Ape in the coat

Karma: 785

Ape in the coat 12 May 2024 7:58 UTC
1 point
0
in reply to: JBlack’s comment on: Semantic Disagreement of Sleeping Beauty Problem
Probabilities are measures on a sigma-algebra of subsets of some set, obeying the usual mathematical axioms for measures together with the requirement that the measure for the whole set is 1.
Not just any set. A sample space. And one of its conditions is that its elements are mutually exclusive, so that one and only one happens in any iteration of probability experiment.
That’s why I need to define a new mathematical entity indexical sample space, for which I’m explicitly lifting this restriction to formally talk about thirdism.
Applying this structure to credence reasoning, the elements of the sample space correspond to relevant states of the universe, the elements of the sigma-algebra correspond to relevant propositions about those states
A minor point is that outcomes and events can both very well be about map not the territory. If elementary event {A} has P(A) = 0, then we can simply not include outcome A into the sample space for simplicity sake.
and the measure (usually called credence for this application) corresponds to a degree of rational belief in the associated propositions. This is a standard probability space structure.
There is a potential source of confusion in the “credence” category. Either you mean it as a synonym for probability, and then it follows all the properties of probability, including the fact that it can only measure formally defined events from the event space, which have stable truth value during an iteration of probability experiment. Or you mean “intuitive feeling about semantic statements which has some relation to betting”, which may or may not be formally definable as probability measure because the statement doesn’t have stable truth value.
People tend to implicitly assume that having a vague feeling about a semantic statements has to mean that there is a way to formally define a probability space where this statement is a coherent event, but it doesn’t actually has to be true. Sleeping Beauty problem is an example of such situation.
In the Sleeping Beauty problem, the participant is obviously uncertain about both what the coin flip was and which day it is. The questions about the coin flip and day are entangled by design, so a sample space that smears whole timelines into one element is inadequate to represent the structure of the uncertainty
I’m not sure what you mean by “smears whole timelines into one element”. We of course should use the appropriate granularity for our outcomes and events. The problem is that we may find ourselves in a situation where we intuitively feel that events has to be even more granular then they formally can.
For example, one of the relevant states of the universe may be “the Sleeping Beauty experiment is going on in which the coin flip was Heads and it is Monday morning and Sleeping Beauty is awake and has just been asked her credence for Heads and not answered yet”.
The fact that something is a semantic statement about the universe doesn’t necessary mean that it’s well-defined event in a probability space.
One of the measurable propositions (i.e. proposition for which Sleeping Beauty may have some rational credence) may be “it is Monday” which includes multiple states of the universe including the previous example.
No it can’t. Semantic statement “Today is Monday” is not a well-defined event in the Sleeping Beauty problem. People can have credence about it in the sense of “vague feeling”, but not in the sense of actual probability value.
You can easily observe yourself that there is no formal way to define “Today” in Sleeping Beauty if you actually engage with the mathematical formalism.
Consider No-Coin-Toss or Single-Awakening problems. If Monday means “Monday awakening happens during this iteration of probability experiment” and, likewise, for Tuesday, we can formally define Today as:
Today = Monday xor Tuesday
On every iteration of probability experiment either Monday or Tuesday awakenings happen. So we can say that the participant knows that “she is awakened Today”, meaning that she knows to be awakened either on Monday or on Tuesday.
P(Today) = P(Monday xor Tuesday) = 1
We can express credence in being awakened on Monday, conditionally on being awakened “Today” as:
P(Monday|Today) = P(Monday|Monday xor Tuesday) = P(Monday)
This is a clear case where Beauty’s uncertainty about which day it is can be expressed via probability theory. Statement “Today is Monday” has stable truth values throughout any iteration of probability experiment
Now consider No-Memory-Loss problem where Sleeping Beauty is completely aware which day it is.
Now statement “Today is Monday” doesn’t have a stable truth value throughout the whole experiment. It’s actually two different statement: Monday is Monday and Tuesday is Tuesday. The first one is always True, the second one is always False. So Beauty’s uncertainty about the question which day it is can’t be expressed via probability theory. Thankfully, she doesn’t have any uncertainty about the day of the week.
So we can do a trick. We can describe No-Memory-Loss problem as two different non-independent probability experiments in a sequential order. First one is Monday-No-Memory-Loss, where the Beauty is sure that it’s is Monday and uncertain about the coin. The second is Tuesday-Tails-No-Memory-Loss where the Beauty is sure that it’s Tuesday and the coin is Tails. The second happens only if the coin was Tails in the first.
In Monday-No-Memory-Loss, Today simply means Monday:
Today = Monday
And statement “Today is Monday” is a well defined event with trivial probability measure:
P(Monday|Today) = P(Monday|Monday) = 1
Similarly with Tuesday-Tails-No-Memory-Loss:
Today = Tuesday
P(Monday|Today) = P(Monday|Tuesday) = 0
And now when we consider regular Sleeping Beauty problem the issue should be clear. If we define Today = Monday xor Tuesday, the Beauty can’t be sure that this event happens, because on Tails both Monday and Tuesday are realized.
And we can’t take advantage of Beauty’s lack of uncertainty about the day as before, because now she has no idea what day it is. And so the statement “Today is Monday” is not a well-defined event of the probability space. It doesn’t have a coherent truth value during the experiment—it’s True xor ( True and False).
We can still talk about events “Monday/Tuesday awakening happens during this iteration of probability experiment”.
P(Monday) = 1
P(Tuesday) = ¹⁄₂
And we can use them in betting schemes. If the Beauty is proposed to bet on the statement “Today is Monday” she can calculate her optimal odds the standard way:
E(Monday) = P(Monday)U(Monday) - P(Tuesday)U(Tuesday)
Solving E(Monday) = 0:
U(Monday) = U(Tuesday)/2
So 1:2 odds.
And the last question is: What was then this intuitive feeling about the semantic statement “Today is Monday”? For which the answer is—it was about weighted probability that Monday happens in the experiment.

