And as I’ve tried to get across, if the two versions are truly isomorphic, and also have faults, one should be able to identify those faults in either one without translating them to the other. But if those faults turn out to depend on a false analysis specific to one, you won’t find them in the other.
The Two Coin version is about what happens on one day. Unlike the Always-Monday-Tails-Tuesday version, the subject can infer no information about coin C1 on another day, which is the mechanism for fault in that version. Each day, in the “world” of the subject, is a fully independent “world” with a mathematically valid sample space that applies to it alone.
“It treats sequential events as mutually exclusive,” No, it treats an observation of a state, when that observation bears no connection to any other, as independent of any other.
“… therefore unlawfully constructs sample space.” What law was broken?
Do you disagree that, on the morning of the observation, there were four equally likely states? Do you think the subject has some information about how the state was observed on another day? That an observer from the outside world has some impact on what is known on the inside? These are the kind of details that produce controversy in the Always-Monday-Tails-Tuesday version. I personally think the inferences about carrying information over between the two days are all invalid, but what I am trying to do is eliminate any basis for doing that.
Yes, each outcome on the first day can be paired with exactly one on the second. But without any information passing to the subject between these two days, she cannot do anything with such pairings. To her, each day is its own, completely independent probability experiment. One where “new information” means she is awakened to see only three of the four possible outcomes.
“Your model treats HH, HT, TH and TT as four individual mutually exclusive outcomes” No, it treats the current state of the coins as four mutually exclusive states.
“However, when you actually do it, you get a different list.”
How so? If your write down the state on the first day that the researchers look at the coins, you will find that {HH, TH, HT, TT} all occur with frequency 1⁄4. Same on the second day. If you write down the frequencies when the subject is awake, you find that {TH, HT, TT} all have frequency 1⁄3.
Here is what you are arguing: Say you repeat this many times and make two lists, one for each day. And each entry on the lists includes the attempt number. You will find that every attempt where the first list says TH, the second list will say TT. And that every attempt where the first list says HT, the second list will not have an entry for that attempt.
What I’m saying is that the subject’s belief cannot cross-correlate the attempt numbers. She could use just the first list with the attempt numbers. Or the second, since each when viewed in isolation has the same properties as the other and so gets the same answer. She can even use a combined list, but if she does she cannot use the attempt numbers to associate the two observations on that attempt. That is what the amnesia drug accomplishes.
What I am saying is that regardless of which list she uses, the probability of heads is 1⁄3. And your arguments that this is wrong require associating the attempts, essentially removing the effect of amnesia. That cannot influence the subject’s belief.
The Two Coin version is about what happens on one day.
Let it be not two different days but two different half-hour intervals. Or even two milliseconds—this doesn’t change the core of the issue that sequential events are not mutually exclusive.
observation of a state, when that observation bears no connection to any other, as independent of any other.
It very much bears a connection. If you are observing state TH it necessary means that either you’ve already observed or will observe state TT.
What law was broken?
The definition of a sample space—it’s supposed to be constructed from mutually exclusive elementary outcomes.
Do you disagree that, on the morning of the observation, there were four equally likely states? Do you think the subject has some information about how the state was observed on another day?
Disagree on both accountsd. You can’t treat HH HT TT TH as individual outcomes and the term “morning of observation” is underspecified. The subject knows that some of them happen sequentially.
what I am trying to do is eliminate any basis for doing that
I noticed, and I applaud your attempts. But you can’t do that because you still have sequential events, anyway, the fact that you call them differently doesn’t change much.
Yes, each outcome on the first day can be paired with exactly one on the second.
Exactly. And the Beauty knows it. Case closed.
But without any information passing to the subject between these two days, she cannot do anything with such pairings. To her, each day is its own, completely independent probability experiment.
She knows that they do not happen at random. This is enough to be sure that each day is not completely independent probability experiment. See Effects of Amnesia section.
No, it treats the current state of the coins as four mutually exclusive states.
Call them “states” if you want. It doesn’t change anything.
How so? If your write down the state on the first day that the researchers look at the coins, you will find that {HH, TH, HT, TT} all occur with frequency 1⁄4. Same on the second day.
