalik suggested in this column (in the comments section), "if you have 'a degree of certainty' of 1/n then you should be willing to accept a bet that offers you n or more dollars for every dollar you bet."

I would challenge the idea that Sleeping Beauty's judgement about the coin flip is merely a judgement about the propensities of that coin. Rather, it's a judgement about her own epistemic relation to the result of the flip. Or you could argue it's a judgement about the propensity of the whole experimental setup to put her in relationship with specific coin flip results. This echoes Werner Heisenberg's comment: "What we observe is not nature in itself but nature exposed to our method of questioning." I believe this to be true generally, not just in the context of quantum theory.

Upon awakening, Sleeping Beauty does gain information about the coin flip result. She knows that (from her own perspective) "if today is Monday, then heads and tails are equally likely." She also learns that "if today is Tuesday, then the coin landed tails." Since she can't rule out "today" being Tuesday, her credence (and willingness to bet) on a tails outcome should be higher than 1/2. "Today" being an indexical, the knowledge expressed pertains to her own situated epistemic perspective (as well as her 'pragmatic' perspective, or affordances to make bets or predictions). But this perspective only is "subjective" in the sense that it is suitable for capturing probabilities, or frequencies, relevant to her personal involvement in the experiment.

But indifference with respect to the joint outcomes (Day, Flip) implies not only

(1) a "posterior" probability for heads that differs from it's prior probability in spite of not learning anything upon awakening and before interacting with the world upon waking up.

but also

(2) different credence assignments for marginals one is equally ignorant of.

So even if (1) is acceptable for Principle of Indifference advocates, how can they justify (2) that flagrantly disregards their beloved principle?

From the premises

P(Flip) = 1/2
P (Day = mon | Flip = heads) = 1
P (Flip = tails | Day = tue) = 1

If a PoI advocate assigns the credence

P(Day = mon, Flip = tails ) = P(Day = mon , Flip = heads) = P(Day = tue, Flip = tails) = 1/3

Then his unconditioned credence that it is Monday is

P(Day = mon) = 2/3

which is clearly not the expression of indifference.

Then for sake of consistency he must ignore the actual information he is given about the coin, by using

P(Flip = tails) = 2/3

Not only does this credence assignment ignore the physical probability of heads that is known to him in advance, but it also indicates different credence assignments for outcomes of Flip when assuming ignorance of Flip! - in flagrant contradiction to PoI.

In general, when a PoI advocates uses PoI for specifying an unknown distribution, they don't possess the epistemic resources to determine the credences that they are implying regarding the values of related variables. And when their implied credences are pointed out to them, they are likely to withdraw their initial credence estimates.

IMO, there isn't a good reason for using the principle of indifference, not even in the case of describing credences. For handling ignorance, it is always better to assign probability intervals than to assign precise probabilities.

If a PoI advocate assigns the credence

P(Day = mon, Flip = tails ) = P(Day = mon , Flip = heads) = P(Day = tue, Flip = tails) = 1/3

Then his unconditioned credence that it is Monday is

P(Day = mon) = 2/3

which is clearly not the expression of indifference. — sime

This assignment is an expression of pairwise indifference between the three possible awakening circumstances. But rather than relying on the Principle of Indiference, I proposed to assign credences on the basis the odds Sleeping Beauty should rationally be willing to accept when betting on those outcomes. The problem's original specification is already precise enough ensures that if Sleeping Beauty accepts 2 to 1 odds on bets on any of these three outcomes upon awakening, she would break even in the long run.

Then for sake of consistency he must ignore the actual information he is given about the coin, by using

P(Flip = tails) = 2/3

There's no inconsistency here. It's precisely because the premise holds that the coin is equally likely to land heads or tails when tossed, and the fact that tails would result in twice as many awakenings, that Sleeping Beauty deduces that she could accept 2 to 1 odds on each awakening on the outcome 'Flip = heads' and break even over the long run.