Ape in the coat 11 May 2024 7:51 UTC
1 point
−2
in reply to: JBlack’s comment on: Semantic Disagreement of Sleeping Beauty Problem
No, introducing the concept of “indexical sample space” does not capture the thirder position, nor language.
And what does it not capture in thirder position, in your opinion?
You do not need to introduce a new type of space, with new definitions and axioms. The notion of credence (as defined in the Sleeping Beauty problem) already uses standard mathematical probability space definitions and axioms.
So thirder think. But they are mistaken, as I show in the previous posts.
Thirder credence fits the No-Coin-Toss problem where Monday and Tuesday don’t happen during the same iteration of the experiment and on awakening the person indeed learns that “they are awaken Today”, which can be formally expressed as an event $M o n d a y x o r T u e s d a y$ .
Not so with Sleeping Beauty, where the participant completely aware that Monday awakening on Tails is followed by Tuesday awakening, therefore, event $M o n d a y x o r T u e s d a y$ doesn’t happen in 50% cases, so instead of learning that the Beauty is awakened today she can only learn that she is awakened at least once.
In Sleeping Beauty problem being awakened Today isn’t a thing you can express via probability space. It’s something that can happen twice in the same iteration of the experiment, just like getting a ball in the example from the post. And so we need a new mathematical model to formally talk about this sort of things, therefore weighted probability space.
I suppose you’ve read all my posts on the topic. What is the crux of our disagreement here?

Ape in the coat 10 May 2024 9:33 UTC
1 point
0
in reply to: Duschkopf’s comment on: Another Non-Anthropic Paradox: The Unsurprising Rareness of Rare Events
If this was true, then we could not observe the event „any other coin sequence“ as well since this event is by definition not being tracked.
When you are tracking event A you are automatically tracking its complement.
In fact, in order to detect a correspondence between a coin sequence that we have in mind and the actual sequence, our brain has to compare them to decide if there is a match. I can hardly imagine how this comparison could work without observing the specific actual sequence in the first place. That we classify and perceive a specific sequence as „any other sequence“ can be the result of the comparison, but is not its starting point.
Oh sure, you are of course completely correct here. But this doesn’t contradict what I’m saying.
The thing is, we observe a particular outcome and then we see which event(s) it corresponds to. Let’s take an example: a series of 3 coin tosses.
So, in the beginning you have sample space which consist of all the elementary outcomes:
${H H H, T T T, H H T, T T H, H T H, T H T, T H H, H T T}$
And an event space, some sigma-algebra of the sample space, which depends on your precommitments. Normally, it would look something like this:
${\emptyset, {H H H, T T T}, {H H T, T T H, H T H, T H T, T H H, H T T}, {H H H, T T T, H H T, T T H, H T H, T H T, T H H, H T T}}$
Because you are intuitively paying attention to whether there all Heads/Tails in a row. So your event space groups individual outcomes in this particular way, separating the event you are tracking and it’s complement.
When a particular combination, say $T H H$ is realized in a iteration of the experiment, your mind works like this:
- Outcome $T H H$ is realized
- Therefore every event from the event space which includes $T H H$ is realized.
- Events ${H H T, T T H, H T H, T H T, T H H, H T T}$ and ${H H H, T T T, H H T, T T H, H T H, T H T, T H H, H T T}$ are realized.
- $P (H H T o r T T H o r H T H o r T H T o r T H H o r H T T) = 2 / 3$
- This isn’t a rare event and so you are not particularly surprised
So, as you see, you do indeed observe an actual sequence, it’s just that observing this sequence isn’t necessary an event in itself.