I’ve specifically explained how. We write down outcomes when the researcher sees the Beauty awake—when they updated on the fact of Beauty’s awakening. The frequency for three outcomes is 1⁄3, moreover they actually go in random order because the observer witnesses only one random awakening per experiment.
If you write down the frequencies when the subject is awake, you find that {TH, HT, TT} all have frequency 1⁄3.
Yep, no one is arguing with that. The problem is that the order isn’t random as your model predicts—TH and TT always go in pairs.
Here is what you are arguing: Say you repeat this many times and make two lists, one for each day.
No, I’m not complicating this with two lists for each day. There is only one list, which documents all the awakenings of the subject, while she is going through the series of experiments. The theory that predicts that two awakening are “completely independent probability experiments” expect that the order of the awakenings is random and it’s proven wrong because there is an order between awakenings. Easy as that.
That is what the amnesia drug accomplishes.
You are mistaken about what the amnesia acomplishes. Once again I send you to reread the Effects of Amnesia section. It’s equally applicable to Two-Coin version of the problem as a regular one.
And your arguments that this is wrong require associating the attempts, essentially removing the effect of amnesia.
According to Beauty’s knowledge, the attempts are already connected. Only if the amnesia removed from her mind the setting of the experiment if she forgot that TT and TH go in pairs, only then she should reason the way you want her to.
On the other hand, if we trully removed the effect of amnesia alltogether, then the Beauty would be 100% confident in Tails when awaken the second time in the same experiment.
So no, I’m talking about the exact knowledge state of the Beauty with the exact level of amnesia that she gets, while you your are talking about a more significant alteration of her mind.
Let it be not two different days but two different half-hour intervals. Or even two milliseconds—this doesn’t change the core of the issue that sequential events are not mutually exclusive.
OUTCOME: A measurable result of a random experiment.
SAMPLE SPACE: a set of exhaustive, mutually exclusive outcomes of a random experiment.
EVENT: Any subset of the sample space of a random experiment.
INDEPENDENT EVENTS: If A and B are events from the same sample space, andthe occurrence of event A does not affect the chances of the occurrence of event B, then A and B are independent events.
The outside world certainly can name the outcomes {HH1_HT2, HT1_HH2, TH1_TT2, TT1_TH2}. But the subject has knowledge of only one pass. So to her, only the current pass exists, because she has no knowledge of the other pass. What happens in that interval can play no part in her belief. The sample space is {HH, HT, TH, TT}.
To her, these four outcomes represent fully independent events, because she has no knowledge of the other pass. To her, the fact that she is awake means the event {HH} has been ruled out. It is still a part of the sample space, but is is one she knows is not happening. That’s how conditional probability works; the sample space is divided into two subsets; one is consistent with the observation, and one is not.
What you are doing, is treating HH (or, in Elga’s implementation, H&Tuesday) as if it ceases to exist as a valid outcome of the experiment. So HH1_HT2 has to be treated differently than TT1_TH2, since HH1_HT2 only “exists” in one pass, while TT1_TH2 “exists” in both. This is not true. Both exist in both passes, but one is unobserved in one pass.
And this really is the fallacy in any halfer argument. They treat the information in the observation as if it applies to both days. Since H&Tuesday “doesn’t exist”, H&Monday fully represents the Heads outcome. So to be consistent, T&Monday has to fully represent the Tails outcome. As does T&Tuesday, so they are fully equivalent.
If you are observing state TH it necessary means that either you’ve already observed or will observe state TT.
You are projecting the result you want onto the process. Say I roll a six-sided die tell you that the result is odd. Then I administer the amnesia drug, and tell you that I previously told you whether th result was even or odd. I then ask you for your degree of belief that the result is a six. Should you say 1⁄6, because as far as you know the sample space is {1,2,3,4,5,6}? Or should you say 0, because “you are [now] observing a state that you’ve already observed is only {1,3,5}?
And if you try to claim that this is different because you don’t know what I told you, that is exactly the point of the Two Coin Version.
The definition of a sample space [was broken] - it’s supposed to be constructed from mutually exclusive elementary outcomes.
It is so constructed.
I’ve specifically explained how. We write down outcomes when the researcher sees the Beauty awake—when they updated on the fact of Beauty’s awakening.
