Sue's reasoning is right for Sue, but wrong for Jane (and vice versa), given that 2/3 of sitters will sit in on a 100 Heads interview but 1/2^100 of participants will have a 100 Heads interview. — Michael

Let's consider another scenario. Suppose Jane is an amnesiac paratrooper set to be dropped behind enemy lines for three separate missions on different nights. Conveniently, she suffers from selective anterograde amnesia which shields mission information in case she's captured. There are two potential drop zones: #1 is dry land, and #2 is a swamp.

Mission Control will toss a coin to decide Jane's destinations. If it lands heads, she will be dropped once over DZ#1 and twice over DZ#2. If it lands tails, she'll be dropped twice over DZ#1 and once over DZ#2. Jane carries a tracking device which will enable Sergeant Sue, who's already operating behind enemy lines, to locate her during one randomly selected mission, collect the mission package, and direct her back home.

Now, what's the probability that the coin landed heads? Before the experiment begins, both Sue and Jane have the same prior credence P(H) = 1/2.

However, suppose Sue locates Jane in DZ#1. What is the updated probability that the coin landed heads? It would be 1/3. Shouldn't Jane update her credence to the same value when she realizes she's landed on dry land?

Suppose the outcome of the coin toss also determines the nature of the package that Jane is carrying. If the coin landed on heads, Jane's package contains a secret explosive; if it landed tails, the package contains a secret medication. Both Jane and Sue know this arrangement beforehand.

When Sue tracks Jane and finds her in DZ#1, they both realize that Sue should deliver the package to a hospital if the coin landed tails (since the package would contain medication), or to a military outpost if the coin landed heads (since it would contain an explosive).

Now, should they both act as though the probability of the coin having landed heads is 1/3? It's not just about winning or losing a bet anymore. The decisions they make based on their credence have immediate and tangible consequences. They must update their credence according to the information they have at hand, including Jane's current location.

So if heads then woken once on Monday and twice on Tuesday, otherwise woken twice on Monday and once on Tuesday.

Sue tells Jane that it's Monday.

What is Jane's credence that the coin landed heads?

I say 1/2.

It's exactly the same reasoning as before.

Sue should reason as if she is randomly selected from the set of all sitters, and 1/3 of sitters sitting in a Monday room are sitting in a heads room.

Jane should reason as it she is randomly selected from the set of all participants, and 1/2 of participants in a Monday room are sitting in a heads room.

This reasoning is much clearer to see in the 100 Heads example, and I don't see how any counterexample example is going to change my mind about 100 Heads. If I'm one of the participants I will only ever reason that P(100 Heads) = $1\over2^{100}$. I am almost certainly not the one participant who will have 2¹⁰¹ interviews.

I should no more update my credence to match my sitter’s than she should update hers to match mine.

So if heads then woken once on Monday and twice on Tuesday, otherwise woken twice on Monday and once on Tuesday.

Sue tells Jane that it's Monday.

What is Jane's credence that the coin landed heads?

I say 1/2.

It's exactly the same reasoning as before.

Sue should reason as if she is randomly selected from the set of all sitters, and 1/3 of sitters sitting in a Monday room are sitting in a heads room.

Jane should reason as it she is randomly selected from the set of all participants, and 1/2 of participants in a Monday room are sitting in a heads room. — Michael

Although you linked my most recent post, I assume you intended to respond to this one.

In my scenario, there was only one room for awakenings. Without loss of generality, we can schedule the occupancy of that room over the three days of the experiment, according to the coin toss result, as follows:

If the coin lands Heads:

Day 1: Jane (first),
Day 2: Helen (first),
Day 3: Helen

If the coin lands Tails:

Day 1: Jane (first),
Day 2: Jane,
Day 3: Helen (first)

I use the label 'first' to signify a participant's first (or only) day of awakening. This is functionally equivalent to Monday in the original problem.

On any given day, whichever candidate doesn't occupy the waking room is moved into the sleeping room.

When Sue is assigned to attend to the participant who is awakened on a randomly chosen day (without being informed if it's Day 1, Day 2 or Day3), her credence in the coin having landed heads is 1/2. After she is informed that the occupant is Jane, she updates her credence P(H) to 1/3. After she is further informed that this is Jane's first or only occasion of awakening, her credence gets updated back to 1/2 again.

I would argue that Jane should update her credence in the same way in light of the same information. But first, I wanted to ensure that we are on the same page regarding both the setup and Sue's credences within it.