Ape in the coat 10 May 2024 8:59 UTC
1 point
0
in reply to: Duschkopf’s comment on: Beauty and the Bets
Sure, I don‘t deny that. What I am saying is, that your probability model don‘t tell you which probability you have to base on a certain decision
It says which probability you have, based on what you’ve observed. If you observed that it’s Monday—you are supposed to use probability conditionally on the fact that it’s Monday, if you didn’t observe that it’s Monday you can’t lawfully use the probability conditionally on the fact that it’s Monday. Simple as that.
There is a possible confusion where people may think that they have observed “this specific thing happened” while actually they observed “any thing from some group of things happened”, which is the technicolor and rare event cases are about.
Suppose a simple experiment where the experimenter flips a fair coin and you have to guess if Tails or Heads, but you are only rewarded for the correct decision if the coin comes up Tails. Then, of course, you should still entertain unconditional probabilities P(Heads)=P(Tails)=1/2. But this uncertainty is completely irrelevant to your decision.
Here you are confusing probability and utility. The fact that P(Heads)=P(Tails)=1/2 is very much relevant to our decision making! The correct reasoning goes like this:
P(Heads) = ¹⁄₂
P(Tails) = ¹⁄₂
U(Heads) = 0
U(Tails) = X,
E(Tails) = P(Tails)U(Tails) - P(Heads)U(Heads) = 1/2X − 0
Solving E(Tails) = 0 for X:
X = 0
Which means that you shouldn’t bet on Heads at any odds
What is relevant, however, is P(Tails/Tails)=1 and P(Heads/Tails)=0, concluding you should follow the strategy always guessing Tails.
And why did you happen to decide that it’s P(Tails|Tails) = 1 and P(Heads|Tails) = 0 instead of
P(Heads|Heads) = 1 and P(Tails|Heads) = 0 which are “relevant” for you decision making?
You seem to just decide the “relevance” of probabilities post hoc, after you’ve already calculated the correct answer the proper way. I don’t think you can formalize this line of thinking, so that you had a way to systematically correctly solve decision theory problems, which you do not yet know the answer to. Otherwise, we wouldn’t need utilities as a concept.
Another way to arrive at this strategy is to calculate expected utilities setting U(Heads)=0 as you would propose. But this is not the only reasonable solution. It’s just a different route of reasoning to take into account the experimental condition that your decision counts only if the coin lands Tails.
This is not “another way”. This is the right way. It has the proper formalization and actually allows us to arrive to the correct answer even if we do not yet know it.
If the optimal betting sheme requires you to rely on P(Heads/Red or Blue)=1/2 when receiving evidence Blue, then the betting sheme demands you to ignore your total evidence.
You do not “ignore your total evidence”—you are never supposed to do that. It’s just that you didn’t actually receive the evidence in the first place. You can observe the fact that the room is blue in the experiment only if you put your mind in a state where you distinguish blue in particular. Until then your event space doesn’t even include “Blue” only “Blue or Red”.
But I suppose it’s better to go to the comment section Another Non-Anthropic Paradox for this particular crux.

Semantic Disagreement of Sleeping Beauty Problem

Ape in the coat8 May 2024 19:09 UTC

9 points

5 comments8 min readLW link

Ape in the coat 2 May 2024 14:56 UTC
6 points
6
on: An explanation of evil in an organized world
God is good*
*for a very specific definition of “goodness”, which doesn’t actually capture the intuition of most people about ethics and is mostly about iteraction of sub-atomic particles.