Beauty is doing the updating, not “they.” She is in an experiment where there are four possible combinations for what the coins are currently showing. She has no ability to infer/anticipate what the coins were/will be showing on another day.
Her observation is that one combination, of what is in the sample space for today, is eliminated.
No, I’m not complicating this with two lists for each day. There is only one list, which documents all the awakenings of the subject,...
Maybe you missed the part where I said you can look at one, or the other, or bot as long as you don’t carry information across.
You are mistaken about what the amnesia acomplishes, Once again I send you to reread the Effects of Amnesia section.
Then you are mistaken in that section.
Her belief can be based only on what she knows. If you create a difference between the two passes, in her knowledge, then maybe you could claim a dependence. I don’t think you can in this case, but to do it requires that difference.
The Two Coin Version does not have a difference. Nothing about what she observed about the outcomes HH1_HT2 or HT1_HH2 in another pass can affect her confidence concerning them in the current pass. (And please, recall that these describe the combinations that are showing.)
The link I use to get here only loads the comments, so I didn’t find the “Effects of Amnesia” section until just now. Editing it:
“But in my two-coin case, the subject is well aware about the setting of the experiment. She knows that her awakening was based on the current state of the coins. It is derived from, but not necessarily the same as, the result of flipping them. She only knows that this wakening was based on their current state, not a state that either precedes or follows from another. And her memory loss prevents her from making any connection between the two. As a good Bayesian, she has to use only the relevant available information that can be applied to the current state.”
The Beauty doesn’t know only about one pass she knows about their relation as well. And because of it she can’t reason as if they happen at random. You need to address this point before we could move on, because all your further reasoning is based on the incorrect premise that beauty knows less than she actually knows.
She has no ability to infer/anticipate what the coins were/will be showing on another day.
She absolutely has this ability as long as she knows the procedure, that TT and TH follow in pairs, she can make such conditional statements: “if the coins are currently TT then they either will be TH tomorrow or were TH yesterday”. It’s very different from not knowing anything whatsoever about the state of the coin on the next day. The fact that you for some reason feel that it should not matter is irrelevant. It’s still clearly more than no information whatsoever and, therefore, she can’t justifiably reason as if she doesn’t have any.
On the other hand, if the memory wipe removed this knowledge from her head as well, if the only thing she truly knew was that she is currently awakened at one of three possible states either TH, HT and TT, and had no idea of the relationship between them, then, only then, she would be justified to reason as you claim she should.
What you are doing, is treating HH (or, in Elga’s implementation, H&Tuesday) as if it ceases to exist
No, I treat is as an event that Beauty doesn’t expect to observe and therefore she doesn’t update when she indeed doesn’t observe it according to the law of conservation of expected evidence. We are talking about Beauty’s perspective after all, not a some outside view.
Suppose an absolutely trustwothy source tells you that the coin is Heads side up. Then you go and look at the coin and indeed it’s Heads side up. What should have been your probability that the coin is Tails side up before you looked at it?
It should be zero. You’ve already known the state of the coin before you looked at it, you got no new information. Does it mean that Tails side of a coin doesn’t exist? No, of course not! It just that you didn’t expect that the coin could possibly be Tails in this particular case based on your knowledge state.
Say I roll a six-sided die tell you that the result is odd. Then I administer the amnesia drug, and tell you that I previously told you whether th result was even or odd. I then ask you for your degree of belief that the result is a six. Should you say 1⁄6, because as far as you know the sample space is {1,2,3,4,5,6}? Or should you say 0, because “you are [now] observing a state that you’ve already observed is only {1,3,5}?
I was going to post a generalized way of reasoning under amnesia in a future post, but here is some: getting memory erased about some evidence just brings you to the state where you didn’t have this particular evidence. And getting an expected memory wipe can only make you less confident in your probability estimate, not more.