There's no inconsistency here. It's precisely because the premise holds that the coin is equally likely to land heads or tails when tossed, and the fact that tails would result in twice as many awakenings, that Sleeping Beauty deduces that she could accept 2 to 1 odds on each awakening and break even in the long run. — Pierre-Normand

Let P(Flip | Today ) represent the conditional probability of the state of coin, relative to the value of SB's indexical "Today". Then

2/3 = P(Flip = Tails | Today = Mon Or Tue) = P(Flip = Tails) = 1/2

What is wrong with this conclusion of inconsistency?

What is wrong with this conclusion of inconsistency? — sime

I'm going to respond tomorrow since I'm now nearing the end of my current period of awakening. Meanwhile, I want to post this answer GPT-4 gave—on behalf of the halfer this time—to a challenge I had imagined. GPT-4 still has the ability to knock my socks off!

PN:

Yesterday, as I was thinking about the halfer's stance that upon awakening Sleeping Beauty's credence on the coin having landed heads should be 1/2, while acknowledging that when the coin landed tails she has two opportunities to guess "tails", thereby acknowledging that a thirder SB would guess right two thirds of the time when betting on tails (and thereby be more willing to bet on tails in order to win more bets), I thought of the following challenge. We suppose that on each awakening a second fair coin also is tossed. The second coin toss is independent and, unlike the first, doesn't impact how the experiment is run. If Sleeping Beauty is more willing to bet on the first coin having landed tails than the second one, does that not imply that the second fair coin now has a less than 1/2 chance to have landed tails?

GPT-4:

Your challenge indeed illustrates a fascinating aspect of this problem. Here is a possible way to respond:

As a halfer, Sleeping Beauty would be more willing to bet on the first coin landing tails, not because she believes it's more likely to land tails, but because there are two opportunities to be correct when betting on tails as opposed to one opportunity when betting on heads. However, this should not influence her betting behavior regarding the second coin, which is independent of the first and does not affect the experiment's procedure.

The crucial point here is that the additional awakenings under the tails outcome for the first coin change the betting odds in favor of tails, but they do not change the underlying physical probabilities of the coin flip, which remain at 1/2 for both heads and tails.

For the second coin, which is not linked to the experiment's procedure and leads to only one awakening regardless of its outcome, there are no such changes in the betting odds, and the physical probabilities remain the only relevant factor for Sleeping Beauty's bets. Therefore, Sleeping Beauty would remain indifferent between betting on heads or tails for the second coin, considering both outcomes equally likely.

In this scenario, the credence Sleeping Beauty assigns to the first coin's outcome upon awakening (1/2 for heads and tails according to the halfer perspective) reflects the physical probabilities of the coin flip, while her willingness to bet on tails rather than heads reflects the altered betting odds due to the additional awakenings under the tails outcome. Conversely, both her credence and her willingness to bet for the second coin directly reflect the physical probabilities, as there are no additional factors affecting the betting odds.

No, the thirder answer is not based on "who am I?" That appears to be just an excuse to reject the logic. In fact, in that post you linked, that question is neither asked nor answered. But there are equivalent ways to get the right answer, with only one SB.

Before I describe one, I need to point out that what most people think is the Sleeping Beauty Problem is actually a more contrived modification of it, invented by Adam Elga to enable his thirder solution. And the controversy is entirely about the parts he added, not the original.

This is the problem as it first ever appeared publicly, in the paper "Self-locating belief and the Sleeping Beauty problem". The two modifications alter words that clearly indicate Elga was thinking about his solution, since they have no impact on this text:

Some researchers are going to put you to sleep. During the [time] that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking. [While you are awake], to what degree ought you believe that the outcome of the coin toss is Heads? — Adam Elga

Elga made the two wakings occur on Monday and Tuesday, and made them different by having Monday be a mandatory waking, and Tuesday's be optional. The first issue you must face, is to decide whether you think Elga's changes alter the correct response.

But there is another way to implement the original problem, as worded, where there is an easy answer. After putting SB to sleep, flip two coins. Call them C1 and C2. If ether is showing Tails, wake SB and ask her "to what degree do you believe that coin C1 is showing Heads?"

After she answers, put her back to sleep with amnesia and turn coin C2 over to show its other side. Then repeat the steps in the previous paragraph, starting with "If either...".