Ape in the coat 2 May 2024 12:01 UTC
1 point
0
in reply to: Duschkopf’s comment on: Beauty and the Bets
First of all, it‘s certainly important to distinguish between a probability model and a strategy. The job of a probability model is simply to suggest the probability of certain events and to describe how probabilities are affected by the realization of other events. A strategy on the other hand is to guide decision making to arrive at certain predefined goals.
Of course. As soon as we are talking about goals and strategies we are not talking about just probabilities anymore. We are also talking about utilities and expected utilities. However, probabilities do not suddenly change because of it. Probabilistic model is the same, there are simply additional considerations as well.
My point is, that the probabilities a model suggests you to have based on the currently available evidence do NOT neccessarily have to match the probabilities that are relevant to your strategy and decisions.
Whether or not your probability model leads to optimal descision making is the test allowing to falsify it. There are no separate “theoretical probabilities” and “decision making probabilities”. Only the ones that guide your behaviour can be correct. What’s the point of a theory that is not applicable to practice, anyway?
If your model claims that the probability based on your evidence is ¹⁄₃ but the optimal decision making happens when you act as if it’s ¹⁄₂, then your model is wrong and you switch to a model that claims that the probability is ¹⁄₂. That’s the whole reason why betting arguments are popular.
If Beauty is awake and doesn‘t know if it is the day her bet counts, it is in fact a rational strategy to behave and decide as if her bet counts today.
Questions of what “counts” or “matters” are not the realm of probability. However, the Beauty is free to adjust her utilities based on the specifics of the betting scheme.
All your model suggests are probabilities conditional on the realization of certain events.
The model says that
P(Heads|Red) = ¹⁄₃
P(Heads|Blue) = ¹⁄₃
but
P(Heads|Red or Blue) = ¹⁄₂
Which obviosly translates in a betting scheme: someone who bets on Tails only when the room is Red wins ²⁄₃ of times and someone who bets on Tails only when the room is Blue wins ²⁄₃ of times, while someone who always bet on Tails wins only ¹⁄₂ of time.
This leads to a conclusion that observing event “Red” instead of “Red or Blue” is possible only for someone who has been expecting to observe event “Red” in particular. Likewise, observing HTHHTTHT is possible for a person who was expecting this particular sequence of coin tosses, instead of any combination with length 8. See Another Non-Anthropic Paradox: The Unsurprising Rareness of Rare Events

Ape in the coat 1 May 2024 6:10 UTC
1 point
0
on: LLMs could be as conscious as human emulations, potentially
From our state of knowledge about consciousness it’s indeed not impossible that modern LLMs are conscious. I wouldn’t say it’s likely, I definitely wouldn’t say that they are as likely to be conscious as uploaded humans. But the point stands. We don’t know for sure and we lack proper way to figure it out.
Previously we could’ve vaguely point towards Turing test, but we are past this stage now. Behavioral analysis of a model at this point is mostly unhelpful. A few tweaks can make the same LLM that previously confidently claimed not to be conscious, to swear that it’s conscious and is suffering. So what a current LLM says about the nature of its consciousness gives us about 0 bit of evidence.
This is another reason to stop making bigger models and spend a lot of time figuring out what we have already created. At some point we may create a conscious LLM, won’t be able to tell the difference and it would be a moral catastrophe.

Ape in the coat 30 Apr 2024 14:28 UTC
1 point
0
in reply to: quetzal_rainbow’s comment on: LLMs seem (relatively) safe
You mean, “ban superintelligence”? Because superintelligences are not human-like.
The kind of superintelligence that doesn’t possess human-likeness that we want it to possess.
That’s the problem with your proposal of “ethics module”. Let’s suppose that we have system of “ethics module” and “nanotech design module”. Nanotech design module outputs 3D-model of supramolecular unholy abomination. What exactly should ethics module do to ensure that this abomination doesn’t kill everyone?
Nanotech design module has to be evaluatable by the ethics module. For that it also be made from multiple sequential LLM calls in explicit natural language. Other type of modules should be banned.