In this dice rolling case, initially my P(6) = 1⁄6, then after you told me that it’s odd, P(6|Odd)=0, and then when I’m memory wiped I’m back to P(6) = 1⁄6 and the knowledge that you’ve already told me whether the result is even or odd doesn’t help P(6|Even or Odd) = 1⁄6
Likewise in Sleeping Beauty I initially have P(Heads) = 1⁄2. Then I awakened exactly as I’ve expected in the experiment and still have P(Heads|Awake) = 1⁄2. Now suppose that I’m awakened once more. If there was no memory wipe I’d learn that I’m a awake a second time which would bring me to P(Heads|Two Awakenings) = 0. But I do not get this evidence due to memory wipe. So due to it, when I’m awakened the second time, I once again learn that I’m awake and still having P(Heads|Awake) = 1⁄2.
What you are implicitly claiming, however, is that getting memory wiped, or even just a possibility of it, makes the Beauty more confident in one outcome over the other! Which is quite bizarre. As if knowing less gives you more knowledge. Moreover, you assume that the person who knowns that their memory was/may be erased, just have to act as if they do not know it.
Suppose a coin is tossed and you received some circumstantial evidence about it’s state. As a result you are currently at 2⁄3 in favor of Heads. And then I tell you: “What odds are you ready to bet on? By the way, I have erased from your memory some crucial evidence in favor of Tails”. Do you really think that you are supposed to agree to bet on 1:2 odds even though you now know that the state of the evidence your currently have may not be trustworthy?
And as I’ve tried to get across, if the two versions are truly isomorphic, and also have faults, one should be able to identify those faults in either one without translating them to the other. But if those faults turn out to depend on a false analysis specific to one, you won’t find them in the other.
The Two Coin version is about what happens on one day. Unlike the Always-Monday-Tails-Tuesday version, the subject can infer no information about coin C1 on another day, which is the mechanism for fault in that version. Each day, in the “world” of the subject, is a fully independent “world” with a mathematically valid sample space that applies to it alone.
“It treats sequential events as mutually exclusive,” No, it treats an observation of a state, when that observation bears no connection to any other, as independent of any other.
“… therefore unlawfully constructs sample space.” What law was broken?
Do you disagree that, on the morning of the observation, there were four equally likely states? Do you think the subject has some information about how the state was observed on another day? That an observer from the outside world has some impact on what is known on the inside? These are the kind of details that produce controversy in the Always-Monday-Tails-Tuesday version. I personally think the inferences about carrying information over between the two days are all invalid, but what I am trying to do is eliminate any basis for doing that.
Yes, each outcome on the first day can be paired with exactly one on the second. But without any information passing to the subject between these two days, she cannot do anything with such pairings. To her, each day is its own, completely independent probability experiment. One where “new information” means she is awakened to see only three of the four possible outcomes.
“Your model treats HH, HT, TH and TT as four individual mutually exclusive outcomes” No, it treats the current state of the coins as four mutually exclusive states.
“However, when you actually do it, you get a different list.”
How so? If your write down the state on the first day that the researchers look at the coins, you will find that {HH, TH, HT, TT} all occur with frequency 1⁄4. Same on the second day. If you write down the frequencies when the subject is awake, you find that {TH, HT, TT} all have frequency 1⁄3.
Here is what you are arguing: Say you repeat this many times and make two lists, one for each day. And each entry on the lists includes the attempt number. You will find that every attempt where the first list says TH, the second list will say TT. And that every attempt where the first list says HT, the second list will not have an entry for that attempt.
What I’m saying is that the subject’s belief cannot cross-correlate the attempt numbers. She could use just the first list with the attempt numbers. Or the second, since each when viewed in isolation has the same properties as the other and so gets the same answer. She can even use a combined list, but if she does she cannot use the attempt numbers to associate the two observations on that attempt. That is what the amnesia drug accomplishes.
What I am saying is that regardless of which list she uses, the probability of heads is 1⁄3. And your arguments that this is wrong require associating the attempts, essentially removing the effect of amnesia. That cannot influence the subject’s belief.
Let it be not two different days but two different half-hour intervals. Or even two milliseconds—this doesn’t change the core of the issue that sequential events are not mutually exclusive.
It very much bears a connection. If you are observing state TH it necessary means that either you’ve already observed or will observe state TT.
The definition of a sample space—it’s supposed to be constructed from mutually exclusive elementary outcomes.
Disagree on both accountsd. You can’t treat HH HT TT TH as individual outcomes and the term “morning of observation” is underspecified. The subject knows that some of them happen sequentially.
