Of course, whether I am the Orignal or the Clone is a matter of fact. There is a definitive answer to it. I also see no problem saying “the probability I am the Original” is essentially the same as ” What is the probability, given that he answers truthfully, that the scientist will say I am the original when I ask? ”.
But does being a matter of fact imply there is a probability to it? Subsequently, what is the justification for 50% being the correct answer?
I’m using “probability” “credence” and “betting odds” mostly interchangeably to refer to my subjective state of knowledge. The 50% number comes from the symmetry of having two indistinguishable experiences being had by two people, one of which is me. Without any additional information to break that symmetry (such as learning that multiple copies were created or that the copying sometimes fails), I should assign equal credence to each possibility.
(to be clear, everything I’ve said also flows from the principle of indifference; if you cannot tell the difference between N states of the world, then the probability 1/N describes your uncertainty about those N states)
It’s not that I wouldn’t care which one I am (they’re identical), but there would be no way for me to differentiate the experiences.
Suppose the scientist told me before the procedure that he would with probability 2⁄3 waken the original in a blue room and the clone in a red room and with probability 1⁄3 reverse the colors. If I were to wake up in a blue room afterward, my credence that I’m the original would be 2⁄3.
I was asking if your reasoning for equal probabilities of Original vs Clone can be summarized as the Principle of Indifference. Not suggesting you do not care which copy you are. Would I be wrong to assume you endorse POI in this problem?
Problem is POI is not a solid theory to rely on and it often lead to paradoxes. In anthropic problems in particular, it is unclear what exactly should be regarded indifferent. See this post for an example.
>if this happened lots of times, and you answered randomly, you would be right roughly 50% of the time
How can this be verified? Like I have outlined in Repeating The Experiment, if you keep participating in the experiment again and again there is no reason to think the relative frequency for you being the Original in each experiment would converge to any particular value. Of course, we can select one copy each time and it will converge to 1⁄2. But that would reflect the probability of the random sample being the Orignal is 1⁄2.
It can be asserted that the two probabilities are the same thing. But at least we should recognize that as an additional assumption.
You know that “probability” doesn’t mean “a number which, if we redid the experiment and substituted ‘pick randomly according to this number’ instead of the actual casual mechanism, would give the same distribution of results”? That it’s a summary of knowledge, not a casual mechanism?
(I’m still trying to figure out where I think you’re confused; from my perspective you keep saying “obviously X, that was never in question, but since X is fundamentally different than X, we can’t just assume the same result holds”. Not trying to make fun, just literally expressing my confusion about the stuff you’re writing. In any case, you’re definitely right about not being able to communicate what you’re talking about very well ;) )
You know that “probability” doesn’t mean “a number which, if we redid the experiment and substituted ‘pick randomly according to this number’ instead of the actual casual mechanism, would give the same distribution of results”?
Er… it doesn’t? Doesn’t it mean exactly that, though? As far as we know? I mean, if you say that P(some outcome) = 0.5, then does it not mean that we think that if we ran the experiment a bunch of times, and also flipped a fair coin the same number of times, then the number of times the given outcome would occur would approximately equal the number of heads we got?
:facepalm: I simplified too much, thank you. The second phrasing is what I meant; “it’s a summary of knowledge, not a causal mechanism”. The first should have illustrated what breaks when substituting a summary for the mechanism, which does require something other than just looking at the summaries with nothing else changed. :D
I guess, let me try to recreate what made me write [the above] and see if I can communicate better.
I think what’s going on is that dadadarren is saying to repeat the experiment. We begin with one person, the Original. Then we split, then split each, then split each again, etc. Now we have 2^n people, with histories [Original, Original, …, Original] and [Original, Original, …, Clone] and etc. There will be n*(n-1)/2 people who have participated in the experiment n times and been the Original n/2 times, they have subjectively seen that they came out the Original 50% of the time. But there have also been other people with different subjective impressions, such as the one who was the Original every time. That one’s subjective impression is “Original 100%!”.