Ape in the coat 30 Apr 2024 9:00 UTC
2 points
0
in reply to: Markvy’s comment on: The Solution to Sleeping Beauty
indeed. But still… if she wonders out loud “what day is it?” at the very moment she says that, it has an answer.
There is no “but”. As long as the Beauty is unable to distinguish between Monday and Tuesday awakenings, as long as the decision process which leads her to say the phrase “what day is it” works the same way, from her perspective there is no one “very moment she says that”. On Tails, there are two different moments when she says this, and the answer is different for them. So there is no answer for her
An experimenter who overhears her knows the answer. It seems to me that you “resolve” this tension is that the two of them are technically asking a different question, even though they are using the same words
Yes, you are correct. From the position of the experimenter, who knows which day it is, or who is hired to work only on one random day this is a coherent question with an actual answer. The words we use are the same but mathematical formalism is different.
For an experimenter who knows that it’s Monday the probability that today is Monday is simply:
P(Monday|Monday) = 1
For an experimenter who is hired to work only on one random day it is:
P(Monday|Monday xor Tuesday) = ¹⁄₂
But still… how surprised should she be if she were to learn that today is Monday? It seems that taking your stance to its conclusion, the answer would be “zero surprise: she knew for sure she would wake up on Monday so no need to be surprised it happened”
And even if she were to learn that the coin landed tails, so she knows that this is just one of a total of two awakenings, she should have zero surprise upon learning the day of the week, since she now knows both awakenings must happen.
Completely correct. Beauty knew that she would be awaken on Monday either way and so she is not surprised. This is a standard thing with non-mutually exclusive events. Consider this:
A coin is tossed and you are put to sleep. On Heads there will be a red ball in your room. On Tails there will be a red and a blue ball in your room. How surprised should you be to find a red ball in your room?
Which seems to violate conservation of expected evidence, except you already said that the there’s no coherent probabilities here for that particular question, so that’s fine too.
The appearance of violation of conservation of expected evidence comes from the belief that awakening on Monday and on Tuesday are mutually exclusive, while they are, in fact sequential.
This makes sense, but I’m not used to it. For instance, I’m used to these questions having the same answer:
1. P(today is Monday)?
2. P(today is Monday | the sleep lab gets hit by a tornado)
Yet here, the second question is fine (assuming tornadoes are rare enough that we can ignore the chance of two on consecutive days) while the first makes no sense because we can’t even define “today”
It makes sense but it’s very disorienting, like incompleteness theorem level of disorientation or even more
I completely understand. It is counterintuitive because evolution didn’t prepare us to deal with situations where an experience is repeated the same while we receive memory loss. As I write in the post:
If I forget what is the current day of the week in my regular life, well, it’s only natural to start from a ¹⁄₇ prior per day and work from there. I can do it because the causal process that leads to me forgetting such information can be roughly modeled as a low probability occurrence which can happen to me at any day.
It wouldn’t be the case, if I was guaranteed to also forget the current day of the week on the next 6 days as well, after I forgot it on the first one. This would be a different causal process, with different properties—causation between forgetting—and it has to be modeled differently. But we do not actually encounter such situations in everyday life, and so our intuition is caught completely flat footed by them.
The whole paradox arises from this issue with our intuition, and just like with incompleteness theorem (thanks for the flattering comparison, btw), what we need to do now is to re-calibrate our intuitions, make it more accustomed to the truth, preserved by the math, instead of trying to fight it.

Ape in the coat 30 Apr 2024 8:44 UTC
3 points
−2
in reply to: quetzal_rainbow’s comment on: LLMs seem (relatively) safe
“here is how to make LLMs more capable but less humanlike, it will be adopted because it makes LLMs more capable”.
Thankfully, this is a class of problems that humanity has an experience dealing with. The solution boils down to regulating all the ways to make LLMs less human-like out of existence.

Ape in the coat 30 Apr 2024 7:48 UTC
5 points
−2
in reply to: quetzal_rainbow’s comment on: LLMs seem (relatively) safe
With every technology there is a way to make it stop working. There are any number of ways to make a plane unable to fly. But the important thing is that we know a way to make a plane fly—therefore humans can fly via a plane.
Likewise, the point that LLM-based-architecture can in principle be safe still stands even if there is a way to make an unsafe LLM-based-architecture.
And this is a huge point. Previously we were in a state where alignment wasn’t even a tractable problem. Where capabilities progressed and alignment stayed in the dirt. Where AI system may understand human values but still not care about them and we didn’t know what to do with it.
But now we can just
have an “ethics module” where the underlying LLM produces text which then feeds into other parts of the system to help guide behavior.
Which makes alignment tractable. Alignment can now be reduced to capability of the ethics module. We know that the system will care about our values as it understands them because we can explicitly code it to do this way via an if-else statement. This is an enormous improvement over the previous status quo.