The issue with the SB problem, is whether to consider the two potential wakings as the same experiment, or different ones. This version resolves that. SB knows that when the researchers looked at the coins, there are four possible arrangements with probability 1/4 each: {HH, HT, TH, TT}. She also knows that, since she is awake, HH is eliminated. She can update her beliefs in the other three to 1/3 each.

It was not my intention to misrepresent your views.

Have a nice day.

there are four possible arrangements with probability 1/4 each: {HH, HT, TH, TT}. — JeffJo

:up:

The issue with the SB problem, is whether to consider the two potential wakings as the same experiment, or different ones. This version resolves that. SB knows that when the researchers looked at the coins, there are four possible arrangements with probability 1/4 each: {HH, HT, TH, TT}. She also knows that, since she is awake, HH is eliminated. She can update her beliefs in the other three to 1/3 each. — JeffJo

I love your variation! However, your conclusion may be a bit premature as halfers can counter it using the same argument with which they counter the thirder position in Elga's setup. Allow me to break down the possible scenarios considering the initial flip results, the rearrangement of the second coin, and the resultant awakening sequence:

TT -> TH -> two awakenings (ww)
TH -> TT -> two awakenings (ww)
HT -> HH -> waking then sleeping (ws)
HH -> HT -> sleeping then waking (sw)

Given these four possible experimental runs following the four possible initial coin flip results, we find that when Sleeping Beauty awakens, she can certainly rule out HH as the current state of the two coins during that specific awakening episode. However, this does not eliminate the possibility of being in either of the last two experimental runs (in addition to, of course, either of the first two).

She could be waking up due to the initial coin flip (which is consistent with run-1, run-2, or run-3), or she could be waking up as a result of the rearrangement of the second coin (run-1, run-2, or run-4). As all these runs are still consistent with her experience, halfers might argue that Sleeping Beauty has gained no new information upon awakening and that these four possible runs remain equiprobable.

Let P(Flip | Today ) represent the conditional probability of the state of coin, relative to the value of SB's indexical "Today". Then

2/3 = P(Flip = Tails | Today = Mon Or Tue) = P(Flip = Tails) = 1/2

What is wrong with this conclusion of inconsistency? — sime

From Sleeping Beauty's perspective, the conditional probability of the coin flip being tails upon her awakening isn't conditioned on 'today' being either Monday or Tuesday, but rather on her actual state of being awakened 'today'. In other words, she conditions this probability on her being in a narrow (day long rather than experimental-run long) centered possible world in which she is awakened. It's this conditioning that yields her updating her prior 1/2 to the posterior 2/3.

So thanks for the enlightening discussion. — Michael

Thanks to you! I'll likely revisit some of your most recent objection just for the sake of clarifying my position (and refine it if needs be). Of course, I'd be delighted if you'd chime in again whenever you feel like it.

Even the word "state" feels too coarse for Sleeping Beauty, since it could denote the situation a robust well-defined subject finds themselves in, or it could denote the very identity of that subject. --- At least, that's how the two main camps look to me. One wonders, where am I? how did I get here? One wonders, what am I? what has made me into this?

As you say, it's all about individuation. Lacking a fixed point of individuation, you can push the lever however you like but you won't actually move anything. — Srap Tasmaner

I agree. I view the states at issue to be irreducibly epistemic states of agents. They are, in other words, relational states defined pragmatically. So, when expressing their credences, thirders and halfers had better properly specify the intended relata of this relation. If credences are explicitly stated as ratios, then one must likewise attend to the way in which the two things being divided are individuated. I'm working on a variation of the Sleeping Beauty problem in which all the options for individuating the relevant relata are laid out so that it becomes clear when halfers and thirders are talking past each other, and when it is that they are led astray through being inconsistent with their individuating assumptions.

Thanks to you! I'll likely revisit some of your most recent objection just for the sake of clarifying my position (and refine it if needs be). Of course, I'd be delighted if you'd chime in again whenever you feel like it. — Pierre-Normand

@Srap Tasmaner @Michael @sime

I also want to thank you all for the same reason. Thank you bandwagon.