But what happens if each person tries to predict what will happen next by using their experimental results (plus maybe a 50% prior) as an expectation of what they think will happen in the next experiment? Then they’ll be much more wrong, collectively, than if they stuck to what they knew about the mechanism than if they plugged in their subjective impression as mechanism. So even the “Original n/(n+1)!” person should assign probability 50% to thinking they’re the Original after the next split; their summary of past observations has no force over how the experiment works, and since they already knew everything about how the experiment works, doesn’t give them any evidence to actually update on.
> I think what’s going on is that dadadarren is saying to repeat the experiment. We begin with one person, the Original. Then we split, then split each, then split each again, etc. Now we have 2^n people, with histories [Original, Original, …, Original] and [Original, Original, …, Clone] and etc. There will be n*(n-1)/2 people who have participated in the experiment n times and been the Original n/2 times, they have subjectively seen that they came out the Original 50% of the time. But there have also been other people with different subjective impressions, such as the one who was the Original every time. That one’s subjective impression is “Original 100%!”.
Ok, slow down here. What you are describing is repeating the experiment, but not from the subject’s first-person perspective. Let’s call this a description from a god’s eye view. There is no “I” in the problem if you describe the experiment this way. Then how do you ask the “probability “I” am the Original?”
What I described in the post is to put yourself inside the subject’s shoes. Imagine you are participating in the experiment from a first-person perspective. Hence, after waking up, you know exactly which one is “I”. Despite there is another copy that is physically indiscernible and you don’t know if you are the Orignal or Clone. This self-identification is primitive.
If this seems convoluted, imagine a case of identical twins. Other people have to differentiate them by some observable features. But for a twin himself, this is not needed. He can inherently tell apart the “I” from the other twin, without needing to know what is the physical difference.
The probability is about “I” being the Original. So in a frequentist analysis, keep the first-person perspective while repeating the experiment. Imagine yourself take part in the same experiment again and again. Focus on “I” throughout these iterations. For your experience, the relative proportion of “I am the Original” has no reason to converge to any value as the iteration increases.
What you are doing is to use the god’s eye model instead. Because there is no “I” in this model, you are substituting “I” with “a random/typical copy”. That’s why I talk about the decision of one person only: the primitively identified “I”. While you are talking about all of the copies as a group. Hence you say “Then they’ll be much more wrong, collectively”
It seems very natural to regard “I” as a randomly selected observer. Doing so will justify self-locating probabilities. Nonetheless, we should recognize that is an additional assumption.
I am undergoing this experiment, repeatedly. The first time I do, there will be two people, both of whom remember prepping for this experiment, both of whom may ask “what is the probability I am the Original?” afterwards, one of whom will unknowingly be the Original, one of whom will unknowingly be the Clone. Subjectively, perhaps I was the Original; in that case if I ask “what is the probability I am the Original?” … okay I’m already stuck. What’s wrong with saying “50%”? Sure, there is a fact of the matter, but I don’t know what it is. In my ignorance, why should I have a problem with saying 50% I’m the original? Certainly if I bet with someone who can find out the true answer I’m going to think my expectation is 0 at 1:1 odds.
But you say that’s meaningless. Fine, let’s go with it. We repeat the experiment. I will focus on “I”. There are 2^n people, but each of them only has the subjective first-person perspective of themselves (and is instructed to ignore the obvious fact that there are another 2^n-1 people in their situation out there, because somehow that’s not “first person” info?? okay). So anyway, there’s just me now, after n experiments. A thought pops up in my head: “am I the Original?” and … well, and I immediately think there’s about a 1/2^n chance I’m the Original, and there’s a 50% chance I’m the first Clone plus n-1 experiments, and there’s a 25% chance I’m the first Clone of the first Clone plus n-2 experiments and a 25% chance I’m the second Clone of the Original plus n-2 experiments and etc.
I have no idea what you mean by “For your experience, the relative proportion of “I am the Original” has no reason to converge to any value as the iteration increases.” Of course it does. It converges to 0% as the number of experiments increases, and it equals 1/2^n at every stage. Why wouldn’t it? You keep saying it doesn’t but your justification is always “in first-person perspective things are different” but as far as I can see they’re not different at all.