Ape in the coat 28 Apr 2024 17:01 UTC
5 points
0
on: Magic by forgetting
Universal guide to magic via anthropics:
1. Be not randomly sampled from a set
2. Assume that you you are randomly sampled from the set anyway
3. Arrive to an absurd conclusion
4. Magic!
Either a strong self-sampling assumption is false
Of course it is false. What are the reasons to even suspect that it might be true?
and-or path-based identity is true.
Note that path-dependent identity also has its own paradoxes: two copies can have different ’weights” depending on how they were created while having the same measure. For example, if in sleep two copies of me will be created and one of the copies will be copied again – when there will be 3 copies in the morning in the same world, but if we calculate chances to be one of them based on paths, they will be ½ and ¼ and ¼.
This actually sounds about right. What’s paradoxical here?

Ape in the coat 23 Apr 2024 7:42 UTC
2 points
0
in reply to: Markvy’s comment on: The Solution to Sleeping Beauty
I knew that not any string of English words gets a probability, but I was naïve enough to think that all statements that are either true or false get one.
Well, I think this one is actually correct. But, as I said in the previous comment, the statement “Today is Monday” doesn’t actually have a coherent truth value throughout the probability experiment. It’s not either True or False. It’s either True or True and False at the same time!
I was hoping they this sequence of posts which kept saying “don’t worry about anthropics, just be careful with the basics and you’ll get the right answer” would show how to answer all possible variations of these “sleep study” questions… instead it turns out that it answers half the questions (the half that ask about the coin) while the other half is shown to be hopeless… and the reason why it’s hopeless really does seem to have an anthropics flavor to it.
We can answer every coherently formulated question. Everything that is formally defined has an answer Being careful with the basics allows to understand which question is coherent and which is not. This is the same principle as with every probability theory problem.
Consider Sleeping-Beauty experiment without memory loss. There, the event Monday xor Tuesday also can’t be said to always happen. And likewise “Today is Monday” also doesn’t have a stable truth value throughout the whole experiment.
Once again, we can’t express Beauty’s uncertainty between the two days using probability theory. We are just not paying attention to it because by the conditions of the experiment, the Beauty is never in such state of uncertainty. If she remembers a previous awakening then it’s Tuesday, if she doesn’t—then it’s Monday.
All the pieces of the issue are already present. The addition of memory loss just makes it’s obvious that there is the problem with our intuition.

Ape in the coat 18 Apr 2024 8:15 UTC
4 points
−1
on: When is a mind me?
“You should anticipate having both experiences” sounds sort of paradoxical or magical, but I think this stems from a verbal confusion.
You can easily clear this confusion if you rephrase it as “You should anticipate having any of these experiences”. Then it’s immediately clear that we are talking about two separate screens. And it’s also clear that our curriocity isn’t actually satisfied. That the question “which one of these two will actually be the case” is still very much on the table.
Rob-y feels exactly as though he was just Rob-x, and Rob-z also feels exactly as though he was just Rob-x
Yes, this is obvious. Still as soon as we got Rob-y and Rob-z they are not “metaphysically the same person”. When Rob-y says “I” he is reffering to Rob-y, not Rob-z and vice versa. More specifically Rob-y is refering to some causal curve through time ans Rob-z is refering to another causal curve through time. These two curves are the same to some point, but then they are not.

Ape in the coat 18 Apr 2024 6:07 UTC
1 point
0
in reply to: Duschkopf’s comment on: Beauty and the Bets
In case the bet is offered on every awakening: do you mean if she gives conflicting answers on Monday and Tuesday that the bet nevertheless is regarded as accepted?
Yes I do.
Of course, if the experiment is run as stated she wouldn’t be able to give conflicting answers, so the point is moot. But having a strict algorithm for resolving such theoretical cases is a good thing anyway.
My initial idea was, that if for example only her Monday answer counts and Beauty knows that, she could reason that when her answer counts it is Monday, arriving at the conclusion that it is reasonable to act as if it was Monday on every awakening, thus grounding her answer on P(H/Monday)=1/2. Same logic holds for rule „last awakening counts“ and „random awakening counts“.
Yes, I got it. As a matter of fact this is unlawful. Probability estimate is about the evidence you receive not about what “counts” for a betting scheme. If the Beauty receives the same evidence when her awakening counts and when it doesn’t count she can’t update her probability estimate. If in order to arrive to the correct answer she needs to behave as if every day is Monday it means that there is something wrong with her model.
Thankfully for thirdism, she does not have to do it. She can just assign zero utility to Tuesday awakening and get the correct betting odds.
Anyway, all this is quite tangental to the question of utility instability. Which is about the Beauty making a bet on Sunday and then reflecting on it during the experiment, even if no bets are proposed. According to thirdism probability of the coin being Heads changes on awakening, so, in order for Beauty not to regret about making an optimal bet on Sunday, her utility has to change as well. Therefore utility instability.