Given these four possible experimental runs following the four possible initial coin flip results, we find that when Sleeping Beauty awakens, she can certainly rule out HH as the current state of the two coins during that specific awakening episode. However, this does not eliminate the possibility of being in either of the last two experimental runs (in addition to, of course, either of the first two). — Pierre-Normand

The points of the variation are:

• There is an "internal" probability experiment that begins when the researchers look at the two coins, and ends when either they see HH or they put SB back to sleep after seeing anything else.
• The effect of the amnesia drug not just that no information is transmitted between "this" SB and a possible version of herself in another internal experiment. It is that there are two such internal experiments in the full experiment, and each is a well-defined, self-contained experiment unto itself.
• The question in the internal probability experiment is about the state of the coins during it, not whether it is the first or second possible internal experiment. So the issue of "indexical 'today'" is completely irrelevant.

But as you point out, halfers are very good at making up reasons for why the answer they intuitively believe must be correct, could be. The "indexical" argument is one. Your example, where they might try to argue that SB must consider the version of herself in the other internal experiment is another - I've seen it used in the classical version.

So, if you need, make one slight change to mine. Always wake SB. If the coins are showing HH, take her on a shopping spree at Saks Fifth Avenue. If not, ask the question. This gets around your suggestion, since there are always two wakings. It is being in a "question" waking that allows an update: Pr(HH|Q)=0 and Pr(HT|Q)=Pr(TH|Q)=Pr(TT|Q)=1/3.

Now ask yourself whether it matters, to the reasons why these updates are possible, if the researchers had simply said "something other than a waking question will happen." The update is valid because SB has observed that the consequence of HH did not happen, regardless of what that consequence could be.

+++++

When I tell you that the card I drew is a Spade, 39 of 52 outcomes are "eliminated" as possibilities. So you can update Pr(Ace of Spades)=1/52 to Pr(Ace of Spades|Spade)=1/13. But "Diamonds" are still a part of the sample space, and their (prior) probabilities are still used for that calculation.

The difficulty with the classic version of the SB Problem arises when we try to physically remove one possibility from the sample space, not just eliminate it from the current instance of the process. That can't be done when the two "internal" parts of the whole problem are different, but SB has to view them as the same. That is the motivation for my variation, to make the internal parts identical in the knowledge basis used o answer the question.

Being left asleep does not remove Tuesday+Heads from whatever sample space you think is appropriate. It is still a possibility, and being awake constitutes an observation that it did not happen.

Here's another criticism of the thirder position: Their reasoning implies that self-induced amnesia is a valid strategy for controlling outcomes:

Suppose that SB gets paid $1 if the coin lands tails, otherwise she must pay $1. Furthermore, suppose that before the experiment begins she is given the choice as to whether or not she will have amnesia during the course of the experiment. According to thirder reasoning, she should choose to have amnesia in order to raise the probability of tails to 2/3

But honestly, all this talk of successes is irrelevant anyway. As I said before, these are two different things:

1. Sleeping Beauty's credence that the coin tossed on Sunday for the current, one-off, experiment landed heads
2. Sleeping Beauty's most profitable strategy for guessing if being asked to guess on heads or tails over multiple games

It's simply a non sequitur to argue that if "a guess of 'tails' wins 2/3 times" is the answer to the second then "1/3" is the answer to the first. — Michael

I took some time to read through the many good arguments on this thread, I agree with the above 2 different things you mentioned above.

The crux of the matter as I see it is whether the question about what the coin "shows" (taking the Scientific American article) is relevant to 1 or 2.

And the way I interpreted what the coin "shows" is that at the moment SB opens her eyes photon leave the coin and hit her retina. What is the probability that those photons contain information for heads? This related to 2 and is 2/3

While as far as I can tell you interpreted "shows" to mean what the coin landed on when it was flipped. This is related to 1 and is 1/2, absolutely.

And I put forward that this kind of difference in interpretation is what is happening in the two different position in the SB problem.

In your example using the thirder position,

If she has amnesia she should guess heads and will will 2/3 of the time.

