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:
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:
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.
Almost any set: only the empty set is excluded. The identities of the elements themselves are irrelevant to the mathematical structure. Any further restrictions are not a part of mathematical definition of probability space, but some particular application you may have in mind.
If elementary event {A} has P(A) = 0, then we can simply not include outcome A into the sample space for simplicity sake.
In some cases this is reasonable, but in others it is impossible. For example, when defining continuous probability distributions you can’t eliminate sample set elements having measure zero or you will be left with the empty set.
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.
It is a synonym for probability in the sense that it is a mathematical probability: that is, a measure over a sigma-algebra for which the axioms of a probability space are satisfied. I use a different term here to denote this application of the mathematical concept to a particular real-world purpose. Beside which, the Sleeping Beauty problem explicit uses the word.
I also don’t quite know what you mean by the phrase “stable truth value”. As defined, a universe state either satisfies or does not satisfy a proposition. If you’re referring to propositions that may vary over space or time, then when modelling a given situation you have two choices: either restrict the universe states in your set to locations or time regions over which all selected propositions have a definite truth value, or restrict the propositions to those that have a definite truth value over the selected universe states. Either way works.
Semantic statement “Today is Monday” is not a well-defined event in the Sleeping Beauty problem.
Of course it is. I described the structure under which it is, and you can verify that it does in fact satisfy the axioms of a probability space. As you’re looking for a crux, this is probably it.
Universe states can be distinguished by time information, and in problems like this where time is part of the world-model, they should be. The mathematical structure of a probability space has nothing to do with it, as the mathematical formalism doesn’t care what the elements of a sample space are.
Otherwise you can’t model even a non-coin flip form of the Sleeping Beauty problem in which Beauty is always awoken twice. If the problem asks “what should be Beauty’s credence that it is Monday” then you can’t even model the question without distinguishing universe states by time.
Almost any set: only the empty set is excluded. The identities of the elements themselves are irrelevant to the mathematical structure. Any further restrictions are not a part of mathematical definition of probability space, but some particular application you may have in mind.
Any non-empty set can be a sample space for some problem. But we are interested in a sample space for a very specific problem—Sleeping Beauty. This “applications we have in mind” is the whole point of the discussion.
Was it really not clear? I have a whole post dedicated towards exploring different probability spaces that people proposed for the Sleeping Beauty problem, where I explicitly notice tha these probability spaces are valid in principle but not sound when applied to the Sleeping Beauty problem—they describe some other problems instead. Likewise, in this post I point that indexical sample space for one problem can be a sample space for another problem—the one where elements of the set indeed are mutually exclusive outcomes. How did you manage to miss it?
Anyway, when we fixed the problem that we are talking about, there are some specific conditions according to which we can say whether a set indeed is a sample space for this problem. And one of them is that the elements of it has to be mutually exclusive outcomes, such as in every iteration of the probability experiment one and only one of them is realized. Are we on the same page here?
So if we define a set in such a way that it consist of outcomes that happen multiple times during the same trial, it can’t be a sample space for this problem anymore. However we can remove this restriction by defining a new entity: indexical sample space for this problem.
In some cases this is reasonable, but in others it is impossible. For example, when defining continuous probability distributions you can’t eliminate sample set elements having measure zero or you will be left with the empty set.
Fair point. No disagreement here. I was talking specifically about discrete case.
It is a synonym for probability in the sense that it is a mathematical probability: that is, a measure over a sigma-algebra for which the axioms of a probability space are satisfied. I use a different term here to denote this application of the mathematical concept to a particular real-world purpose.
Our particulr real world purpose is the Sleeping Beuaty problem. Credence that the Beauty has isn’t just a value of measure function from some probability space it’s the value of probability function from the probability space specifically for this problem. Which sample space has to consist from mutually exclusive outcomes for this problem.
I also don’t quite know what you mean by the phrase “stable truth value”
There is this thing called probability experiment. Which I provided you a wiki link to in a previous comment. According to Kolmogorov, it’s some complex of conditions that can be repeated and on repetition outputs different results all of which belong to some set. On any iteration of the experiment exactly one result is output. This is how we merge our mathematical model with the phisical universe. We say that these results are elements of the sample space. And so one outcome of the sample space is realized at every iteration of the experiment. And every element of the event space to which the realized outcome belongs to is also considered to be realized. Thus we can say that for every iteration of probability experiment every well-defined event is either realized or not realized—has a stable truth value.