Ape in the coat 17 Apr 2024 8:05 UTC
1 point
0
in reply to: Duschkopf’s comment on: Beauty and the Bets
There are indeed ways to obfuscate the utility instability under thirdism by different betting schemes where it’s less obvious, as the probability relevant to betting isn’t P(Heads|Awake) = ¹⁄₃ but one of thoses you meantion which equal ¹⁄₂.
The way to define the scheme specifically for P(Heads|Awake), is this: you get asked to bet on every awakening. One agreement is sufficient, and only one agreement counts. No random selecting takes place.
This way the Beauty doesn’t get any extra evidence when she is asked to bet, therefore she can’t update her credence for the coin being Heads based on the sole fact of being asked to bet, the way you propose.

Ape in the coat 17 Apr 2024 6:37 UTC
3 points
0
in reply to: Markvy’s comment on: The Solution to Sleeping Beauty
This makes me uncomfortable. From the perspective of sleeping beauty, who just woke up, the statement “today is Monday” is either true or false (she just doesn’t know which one). Yet you claim she can’t meaningfully assign it a probability. This feels wrong, and yet, if I try to claim that the probability is, say, ²⁄₃, then you will ask me “in what sample space?” and I don’t know the answer.
Where does the feeling of wrongness come from? Were you under impression that probability theory promised us to always assign some measure to any statement in natural language? It just so happens that most of the time we can construct an appropriate probability space. But the actual rule is about whether or not we can construct a probability space, not whether or not something is a statement in natural language.
Is it really so surprising that a person who is experiencing amnesia and the repetetion of the same experience, while being fully aware of the procedure can’t meaningfully assign credence to “this is the first time I have this experience”? Don’t you think there has to be some kind of problems with Beauty’s knowledge state? The situation whre due to memory erasure the Beauty loses the ability to coherently reason about some statements makes much more sense than the alternative proposed by thirdism—according to which the Beauty becomes more confident in the state of the coin than she would’ve been if she didn’t have her memory erased.
Another intuition pump is that “today is Monday” is not actually True xor False from the perspective of the Beauty. From her perspective it’s True xor (True and False). This is because on Tails, the Beauty isn’t reasoning just for some one awakening—she is reasoning for both of them at the same time. When she awakens the first time the statement “today is Monday” is True, and when she awakens the second time the same statement is False. So the statement “today is Monday” doesn’t have stable truth value throughout the whole iteration of probability experiment. Suppose that Beauty really does not want to make false statements. Deciding to say outloud “Today is Monday”, leads to making a false statement in 100% of iterations of experiemnt when the coin is Tails.
P(today is Monday | heads) = 100% is fine. (Or is that tails? I keep forgetting.) P(today is Monday | tails) = 50% is fine too. (Or maybe it’s not? Maybe this is where I’m going working? Needs a bit of work but I suspect I could formalize that one if I had to.) But if those are both fine, we should be able to combine them, like so: heads and tails are mutually exclusive and one of them must happen, so: P(today is Monday) = P(heads) • P(today is Monday | heads) + P(tails) • P(today is Monday | tails) = 0.5 + .25 = 0.75 Okay, I was expecting to get ²⁄₃ here. Odd. More to the point, this felt like cheating and I can’t put my finger on why. maybe need to think more later
Here you are describing Lewis’s model which is appropriate for Single Awakening Problem. There the Beauty is awakened on Monday if the coin is Heads, and if the coin is Tails, she is awakened either on Monday or on Tuesday (not both). It’s easy to see that 75% of awakening in such experiment indeed happen on Monday.
It’s very good that you notice this feeling of cheating. This is a very important virtue. This is what helped me construct the correct model and solve the problem in the first place—I couldn’t accept any other—they all were somewhat off.
I think, you feel this way, because you’ve started solving the problem from the wrong end, started arguing with math, instead of accepting it. You noticed that you can’t define “Today is Monday” normally so you just assumed as an axiom that you can.
But as soon as you assume that “Today is Monday” is a coherent event with a stable truth value throughout the experiment, you inevitably start talking about a different problem, where it’s indeed the case. Where there is only one awakening in any iteration of probability experiment and so you can formally construct a sample space where “Today is Monday” is an elementary mutually exclusive outcome. There is no way around it. Either you model the problem as it is, and then “Today is Monday” is not a coherent event, or you assume that it is coherent and then you are modelling some other problem.