Maybe you object to me thinking there are 2^n-1 others around? I’m fine with changing the experiment to randomly kill off one of the two after any experiments so that there’s always only one around. Doesn’t change my first-person perspective answers in the slightest. Still a 1/2^n chance my history was [not-cloned, not-cloned, not-cloned, …] and a 1/2^n chance my history was [cloned, not-cloned, not-cloned, …] and a 1/2^n chance my history was [not-cloned, cloned, not-cloned, …] and a … etc.
You say you are using the first-person perspective to answer the probability “I am the Original”, and focusing on yourself in the analysis. However, you keep bring up there are two copies. That “one is the original, the other is the clone.” So the probability “I am the Original” is 50%.
Do you realize that you are equating “I” with “a random one of the two” in this analysis? There is an underlying assumption of “I am a random sample” or “I am a typical observer” here.
For repeating the experiment, I am talking about being the Original in each iteration. You may come out as the Clone from the first experiment. You can still participate in a second experiment, after waking up from the second experiment, you may be the Original (or the Clone) of the second experiment. And no matter which one you are, you can take part in a third experiment. You can come out of the third experiment as the Orignal (or the Clone) of the third experiment. And so on. Keep doing this, and keep counting how many times you came out as the Orignal vs the Clone. What is the rationale that they will become roughly equal? I.E. as you repeat more experiments you will experience being the Original roughly half of the time. Again, the justification would be “I” am a random copy.
I am not saying the existence of other copies must be ignored. I am saying if you reason from the first-person perspective, imagine yourself waking up from the experiments, then it is primitively clear all other copies are not the “I” or “myself” questioned by self-locating probability. Because you are very used to take the god’s eye view and consider all copies together (and treating “I” as a random sample of all copies) I suggested to not pay attention to anyone else but imagine you as a participant, and focus on yourself. But evidently, this doesn’t work.
It is a tricky matter to communicate for sure. If this still seems convoluted maybe I shall use examples with solid numbers and bets to highlight the paradox of self-locating probability. Would you be interested in that?
Of course, whether I am the Orignal or the Clone is a matter of fact. There is a definitive answer to it. I also see no problem saying “the probability I am the Original” is essentially the same as ” What is the probability, given that he answers truthfully, that the scientist will say I am the original when I ask? ”.
But does being a matter of fact imply there is a probability to it? Subsequently, what is the justification for 50% being the correct answer?
I’m using “probability” “credence” and “betting odds” mostly interchangeably to refer to my subjective state of knowledge. The 50% number comes from the symmetry of having two indistinguishable experiences being had by two people, one of which is me. Without any additional information to break that symmetry (such as learning that multiple copies were created or that the copying sometimes fails), I should assign equal credence to each possibility.
Would you say your reasoning is a principle of indifference between “I am the Original” vs “I am the Clone”?
(to be clear, everything I’ve said also flows from the principle of indifference; if you cannot tell the difference between N states of the world, then the probability 1/N describes your uncertainty about those N states)
It’s not that I wouldn’t care which one I am (they’re identical), but there would be no way for me to differentiate the experiences.
Suppose the scientist told me before the procedure that he would with probability 2⁄3 waken the original in a blue room and the clone in a red room and with probability 1⁄3 reverse the colors. If I were to wake up in a blue room afterward, my credence that I’m the original would be 2⁄3.
I was asking if your reasoning for equal probabilities of Original vs Clone can be summarized as the Principle of Indifference. Not suggesting you do not care which copy you are. Would I be wrong to assume you endorse POI in this problem?
I would say POI applies here.
Problem is POI is not a solid theory to rely on and it often lead to paradoxes. In anthropic problems in particular, it is unclear what exactly should be regarded indifferent. See this post for an example.
Justifications for 50% being the correct answer:
if this happened lots of times, and you answered randomly, you would be right roughly 50% of the time
if you tried to make a wager, a bookie would give you near-1:1 odds
50% is the correct answer to all of the equivalent questions which you accept are probabilities
:shrug:
>if this happened lots of times, and you answered randomly, you would be right roughly 50% of the time
How can this be verified? Like I have outlined in Repeating The Experiment, if you keep participating in the experiment again and again there is no reason to think the relative frequency for you being the Original in each experiment would converge to any particular value. Of course, we can select one copy each time and it will converge to 1⁄2. But that would reflect the probability of the random sample being the Orignal is 1⁄2.