(I'll address your extreme example separately.)

Although you linked to my most recent post, I assume you intended to respond to this one. — Pierre-Normand

No, I was just trying to rephrase your secret mission example into a way that I could understand better. Did I misinterpret it?

If not then it appears to be saying the same thing as the above?

No, I was just trying to rephrase your secret mission example into a way that I could understand better. Did I misinterpret it? — Michael

In the secret mission example, Sergeant Sue is tasked with locating Jane on one of her three scheduled missions selected at random. Neither Sue nor Jane know if this mission is the first, second or third one or, if it's not the first, in what zone(s) Jane landed before.

Was my rephrasing of it wrong? I'm treating DZ#1 as Monday and DZ#2 as Tuesday. If twice at DZ#1 then twice on Monday, if once at DZ#2 then once on Tuesday. If you know that it's DZ#1 then you know that it's Monday.

I would argue that Jane should update her credence in the same way in light of the same information. — Pierre-Normand

Jane should reason as if she is randomly selected from the set of all participants, because she is.
Sue should reason as if she is randomly selected from the set of all sitters, because she is.

Jane shouldn't update her credence to match Sue and Sue shouldn't update her credence to match Jane.

I think my extreme example shows why. It's just not rational for each participant to reason that the coin most likely landed heads 100 times but it is rational for each sitter to reason that they are sitting in on a 100 heads experiment.

Was my rephrasing of it wrong? I'm treating DZ#1 as Monday and DZ#2 as Tuesday. If twice at DZ#1 then twice on Monday, if once at DZ#2 then once on Tuesday. If you know what it's DZ#1 then you know that it's Monday. — Michael

Structurally, DZ#1 is similar to awake and DZ#2 is similar to asleep. In this way, the Jane&Helen scenario perfectly maps on the secret mission scenario, with DZ#2 and Helen fulfilling the roles of dummy place holders, as it were.

Jane shouldn't update her credence to match Sue and Sue shouldn't update her credence to match Jane. — Michael

But we are agreed on the validity of Sue's credences in both scenarios, right?

But we are agreed on the validity of Sue's credences in both scenarios, right? — Pierre-Normand

Yes, I said as much with my extreme example.

Given that $2\over3$ of sitters will sit in on a 100 Heads interview it is rational for each sitter to reason that they are likely sitting in on a 100 Heads interview.

But given that $1\over2^{100}$ of participants will have a 100 Heads interview it is rational for each participant to reason that their interview is likely not a 100 Heads interview.

Neither the sitter nor the participant should update their credence to match the other's. They each belong to a different reference class.

And if they reason this way then 2¹⁰¹ sitters will be right (once) and 2¹⁰⁰ - 1 will be wrong (once), and 2¹⁰⁰ - 1 participants will be right (once) and 1 will be wrong (2¹⁰¹ times).

I'd say that's all the evidence I need to justify my credence.

Neither the sitter nor the participant should update their credence to match the other's. — Michael

In that case, in the secret mission scenario, it looks like Sergeant Sue and G. I. Jane are poised to having a fight—or at least a heated argument—regarding where the secret package should be delivered. Sergeant Sue would advocate for its delivery to the allied field hospital, relying on her 1/3 credence, while G.I Jane, holding a 1/2 credence, would maintain that the destination of the package doesn't make a difference. (The parameters of the scenario could be tweaked to make the dispute more pronounced, through yielding 1/6 vs 6/11 credences, say).

Alternatively, Sue and Jane could come to realize that their apparently divergent credences target different classes of events (which is what I have argued) and conclude that both of them are true, but that it's the 1/3 credence that is relevant for determining the likely contents of the package they are presently dealing with.

Here is another discussion with GPT-4 that rehearses my pragmatist conception of the dispute between halfers and thirders.

PN:

Let's explore a variation of the Sleeping Beauty problem. In this version, a six-sided die is tossed to determine the course of the experiment. Should the die result in 'six', Jane will be awakened six times consecutively. Otherwise, she will experience only a single awakening. This sequence will unfold over six days, with a single run of the experiment.

In the case where the die does not land on six, Jane's lone awakening will occur on the sixth day of the experiment, and she will be released on the seventh day. The die outcome also determines the location of Jane's stay: 'six' results in her lodging in the East Wing, while any other outcome places her in the West Wing.