Ape in the coat 15 Apr 2024 9:30 UTC
1 point
0
in reply to: Markvy’s comment on: The Solution to Sleeping Beauty
The second one looks “obvious” from symmetry considerations but actually formalizing seems harder than expected.
Exactly! I’m glad that you actually engaged with the problem.
The first step is to realize that here “today” can’t mean “Monday xor Tuesday” because such event never happens. On every iteration of experiment both Monday and Tuesday are realized. So we can’t say that the participant knows that they are awakened on Monday xor Tuesday.
Can we say that participant knows that they are awakened on Monday or Tuesday? Sure. As a matter of fact:
P(Monday or Tuesday) = 1
P(Heads|Monday or Tuesday) = P(Heads) = ¹⁄₂
This works, here probability that the coin is Heads in this iteration of the experiment happens to be the same as what our intuition is telling us P(Heads|Today) is supposed to be, however we still can’t define “Today is Monday”:
P(Monday|Monday or Tuesday) = P(Monday) = 1
Which doesn’t fit our intuition.
How can this be? How can we have a seeminglly well-defined probability for “Today the coin is Heads” but not for “Today is Monday”? Either “Today” is well-defined or it’s not, right? Take some time to think about it.
What do we actually mean when we say that on an awakening the participant supposed to believe that the coin is Heads with 50% probability? Is it really about this day in particular? Or is it about something else?
The answer is: we actually mean, that on any day of the experiment be it Monday or Tuesday the participant is supposed to believe that the coin is Heads with 50% probability. We can not formally specify “Today” in this problem but there is a clever, almost cheating way to specify “Anyday” without breaking anything.
This is not easy. It requires a way to define P(A|B), when P(B) itself is undefined which is unconventional. But, moreover, it requires symmetry. P(Heads|Monday) has to be equal to P(Heads|Tuesday) only then we have a coherent P(Heads|Anyday).

Ape in the coat 14 Apr 2024 18:52 UTC
2 points
0
in reply to: Markvy’s comment on: The Solution to Sleeping Beauty
First of all, that can’t possibly be right.
I understand that it all may be somewhat counterintuitive. I’ll try to answer whatever questions you have. If you think you have some way to formally define what “Today” means in Sleeping Beauty—feel free to try.
Second of all, it goes against everything you’ve been saying for the entire series.
Seems very much in accordance with what I’ve been saying.
Throughout the series I keep repeating the point that all we need to solve anthropics is to follow probability theory where it leads and then there will be no paradoxes. This is exactly what I’m doing here. There is no formal way to define “Today is Monday” in Sleeping Beauty and so I simply accept this, as the math tells me to, and then the “paradox” immediately resolves.
Suppose someone who has never heard of the experiment happens to call sleeping beauty on her cell phone during the experiment and ask her “hey, my watch died and now I don’t know what day it is; could you tell me whether today is Monday or Tuesday?” (This is probably a breach of protocol and they should have confiscated her phone until the end, but let’s ignore that.).
Are you saying that she has no good way to reason mathematically about that question? Suppose they told her “I’ll pay you a hundred bucks if it turns out you’re right, and it costs you nothing to be wrong, please just give me your best guess”. Are you saying there’s no way for her to make a good guess? If you’re not saying that, then since probabilities are more basic than utilities, shouldn’t she also have a credence?
Good question. First of all, as we are talking about betting I recommend you read the next post, where I explore it in more details, especially if you are not fluent in expected utility calculations.
Secondly, we can’t ignore the breach of the protocol. You see, if anything breaks the symmetry between awakening, the experiment changes in a substantial manner. See Rare Event Sleeping Beauty, where probability that the coin is Heads can actually be ¹⁄₃.
But we can make a similar situation without breaking the symmetry. Suppose that on every awakening a researcher comes to the room and proposes the Beauty to bet on which day it currently is. At which odds should the Beauty take the bet?

This is essentially the same betting scheme as ice-cream stand, which I deal with in the end of the previous comment.

Ape in the coat

Se­man­tic Disagree­ment of Sleep­ing Beauty Problem

God is good*

Semantic Disagreement of Sleeping Beauty Problem