It can be asserted that the two probabilities are the same thing. But at least we should recognize that as an additional assumption.
What? Yeah, still missing something.
You know that “probability” doesn’t mean “a number which, if we redid the experiment and substituted ‘pick randomly according to this number’ instead of the actual casual mechanism, would give the same distribution of results”? That it’s a summary of knowledge, not a casual mechanism?
(I’m still trying to figure out where I think you’re confused; from my perspective you keep saying “obviously X, that was never in question, but since X is fundamentally different than X, we can’t just assume the same result holds”. Not trying to make fun, just literally expressing my confusion about the stuff you’re writing. In any case, you’re definitely right about not being able to communicate what you’re talking about very well ;) )
Er… it doesn’t? Doesn’t it mean exactly that, though? As far as we know? I mean, if you say that P(some outcome) = 0.5, then does it not mean that we think that if we ran the experiment a bunch of times, and also flipped a fair coin the same number of times, then the number of times the given outcome would occur would approximately equal the number of heads we got?
:facepalm: I simplified too much, thank you. The second phrasing is what I meant; “it’s a summary of knowledge, not a causal mechanism”. The first should have illustrated what breaks when substituting a summary for the mechanism, which does require something other than just looking at the summaries with nothing else changed. :D
I guess, let me try to recreate what made me write [the above] and see if I can communicate better.
I think what’s going on is that dadadarren is saying to repeat the experiment. We begin with one person, the Original. Then we split, then split each, then split each again, etc. Now we have 2^n people, with histories [Original, Original, …, Original] and [Original, Original, …, Clone] and etc. There will be n*(n-1)/2 people who have participated in the experiment n times and been the Original n/2 times, they have subjectively seen that they came out the Original 50% of the time. But there have also been other people with different subjective impressions, such as the one who was the Original every time. That one’s subjective impression is “Original 100%!”.
But what happens if each person tries to predict what will happen next by using their experimental results (plus maybe a 50% prior) as an expectation of what they think will happen in the next experiment? Then they’ll be much more wrong, collectively, than if they stuck to what they knew about the mechanism than if they plugged in their subjective impression as mechanism. So even the “Original n/(n+1)!” person should assign probability 50% to thinking they’re the Original after the next split; their summary of past observations has no force over how the experiment works, and since they already knew everything about how the experiment works, doesn’t give them any evidence to actually update on.
> I think what’s going on is that dadadarren is saying to repeat the experiment. We begin with one person, the Original. Then we split, then split each, then split each again, etc. Now we have 2^n people, with histories [Original, Original, …, Original] and [Original, Original, …, Clone] and etc. There will be n*(n-1)/2 people who have participated in the experiment n times and been the Original n/2 times, they have subjectively seen that they came out the Original 50% of the time. But there have also been other people with different subjective impressions, such as the one who was the Original every time. That one’s subjective impression is “Original 100%!”.
Ok, slow down here. What you are describing is repeating the experiment, but not from the subject’s first-person perspective. Let’s call this a description from a god’s eye view. There is no “I” in the problem if you describe the experiment this way. Then how do you ask the “probability “I” am the Original?”
What I described in the post is to put yourself inside the subject’s shoes. Imagine you are participating in the experiment from a first-person perspective. Hence, after waking up, you know exactly which one is “I”. Despite there is another copy that is physically indiscernible and you don’t know if you are the Orignal or Clone. This self-identification is primitive.
If this seems convoluted, imagine a case of identical twins. Other people have to differentiate them by some observable features. But for a twin himself, this is not needed. He can inherently tell apart the “I” from the other twin, without needing to know what is the physical difference.
The probability is about “I” being the Original. So in a frequentist analysis, keep the first-person perspective while repeating the experiment. Imagine yourself take part in the same experiment again and again. Focus on “I” throughout these iterations. For your experience, the relative proportion of “I am the Original” has no reason to converge to any value as the iteration increases.
What you are doing is to use the god’s eye model instead. Because there is no “I” in this model, you are substituting “I” with “a random/typical copy”. That’s why I talk about the decision of one person only: the primitively identified “I”. While you are talking about all of the copies as a group. Hence you say “Then they’ll be much more wrong, collectively”
It seems very natural to regard “I” as a randomly selected observer. Doing so will justify self-locating probabilities. Nonetheless, we should recognize that is an additional assumption.