If she doesn't have amnesia she should guess either on the first wake up (1/2 probability so doesn't matter which she guesses) and she should guess tail with absolute certainty if she remembers having woken up before (ie on her second wake up). Again she will win 2/3 of the time.

So having amnesia or not does not change the probability that she will win, but the tactics she should use are different.

If she has amnesia she should guess heads and will will 2/3 of the time.

If she doesn't have amnesia she should guess either on the first wake up (1/2 probability so doesn't matter which she guesses) and she should guess tail with absolute certainty if she remembers having woken up before (ie on her second wake up). Again she will win 2/3 of the time.

So having amnesia or not does not change the probability that she will win, but the tactics she should use are different. — PhilosophyRunner

In my argument, SB isn't asked to guess anything during the experiment. To make things really simple, let us only suppose that SB will lose $1 if the coin lands heads, without a wager being involved.

According to thirders, if she has amnesia then

P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday) = 1/3

So, according to thirders her probability of losing $1 when awakened on monday in a state of amnesia is 1/3.

On the other hand, if she doesn't have amnesia when waking up on Monday then thirders will agree that her probability of losing $1 is 1/2.

So according to thirder logic, it is rational for SB to consent to having amnesia before the experiment begins, in order to bias the coin's outcome towards tails.

Alternatively, she loses $1 each time she wakes. What is her credence that she will lose $2?

It’s not the case that she loses $2 2/3 of the time, although it is the case that 2/3 of the time she wakes up she’s in an experiment where she’s going to lose $2.

I think thirders conflate the two. The latter is obvious but not what is asked about when asked her credence that the coin landed tails.

According to thirders, if she has amnesia then

P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday) = 1/3 — sime

More precisely, the thirder view could be expressed as the claim that whenever Sleeping Beauty awakens, from her epistemic perspective, P(today is Tuesday and the coin landed tails) = P(today is Monday and the coin landed tails) = P(today is Monday and the coin landed heads) = 1/3

So, according to thirders her probability of losing $1 when awakened on monday in a state of amnesia is 1/3.

1/3 would be her prior upon awakening and before being informed that the day is Monday. Upon being informed of this, Sleeping Beauty would update her credence to 1/2.

1/3 would be her prior upon awakening and before being informed that the day is Monday. Upon being informed of this, Sleeping Beauty would update her credence to 1/2. — Pierre-Normand

That is true according to thirder's logic, but it doesn't alter their fallacious implication that consenting to amnesia before the experiment improves SB's chances of getting tails. If SB accepts that amnesia isn't a relevant causal factor of the coin's outcome, then she must avoid using a self-location credence that has that implication.

Similar magical thinking is demonstrated in avoidance coping. A poor person who has financial anxiety might raise their credence that their bank balance is in the black by choosing not to read their financial statements. Having accurate credences can be painful and reduce one's ability to cope in the short term, so there are short-term psychological benefits in choosing ignorance.

This is why I refuse to use betting arguments.

Suppose that SB gets paid $1 if the coin lands tails, otherwise she must pay $1. Furthermore, suppose that before the experiment begins she is given the choice as to whether or not she will have amnesia during the course of the experiment. According to thirder reasoning, she should choose to have amnesia in order to raise the probability of tails to 2/3 — sime

Is/does she paid/pay this $1 on both days, or on Wednesday after the experiment is over? In the latter case, can she choose not to have amnesia, and then choose "Heads" if she recalls no other waling but change that to Tails if she does?

This is why I tried to use a different setup, where this illogical juxtaposition of two different states is not an issue.

+++++

Probability is not a property of the system, it is a property of what is known about the system. Say I draw a card. After I look at it, I tell Andy that it is a black card, Betty that it is a spade, Cindy that its value is less than 10, and David that it is a seven (all separately). I ask each what they think the probability is that it is the Seven of Spades. Andy says 1/26, Betty says 1/13, Cindy says 1/32, and David says 1/4. All are right, but that does not affect my draw. I had a 1/52 chance to draw it.