Aunt Betsy, who plans to pick Jane up upon her release, is aware that Jane checked into the facility on June 30th and will be released on July 7th. On July 1st, after the die has been rolled but before any awakenings, Betsy calls the facility to learn where she should pick up Jane. The manager informs her that this information cannot be disclosed directly but can be communicated to Jane, who is allowed to send a response.

To ensure that Jane's knowledge doesn't impact her belief in the probability of rolling a 'six', the manager forwards Betsy's inquiry to Jane during each awakening, irrespective of the die result. Jane's responses are collected, and only one is sent to Aunt Betsy on the evening of July 6th, during Jane's final sleep.

Upon receiving Betsy's inquiry, Jane estimates her belief in the probability of the die landing on 'six', which she deduces to be 6/11. She reasons that Betsy should therefore wait for her at the East Wing, where she believes there is a 6/11 chance she is being housed.

By contrast, from Betsy's perspective, there should be a 5/6 probability that the die didn't land on 'six'. This perspective, grounded in observations from multiple hypothetical experiment runs, indicates Jane is likely housed in the West Wing with a 5/6 probability. Despite stemming from the same fundamental setup, their credences appear to conflict: Jane and Betsy can't both be correct about where Jane is most likely to be released from. What went wrong?

GPT-4:

This is an interesting variation of the Sleeping Beauty Problem, and the apparent conflict arises from a fundamental difference in the knowledge and perspectives of Aunt Betsy and Jane.

From Jane's perspective, each awakening she experiences is a random draw from all her potential awakenings. If the die landed on 'six', she will be awakened six times, and if the die did not land on 'six', she will be awakened once. So when she wakes up and contemplates her situation, she reasons that there are six times more ways she could have found herself awake in the East Wing compared to the West Wing, and thus, she assigns a 6/11 credence to being in the East Wing.

On the other hand, Aunt Betsy's perspective is fundamentally different. She knows that the experiment will last for seven days and that Jane will only be released on the seventh day. She does not know whether Jane was awakened multiple times or just once, and she is not experiencing the awakenings as Jane is. Her 5/6 credence that Jane is in the West Wing is based on the probability of the die roll, independent of Jane's awakening experiences.

So, the difference in their credences comes from the different ways they experience or perceive the experiment. Jane's credence is based on her waking experiences, while Aunt Betsy's is based on the initial conditions of the experiment. While this might appear to be a conflict, it's simply a result of their different perspectives on the same situation. This situation also underscores the subjective nature of Bayesian probabilities, which are personal degrees of belief and may vary between individuals even when considering the same event, based on their differing perspectives or knowledge about the situation.

PN:

I very much agree with everything that you said, but the solution of the puzzle involves the reconciliation of Jane's credence with the fact that her aunt will indeed pick her up at the wrong wing five times out of six if she would rely on Jane's message (and hence why Jane should rethink it).

GPT-4:

This is a great thought experiment. It reminds us that although probabilities might be personally reasonable, they can't always be used interchangeably between different people or contexts, especially when the information set or the perspective changes.

The solution to this puzzle lies in realizing that Jane, when she sends the message, is aggregating over multiple awakenings. The 6/11 credence that Jane assigns to being in the East Wing is not tied to any specific day of the experiment but is an average over all her possible awakenings. On each awakening, she updates her belief based on the premise that she's equally likely to be awake on any given day.

However, Aunt Betsy doesn't care about Jane's experience across all her awakenings. She cares about Jane's location on a specific day, the seventh day. Jane needs to acknowledge this and should instead reason as follows:

If the die landed on 'six', Jane will have six awakenings in the East Wing.
If the die didn't land on 'six', Jane will have one awakening in the West Wing.
However, from Jane's perspective, each of her awakenings is equally likely. Thus, if Jane knew it was her final awakening, her credence in being in the East Wing would only be 6/7, not 6/11. If it is not her final awakening, Jane can deduce that she must be in the East Wing. Therefore, Jane should send two types of messages: one indicating East Wing if it is not her final awakening, and another indicating a 6/7 chance of being in the East Wing if it is her final awakening.

From Aunt Betsy's perspective, she would need to know if it's Jane's last awakening, which she doesn't. But if the manager were to choose to send the message indicating the East Wing, if any, or the other message otherwise, Aunt Betsy's best strategy would be to follow the message's instructions and pick Jane up at the East Wing if the message indicates so, otherwise at the West Wing. This strategy would result in Aunt Betsy waiting at the correct wing six out of seven times. This may not match perfectly Jane's 6/11 credence but it's the best strategy that can be devised from their shared perspective.