Okay, let me try again, then.
I am undergoing this experiment, repeatedly. The first time I do, there will be two people, both of whom remember prepping for this experiment, both of whom may ask “what is the probability I am the Original?” afterwards, one of whom will unknowingly be the Original, one of whom will unknowingly be the Clone. Subjectively, perhaps I was the Original; in that case if I ask “what is the probability I am the Original?” … okay I’m already stuck. What’s wrong with saying “50%”? Sure, there is a fact of the matter, but I don’t know what it is. In my ignorance, why should I have a problem with saying 50% I’m the original? Certainly if I bet with someone who can find out the true answer I’m going to think my expectation is 0 at 1:1 odds.
But you say that’s meaningless. Fine, let’s go with it. We repeat the experiment. I will focus on “I”. There are 2^n people, but each of them only has the subjective first-person perspective of themselves (and is instructed to ignore the obvious fact that there are another 2^n-1 people in their situation out there, because somehow that’s not “first person” info?? okay). So anyway, there’s just me now, after n experiments. A thought pops up in my head: “am I the Original?” and … well, and I immediately think there’s about a 1/2^n chance I’m the Original, and there’s a 50% chance I’m the first Clone plus n-1 experiments, and there’s a 25% chance I’m the first Clone of the first Clone plus n-2 experiments and a 25% chance I’m the second Clone of the Original plus n-2 experiments and etc.
I have no idea what you mean by “For your experience, the relative proportion of “I am the Original” has no reason to converge to any value as the iteration increases.” Of course it does. It converges to 0% as the number of experiments increases, and it equals 1/2^n at every stage. Why wouldn’t it? You keep saying it doesn’t but your justification is always “in first-person perspective things are different” but as far as I can see they’re not different at all.
Maybe you object to me thinking there are 2^n-1 others around? I’m fine with changing the experiment to randomly kill off one of the two after any experiments so that there’s always only one around. Doesn’t change my first-person perspective answers in the slightest. Still a 1/2^n chance my history was [not-cloned, not-cloned, not-cloned, …] and a 1/2^n chance my history was [cloned, not-cloned, not-cloned, …] and a 1/2^n chance my history was [not-cloned, cloned, not-cloned, …] and a … etc.
Ok.
You say you are using the first-person perspective to answer the probability “I am the Original”, and focusing on yourself in the analysis. However, you keep bring up there are two copies. That “one is the original, the other is the clone.” So the probability “I am the Original” is 50%.
Do you realize that you are equating “I” with “a random one of the two” in this analysis? There is an underlying assumption of “I am a random sample” or “I am a typical observer” here.
For repeating the experiment, I am talking about being the Original in each iteration. You may come out as the Clone from the first experiment. You can still participate in a second experiment, after waking up from the second experiment, you may be the Original (or the Clone) of the second experiment. And no matter which one you are, you can take part in a third experiment. You can come out of the third experiment as the Orignal (or the Clone) of the third experiment. And so on. Keep doing this, and keep counting how many times you came out as the Orignal vs the Clone. What is the rationale that they will become roughly equal? I.E. as you repeat more experiments you will experience being the Original roughly half of the time. Again, the justification would be “I” am a random copy.
I am not saying the existence of other copies must be ignored. I am saying if you reason from the first-person perspective, imagine yourself waking up from the experiments, then it is primitively clear all other copies are not the “I” or “myself” questioned by self-locating probability. Because you are very used to take the god’s eye view and consider all copies together (and treating “I” as a random sample of all copies) I suggested to not pay attention to anyone else but imagine you as a participant, and focus on yourself. But evidently, this doesn’t work.
It is a tricky matter to communicate for sure. If this still seems convoluted maybe I shall use examples with solid numbers and bets to highlight the paradox of self-locating probability. Would you be interested in that?
Well, yes, sorry, for the snark, but… obviously! If you know how to make it concrete with numbers instead of wishy-washy with words, please do so!
Alright, please see this post. Which camp you are in? And how do you answer the related problem.