If something doesn’t have a stable truth value during an iteration of the experiment, it’s not a well defined event. This is exactly what happens with statement “Today is Monday” in Sleeping Beauty experiment.
either restrict the universe states in your set to locations or time regions over which all selected propositions have a definite truth value, or restrict the propositions to those that have a definite truth value over the selected universe states. Either way works.
What you are saying is that we can either modify the setting of the experiment so that the statement have a stable truth value or use different statements. This is true. But we want to talk about a specific experimental setting—Sleeping Beauty problem—we can’t modify it, otherwise we would be talking about something else, like No-Coin-Toss problem. Therefore, the only way is to acknowledge that some statements are not coherent in the current experimental setting and not use them. This is exactly what I do.
I described the structure under which it is, and you can verify that it does in fact satisfy the axioms of a probability space.
The fact that you can describe such mathematical structure means that there is an experiment where statement “Today is Monday” is a well-defined event. Here we are in agreement. More specifically this is the experiment where awakenings on Monday and Tuesday are mutually exclusive during one trial, such as No-Coin-Toss or Single-Awakening.
But these experiments are not Sleeping Beauty experiment. And this is what I’ve been talking about the whole time that specifically in Sleeping Beauty experiment, from the perspective of the Beauty “Today is Monday” isn’t a well-defined event, it doesn’t have a stable truth value.
You seem to be thinking that if something is a well-defined event in some experiment, it has to be a well defined event in Sleeping Beauty experiment as well. Is this our crux?
Otherwise you can’t model even a non-coin flip form of the Sleeping Beauty problem in which Beauty is always awoken twice. If the problem asks “what should be Beauty’s credence that it is Monday” then you can’t even model the question without distinguishing universe states by time.
Yes. This is a completely correct conclusion which I’m planning to talk about in the next post. If a person goes through memory loss and the repetition of the same experience, there is no coherent credence/probability that this experience happens the first time that this person can have.
P(Monday) = 1
P(Tuesday) = 1
The vague inuitive feeling that it has to be 1⁄2 is once again pointing to weighted probability, which renormalizes the measure function.
More specifically this is the experiment where awakenings on Monday and Tuesday are mutually exclusive during one trial, such as No-Coin-Toss or Single-Awakening
No, I specifically was referring to the Sleeping Beauty experiment. Re-read my comment. Or not. At this point it’s quite clear that we are failing to communicate in a fundamental way. I’m somewhat frustrated that you don’t even comment on those parts where I try to communicate the structure of the question, but only on the parts which seem tangential or merely about terminology. There is no need to reply to this comment, as I probably won’t continue participating in this discussion any further.
I find this situation quite ironic. From my perspective, I painstakingly cited and answered your comments piece by piece even though you didn’t engage much neither with the arguments in my posts nor with any of my replies.
I’m not sure how I could have missed “parts where you try to communicate the structure of the question”. The only things which I haven’t directly cited in your previous comment are:
Beside which, the Sleeping Beauty problem explicit uses the word.
and
As defined, a universe state either satisfies or does not satisfy a proposition. If you’re referring to propositions that may vary over space or time, then when modelling a given situation you have two choices
Which I neither disagree nor have any interesting to add.
Universe states can be distinguished by time information, and in problems like this where time is part of the world-model, they should be. The mathematical structure of a probability space has nothing to do with it, as the mathematical formalism doesn’t care what the elements of a sample space are.
Which does not have any argument in it. You just make statements without explaining why they are supposed to be true. I obviously do not agree that in Sleeping Beauty case our model should be treating time states as individual outcomes. But simply proclaming this disagreement doesn’t serve any purpose after I’ve already presented comprehensive explanation why we can’t lawfully reason about time states in this particular case as if they are mutually exclusive outcomes, which you, once again, failed to engage with.
I would appreciate if you addressed this meta point because I’m honestly confused about your perspective on this failure of our communication.
Meta ended.
No, I specifically was referring to the Sleeping Beauty
If you were specifically referring to Sleeping Beauty problem then your previous comment doesn’t make sense.
Either you are logically pinpointing any sample space for at least some problem, and then you can say that any non-empty set fits this description.
Or you are logically pinpointing sample space specifically for the Sleeping Beauty problem, then you can’t dismiss the condition of mutually exclusivity of outcomes which are relevant to the particular application of the sample space.
Eh, I’m not doing anything else important right now, so let’s beat this dead horse further.
“As defined, a universe state either satisfies or does not satisfy a proposition. If you’re referring to propositions that may vary over space or time, then when modelling a given situation you have two choices”
Which I neither disagree nor have any interesting to add.
This is the whole point! That’s why I pointed it out as the likely crux, and you’re saying that’s fine, no disagreement there. Then you reject one of the choices.