I noticed, and I applaud your attempts. But you can’t do that because you still have sequential events, anyway, the fact that you call them differently doesn’t change much.
Exactly. And the Beauty knows it. Case closed.
She knows that they do not happen at random. This is enough to be sure that each day is not completely independent probability experiment. See Effects of Amnesia section.
Call them “states” if you want. It doesn’t change anything.
I’ve specifically explained how. We write down outcomes when the researcher sees the Beauty awake—when they updated on the fact of Beauty’s awakening. The frequency for three outcomes is 1⁄3, moreover they actually go in random order because the observer witnesses only one random awakening per experiment.
Yep, no one is arguing with that. The problem is that the order isn’t random as your model predicts—TH and TT always go in pairs.
No, I’m not complicating this with two lists for each day. There is only one list, which documents all the awakenings of the subject, while she is going through the series of experiments. The theory that predicts that two awakening are “completely independent probability experiments” expect that the order of the awakenings is random and it’s proven wrong because there is an order between awakenings. Easy as that.
You are mistaken about what the amnesia acomplishes. Once again I send you to reread the Effects of Amnesia section. It’s equally applicable to Two-Coin version of the problem as a regular one.
According to Beauty’s knowledge, the attempts are already connected. Only if the amnesia removed from her mind the setting of the experiment if she forgot that TT and TH go in pairs, only then she should reason the way you want her to.
On the other hand, if we trully removed the effect of amnesia alltogether, then the Beauty would be 100% confident in Tails when awaken the second time in the same experiment.
So no, I’m talking about the exact knowledge state of the Beauty with the exact level of amnesia that she gets, while you your are talking about a more significant alteration of her mind.
OUTCOME: A measurable result of a random experiment.
SAMPLE SPACE: a set of exhaustive, mutually exclusive outcomes of a random experiment.
EVENT: Any subset of the sample space of a random experiment.
INDEPENDENT EVENTS: If A and B are events from the same sample space, and the occurrence of event A does not affect the chances of the occurrence of event B, then A and B are independent events.
The outside world certainly can name the outcomes {HH1_HT2, HT1_HH2, TH1_TT2, TT1_TH2}. But the subject has knowledge of only one pass. So to her, only the current pass exists, because she has no knowledge of the other pass. What happens in that interval can play no part in her belief. The sample space is {HH, HT, TH, TT}.
To her, these four outcomes represent fully independent events, because she has no knowledge of the other pass. To her, the fact that she is awake means the event {HH} has been ruled out. It is still a part of the sample space, but is is one she knows is not happening. That’s how conditional probability works; the sample space is divided into two subsets; one is consistent with the observation, and one is not.
What you are doing, is treating HH (or, in Elga’s implementation, H&Tuesday) as if it ceases to exist as a valid outcome of the experiment. So HH1_HT2 has to be treated differently than TT1_TH2, since HH1_HT2 only “exists” in one pass, while TT1_TH2 “exists” in both. This is not true. Both exist in both passes, but one is unobserved in one pass.
And this really is the fallacy in any halfer argument. They treat the information in the observation as if it applies to both days. Since H&Tuesday “doesn’t exist”, H&Monday fully represents the Heads outcome. So to be consistent, T&Monday has to fully represent the Tails outcome. As does T&Tuesday, so they are fully equivalent.
You are projecting the result you want onto the process. Say I roll a six-sided die tell you that the result is odd. Then I administer the amnesia drug, and tell you that I previously told you whether th result was even or odd. I then ask you for your degree of belief that the result is a six. Should you say 1⁄6, because as far as you know the sample space is {1,2,3,4,5,6}? Or should you say 0, because “you are [now] observing a state that you’ve already observed is only {1,3,5}?
And if you try to claim that this is different because you don’t know what I told you, that is exactly the point of the Two Coin Version.
It is so constructed.
Beauty is doing the updating, not “they.” She is in an experiment where there are four possible combinations for what the coins are currently showing. She has no ability to infer/anticipate what the coins were/will be showing on another day.
Her observation is that one combination, of what is in the sample space for today, is eliminated.
Maybe you missed the part where I said you can look at one, or the other, or bot as long as you don’t carry information across.
Then you are mistaken in that section.