The thirder argument is that SB knows of only three possible states. To her knowledge, there is no functional difference between them, so each has the same probability. Since they must sum to 1, each is 1/3. The fault they find with the halfer argument, is that it implies that to her, Monday&Tails and Tuesday&Tails represent the same state of possible knowledge. And they don't. They represent the same future, as determined on Sunday, but not the same state on a waking day.

The latter is obvious but not what is asked about when asked her credence that the coin landed tails. — Michael

This right here is the key disagreement.

Taking the wording from the Scientific American article, it says "What is the probability that the coin shows heads?"

It does not say "what is the probability that the coin landed heads."

The linguistic difference between the coin showing heads and landing on heads when flipped is the disagreement.

Thirders take the position that "shows" is referring to the event of light reflecting from the coin and then entering SB retina, which is dependent on when and how many times she opens her eyes. The showing happens multiple times in the experiment if the coin lands on heads, it is not a one off event.

Halfers take the position "shows" is only referring to how the coin landed when it was flipped. That is independent to when and how many times SB opens her eyes. The coin lands only once per experiment - it is a one off event as you say.

The disagreement is linguistic rather than statistical.

I'm not sure what your point was, but you can't make a valid one by leaving out half of the quote.

when the researchers looked at the coins, there are four possible arrangements with probability 1/4 each: {HH, HT, TH, TT}. — JeffJo

The state of the system at that time consisted of four equally-likely possibilities. SB knows, from the experiment set up, about these four equally-likely possibilities. But she also knows that the current state of the system does not include HH. It matters not what the state would be (asleep, gone shopping, whatever) if it did include HH, because she knows that it does not.

Except for the unusual setup, this is a classic example of an introductory-level conditional probability problem. Within her knowledge, the conditional probability of HT is now 1/3.

The fallacy in the halfer argument is that they confuse the changing state of the system, with what SB was aware could be possible states when the experiment started on Sunday. They do this because they can't describe the current state in terms of SB's knowledge when she is awake.

The point of my variation is to make the current state of the system functionally the same anytime she is asked the question.

You are missing out that she will also wake up on the Tuesday if the coin landed on heads - you are only looking at Monday.

When you add in the probabilities of the Tuesday as well, then you find that amnesia is no better or worse as per my previous post.

Is/does she paid/pay this $1 on both days, or on Wednesday after the experiment is over? In the latter case, can she choose not to have amnesia, and then choose "Heads" if she recalls no other waling but change that to Tails if she does? — JeffJo

I simplified my thought experiment to say that she loses $1 on Monday if heads comes up, otherwise she loses nothing. No stake is involved, and she isn't asked to bet on the outcome during the course of the experiment.

It boils down to the following question. Should she risk a single draw from

P(Flip = Heads | Today = Monday) = 1/2 (no amnesia)

Or should she risk a single draw from

P(Flip = Heads | Today = Monday Or Tuesday) = x (amnesia)

According to halfers, x = 1/2, implying that she gains nothing by choosing amnesia.

According to thirders, x = 1/3, indicating that she gains from choosing amnesia.

Say I draw a card. After I look at it, I tell Andy that it is a black card, Betty that it is a spade, Cindy that its value is less than 10, and David that it is a seven (all separately). I ask each what they think the probability is that it is the Seven of Spades. Andy says 1/26, Betty says 1/13, Cindy says 1/32, and David says 1/4. All are right, but that does not affect my draw. I had a 1/52 chance to draw it. — JeffJo

If you are referring to a context involving repeated trials, then all of your probabilities are physically meaningful posterior probabilities , for we are confident on the basis of our past experience with card games that the frequencies will roughly obtain. In which case none of your above probabilities are appealing to the controversial principle of indifference.

In stark contrast, Elga (for example) invokes the principle of indifference on the basis of ignorance to assert

P(Monday | Tails) = P(Tuesday | Tails)

Leading to him to a conclusion the conflicts with his knowledge of fair coins and that encourages avoidance coping. What he ought to have done, is to represent his ignorance with the maximal set of permissible distributions and to assign confidence bounds, until as and when he has reason to whittle the set down to a smaller subset and make more specific predictions.

Or if he really must assign a single distribution (i can't think of a good reason), to choose one whose deductive implications cohere with his broader state of knowledge.