You agree that any non-empty set can be the sample space for some probability space. I described a set: those of universe states labelled by time of being asked about a credence.
I chose my set to be a cartesian product of the two relevant properties that Beauty is uncertain of on any occasion when she awakens and is asked for her credence: what day it is on that occasion of being asked (Monday or Tuesday), and what the coin flip result was (Heads or Tails). On any possible occasion of being asked, it is either Monday or Tuesday (but not both), and either Heads or Tails (but not both). I can set the credence for (Tuseday,Heads) to zero since Beauty knows that’s impossible by the setup of the experiment.
If Beauty knew which day it was on any occasion when she is asked, then she should give one of two different answers for credences. These correspond to the conditional credences P(Heads | Monday) and P(Heads | Tuesday). Likewise, knowing what the coin flip was would give different conditional credences P(Monday | Heads) and P(Monday | Tails).
All that is mathematically required of these credences is that they obey the axioms of a measure space with total measure 1, because that’s exactly the definition of a probability space. My only claim in this thread—in contrast to your post—is that they can.
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.
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.
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.
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.
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.
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.
Almost any set: only the empty set is excluded. The identities of the elements themselves are irrelevant to the mathematical structure. Any further restrictions are not a part of mathematical definition of probability space, but some particular application you may have in mind.
In some cases this is reasonable, but in others it is impossible. For example, when defining continuous probability distributions you can’t eliminate sample set elements having measure zero or you will be left with the empty set.
It is a synonym for probability in the sense that it is a mathematical probability: that is, a measure over a sigma-algebra for which the axioms of a probability space are satisfied. I use a different term here to denote this application of the mathematical concept to a particular real-world purpose. Beside which, the Sleeping Beauty problem explicit uses the word.
I also don’t quite know what you mean by the phrase “stable truth value”. As defined, a universe state either satisfies or does not satisfy a proposition. If you’re referring to propositions that may vary over space or time, then when modelling a given situation you have two choices: either restrict the universe states in your set to locations or time regions over which all selected propositions have a definite truth value, or restrict the propositions to those that have a definite truth value over the selected universe states. Either way works.
Of course it is. I described the structure under which it is, and you can verify that it does in fact satisfy the axioms of a probability space. As you’re looking for a crux, this is probably it.
Universe states can be distinguished by time information, and in problems like this where time is part of the world-model, they should be. The mathematical structure of a probability space has nothing to do with it, as the mathematical formalism doesn’t care what the elements of a sample space are.
Otherwise you can’t model even a non-coin flip form of the Sleeping Beauty problem in which Beauty is always awoken twice. If the problem asks “what should be Beauty’s credence that it is Monday” then you can’t even model the question without distinguishing universe states by time.
Any non-empty set can be a sample space for some problem. But we are interested in a sample space for a very specific problem—Sleeping Beauty. This “applications we have in mind” is the whole point of the discussion.
Was it really not clear? I have a whole post dedicated towards exploring different probability spaces that people proposed for the Sleeping Beauty problem, where I explicitly notice tha these probability spaces are valid in principle but not sound when applied to the Sleeping Beauty problem—they describe some other problems instead. Likewise, in this post I point that indexical sample space for one problem can be a sample space for another problem—the one where elements of the set indeed are mutually exclusive outcomes. How did you manage to miss it?
Anyway, when we fixed the problem that we are talking about, there are some specific conditions according to which we can say whether a set indeed is a sample space for this problem. And one of them is that the elements of it has to be mutually exclusive outcomes, such as in every iteration of the probability experiment one and only one of them is realized. Are we on the same page here?
So if we define a set in such a way that it consist of outcomes that happen multiple times during the same trial, it can’t be a sample space for this problem anymore. However we can remove this restriction by defining a new entity: indexical sample space for this problem.
Fair point. No disagreement here. I was talking specifically about discrete case.
Our particulr real world purpose is the Sleeping Beuaty problem. Credence that the Beauty has isn’t just a value of measure function from some probability space it’s the value of probability function from the probability space specifically for this problem. Which sample space has to consist from mutually exclusive outcomes for this problem.
There is this thing called probability experiment. Which I provided you a wiki link to in a previous comment. According to Kolmogorov, it’s some complex of conditions that can be repeated and on repetition outputs different results all of which belong to some set. On any iteration of the experiment exactly one result is output. This is how we merge our mathematical model with the phisical universe. We say that these results are elements of the sample space. And so one outcome of the sample space is realized at every iteration of the experiment. And every element of the event space to which the realized outcome belongs to is also considered to be realized. Thus we can say that for every iteration of probability experiment every well-defined event is either realized or not realized—has a stable truth value.