PN:

Aunt Betsy doesn't know the experimental setup, so I equate her perspective with the observed result that (were she placed in this situation multiple times) she would find Jane coming out of the West Wing five out of six times. This is the same observation that Jane would make. (The seventh day doesn't count in the calculations since it's not a day of awakening and interview, it's just the day of discharge.)

Jane's mistake was to infer from the fact that her being located in the East Wing six out of eleven times when she awakens translates in her coming out of the facility six out of eleven times when she is discharged. It's as simple as that.

Six of her awakenings that occur in the East Wing result in one single discharge from that wing, which is why she should direct her aunt to the East Wing where 5 out of 6 of her discharges occur (following a single awakening), despite the fact that it is *now* (on the occasion of her present awakening) more likely than not that she is in the East Wing.

GPT-4:

Ah, I see! Thank you for the clarification. Yes, Jane seems to have confused her credence for being in the East Wing at the time of any awakening (6/11) with the probability of being discharged from the East Wing at the end of the experiment. As you mentioned, the error arises because six awakenings in the East Wing all lead to one single discharge from the East Wing, while one awakening in the West Wing leads to one discharge from the West Wing.

Therefore, the probability that Jane will be discharged from the East Wing is indeed 1/6, not 6/11. Conversely, the probability that she will be discharged from the West Wing is 5/6. So, Jane should inform Aunt Betsy to wait for her at the West Wing, since that is where she will most likely be discharged from. This aligns with the observed result that Aunt Betsy would find Jane coming out of the West Wing five out of six times if this scenario were to be repeated multiple times.

Thank you for presenting this interesting variation of the Sleeping Beauty problem. It indeed reinforces the point that probabilities can't be used interchangeably between different contexts and reminds us to always consider the exact event or outcome that the probability is referring to.

PN:

My variation of the Sleeping Beauty problem is designed to demonstrate that conflicting credences don't necessarily stem from incompatible epistemic viewpoints. Rather, they often arise from differing interpretations of what the credences actually represent. In the Sleeping Beauty problem, the confusion is compounded by the fact that different credences—1/2 and 1/3—are both assigned to the same 'state' (Heads) of an object (a fair coin). However, what truly distinguishes these credences, and explains their different values, is the uniquely individualted sets of events each considers. For a 'halfer', the credence is based on the ratio of Heads-experimental runs to total experimental runs. Conversely, for a 'thirder', it's about the ratio of Heads-awakenings to total awakenings. The apparent conflict is not a matter of differing epistemic perspectives, but rather differing definitions of the credences in question.

GPT-4:

I understand your point and agree with it. The distinction you've made is essential and helps clear up the confusion surrounding the Sleeping Beauty problem. Many debates about probability and credence essentially boil down to differences in the definitions of what constitutes an event or an experimental trial. In this context, it's clear that what one person considers as an 'event' might differ substantially from another person's definition. By drawing attention to these differences in definition, we can gain a clearer understanding of why different credences are assigned in the first place and clarify seemingly conflicting probabilities. It's a perspective that adds a significant depth to the discussion and makes the resolution of the paradox more nuanced and comprehensive.

PN:

Coming back to the new variation, which was intended to highlight a particular pragmatic context in which Jane can—and should—make use of a 'halfer' conception of her credence P('six') = 1/6, there are other pragmatic concerns that can make her rely on the 'thirder' conception. One such context would be for her to find an opportunity to escape from the facility and needing to prepare accordingly depending on her being hosted in the East or West wing. One of them might be surrounded by a moat filled with crocodiles and the other with a forest infested with lions. Since, in this case, the escape opportunities are distributed evenly across occasions of awakening, Jane should prepare her escape on the basis of a 'thirder' definition of her credence P('six') = 6/11.

GPT-4

Yes, that's an excellent extension of the scenario. If Jane finds herself in a situation where she must make a decision based on her current location during each awakening, then it is indeed more appropriate for her to use the 'thirder' credence. Given the uniform distribution of escape opportunities across awakenings, Jane's best strategy would be to prepare for the highest probability scenario in her current awakening, which is in the East Wing according to 'thirder' reasoning.

Your point here further underlines the importance of considering the context and its implications when deciding which credence to rely on. The 'halfer' and 'thirder' perspectives are not mutually exclusive or inherently right or wrong; instead, they offer different lenses through which to view the problem, each one more useful in different situations. It's the specifics of the situation and the nature of the decision to be made that determine which perspective should be used.