Her belief can be based only on what she knows. If you create a difference between the two passes, in her knowledge, then maybe you could claim a dependence. I don’t think you can in this case, but to do it requires that difference.
The Two Coin Version does not have a difference. Nothing about what she observed about the outcomes HH1_HT2 or HT1_HH2 in another pass can affect her confidence concerning them in the current pass. (And please, recall that these describe the combinations that are showing.)
The link I use to get here only loads the comments, so I didn’t find the “Effects of Amnesia” section until just now. Editing it:
“But in my two-coin case, the subject is well aware about the setting of the experiment. She knows that her awakening was based on the current state of the coins. It is derived from, but not necessarily the same as, the result of flipping them. She only knows that this wakening was based on their current state, not a state that either precedes or follows from another. And her memory loss prevents her from making any connection between the two. As a good Bayesian, she has to use only the relevant available information that can be applied to the current state.”
This is the crux of our disagreement.
The Beauty doesn’t know only about one pass she knows about their relation as well. And because of it she can’t reason as if they happen at random. You need to address this point before we could move on, because all your further reasoning is based on the incorrect premise that beauty knows less than she actually knows.
She absolutely has this ability as long as she knows the procedure, that TT and TH follow in pairs, she can make such conditional statements: “if the coins are currently TT then they either will be TH tomorrow or were TH yesterday”. It’s very different from not knowing anything whatsoever about the state of the coin on the next day. The fact that you for some reason feel that it should not matter is irrelevant. It’s still clearly more than no information whatsoever and, therefore, she can’t justifiably reason as if she doesn’t have any.
On the other hand, if the memory wipe removed this knowledge from her head as well, if the only thing she truly knew was that she is currently awakened at one of three possible states either TH, HT and TT, and had no idea of the relationship between them, then, only then, she would be justified to reason as you claim she should.
No, I treat is as an event that Beauty doesn’t expect to observe and therefore she doesn’t update when she indeed doesn’t observe it according to the law of conservation of expected evidence. We are talking about Beauty’s perspective after all, not a some outside view.
Suppose an absolutely trustwothy source tells you that the coin is Heads side up. Then you go and look at the coin and indeed it’s Heads side up. What should have been your probability that the coin is Tails side up before you looked at it?
It should be zero. You’ve already known the state of the coin before you looked at it, you got no new information. Does it mean that Tails side of a coin doesn’t exist? No, of course not! It just that you didn’t expect that the coin could possibly be Tails in this particular case based on your knowledge state.
I was going to post a generalized way of reasoning under amnesia in a future post, but here is some: getting memory erased about some evidence just brings you to the state where you didn’t have this particular evidence. And getting an expected memory wipe can only make you less confident in your probability estimate, not more.
In this dice rolling case, initially my P(6) = 1⁄6, then after you told me that it’s odd, P(6|Odd)=0, and then when I’m memory wiped I’m back to P(6) = 1⁄6 and the knowledge that you’ve already told me whether the result is even or odd doesn’t help P(6|Even or Odd) = 1⁄6
Likewise in Sleeping Beauty I initially have P(Heads) = 1⁄2. Then I awakened exactly as I’ve expected in the experiment and still have P(Heads|Awake) = 1⁄2. Now suppose that I’m awakened once more. If there was no memory wipe I’d learn that I’m a awake a second time which would bring me to P(Heads|Two Awakenings) = 0. But I do not get this evidence due to memory wipe. So due to it, when I’m awakened the second time, I once again learn that I’m awake and still having P(Heads|Awake) = 1⁄2.
What you are implicitly claiming, however, is that getting memory wiped, or even just a possibility of it, makes the Beauty more confident in one outcome over the other! Which is quite bizarre. As if knowing less gives you more knowledge. Moreover, you assume that the person who knowns that their memory was/may be erased, just have to act as if they do not know it.
Suppose a coin is tossed and you received some circumstantial evidence about it’s state. As a result you are currently at 2⁄3 in favor of Heads. And then I tell you: “What odds are you ready to bet on? By the way, I have erased from your memory some crucial evidence in favor of Tails”. Do you really think that you are supposed to agree to bet on 1:2 odds even though you now know that the state of the evidence your currently have may not be trustworthy?