If something doesn’t have a stable truth value during an iteration of the experiment, it’s not a well defined event. This is exactly what happens with statement “Today is Monday” in Sleeping Beauty experiment.
What you are saying is that we can either modify the setting of the experiment so that the statement have a stable truth value or use different statements. This is true. But we want to talk about a specific experimental setting—Sleeping Beauty problem—we can’t modify it, otherwise we would be talking about something else, like No-Coin-Toss problem. Therefore, the only way is to acknowledge that some statements are not coherent in the current experimental setting and not use them. This is exactly what I do.
The fact that you can describe such mathematical structure means that there is an experiment where statement “Today is Monday” is a well-defined event. Here we are in agreement. More specifically this is the experiment where awakenings on Monday and Tuesday are mutually exclusive during one trial, such as No-Coin-Toss or Single-Awakening.
But these experiments are not Sleeping Beauty experiment. And this is what I’ve been talking about the whole time that specifically in Sleeping Beauty experiment, from the perspective of the Beauty “Today is Monday” isn’t a well-defined event, it doesn’t have a stable truth value.
You seem to be thinking that if something is a well-defined event in some experiment, it has to be a well defined event in Sleeping Beauty experiment as well. Is this our crux?
Yes. This is a completely correct conclusion which I’m planning to talk about in the next post. If a person goes through memory loss and the repetition of the same experience, there is no coherent credence/probability that this experience happens the first time that this person can have.
P(Monday) = 1
P(Tuesday) = 1
The vague inuitive feeling that it has to be 1⁄2 is once again pointing to weighted probability, which renormalizes the measure function.
No, I specifically was referring to the Sleeping Beauty experiment. Re-read my comment. Or not. At this point it’s quite clear that we are failing to communicate in a fundamental way. I’m somewhat frustrated that you don’t even comment on those parts where I try to communicate the structure of the question, but only on the parts which seem tangential or merely about terminology. There is no need to reply to this comment, as I probably won’t continue participating in this discussion any further.
Meta:
I find this situation quite ironic. From my perspective, I painstakingly cited and answered your comments piece by piece even though you didn’t engage much neither with the arguments in my posts nor with any of my replies.
I’m not sure how I could have missed “parts where you try to communicate the structure of the question”. The only things which I haven’t directly cited in your previous comment are:
and
Which I neither disagree nor have any interesting to add.
Which does not have any argument in it. You just make statements without explaining why they are supposed to be true. I obviously do not agree that in Sleeping Beauty case our model should be treating time states as individual outcomes. But simply proclaming this disagreement doesn’t serve any purpose after I’ve already presented comprehensive explanation why we can’t lawfully reason about time states in this particular case as if they are mutually exclusive outcomes, which you, once again, failed to engage with.
I would appreciate if you addressed this meta point because I’m honestly confused about your perspective on this failure of our communication.
Meta ended.
If you were specifically referring to Sleeping Beauty problem then your previous comment doesn’t make sense.
Either you are logically pinpointing any sample space for at least some problem, and then you can say that any non-empty set fits this description.
Or you are logically pinpointing sample space specifically for the Sleeping Beauty problem, then you can’t dismiss the condition of mutually exclusivity of outcomes which are relevant to the particular application of the sample space.
Eh, I’m not doing anything else important right now, so let’s beat this dead horse further.
This is the whole point! That’s why I pointed it out as the likely crux, and you’re saying that’s fine, no disagreement there. Then you reject one of the choices.
You agree that any non-empty set can be the sample space for some probability space. I described a set: those of universe states labelled by time of being asked about a credence.
I chose my set to be a cartesian product of the two relevant properties that Beauty is uncertain of on any occasion when she awakens and is asked for her credence: what day it is on that occasion of being asked (Monday or Tuesday), and what the coin flip result was (Heads or Tails). On any possible occasion of being asked, it is either Monday or Tuesday (but not both), and either Heads or Tails (but not both). I can set the credence for (Tuseday,Heads) to zero since Beauty knows that’s impossible by the setup of the experiment.
If Beauty knew which day it was on any occasion when she is asked, then she should give one of two different answers for credences. These correspond to the conditional credences P(Heads | Monday) and P(Heads | Tuesday). Likewise, knowing what the coin flip was would give different conditional credences P(Monday | Heads) and P(Monday | Tails).
All that is mathematically required of these credences is that they obey the axioms of a measure space with total measure 1, because that’s exactly the definition of a probability space. My only claim in this thread—in contrast to your post—is that they can.