Indexical uncertainty and subjective randomness

Question

 @user1247 wrote in the context of a many worlds discussion

given that indexical uncertainty in the space of all worlds leads to subjective randomness. 

I'd like to have an explanation of what this means and in which sense the first leads to the other. 

Comments

@dimension10: But only if they are on a graduate+ level, which means, argued mathematically or experimentally. For example, Everett's paper introducing the MWI (and the Ph.D. thesis underlying it) is physics, published in a physics journal, and one can argue about its deductions in a logical way. Thus this question is fine for Q&A. But indexical uncertainty is a purely philosophical concept, and the conclusions have nothing to do with physics as a science.

Answer 1

Indexical uncertainties arise in philosophical discussions about Boltzmann brains, multiverse scenarios, cloning thought experiments, and Doomsday arguments, and really any case where you consider an observer trying to use Bayesian reasoning to determine his or her reference class. Other key-words are "self-locating uncertainty", "anthropic self-selection", "self-sampling", and here and here are a couple wikipedia articles that touch on the subject. A typical example of how it leads to subjective randomness in a deterministic scenario:

Suppose you enroll in an experiment in which you are blindfolded, taken into a room, and cloned. The clone always ends up in a red room, the original in a blue room. After the cloning operation you are then asked to remove your blindfold. At this point you have "indexical uncertainty" about whether you are the clone or the original, and so you predict a 50% chance of being in a red or blue room. Indeed, if you repeat the experience many times, you will find yourself in a red or blue room with a purely random 50% probability. 

Now there are many puzzles and questions and debates about all of this, but as far as I know there is little debate about the basic concept, that subjectively pure randomness can arise from purely deterministic cloning scenarios. And of course in the MWI the "clones" are just branches of a universal wave function. 

Comments

Thanks for the answer. Let me challenge the basic concept, that subjectively pure randomness can arise from purely deterministic cloning scenarios - if there hasn't been a debate about it so far (the concept is very young!) then there is one now. 

After the cloning operation, there will be two selves, not one, since the self is of course being cloned, too. The original self will thus find itself with 100% probability in the blue room, the cloned self will find itself with 100% probability in the red room. Repeating the experiment $n$times with some of the intermediate selves will produce up to $2^n$ different selves, and the history of each one will be determined with 100% probability by the binary sequence indicating at which experiment it was the original or the cloned self. Each self will get a different probability statistics (the only statistics available to the self), and it depends on the process selecting for cloning whether in the long run these statistics converge, and to which distribution.

Now you might understand better my answer in the other thread - you may interpret each 1 as being cloned, and each 0 as being original after the corresponding cloning experiment.

In terms of MWI, to turn the philosophy into physics you need to answer my question there: 

What is the probability distribution on the space of all worlds underlying Everett's interpretation? Without an assumed distribution, nothing definite follows.  

More precisely, you'd have to specify a stochastic process that specifies the branching behavior at every time $t$. Only then one can begin to consider the physics behind the philosophical ''indexical uncertainty" argument.

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@Arnold, I'm afraid I still cannot make out what your argument is. First of all let's stick to the cloning example, since your objection to it is more fundamental than your objection about branching behavior in MWI. For the cloning example you didn't supply any new argument, you simply explained something self-evident and then linked to your answer in another thread. Your answer in the other thread seems to be doing nothing more than asserting that the underlying process is deterministic and that therefore there can be no subjective randomness. And this just misses the point to such a degree that you are not engaging at all with the proposition under discussion. 

Maybe it will help to consider a concrete example. The experiment consists of three cloning repetitions, with each participant (both the clone and the original) re-enrolling at each step. There are eight possible histories:

[111],[110],[011],[101],[100],[010],[001],[000], where '1' stands for red room, and '0' for blue room. Note that in 25% of histories the participant finds himself in a room of constant color, while in 75% of histories, the color ratio is 2:1.

From a "global perspective", of course the process is entirely deterministic. That is the point. But from the perspective of a participant in the experiment the situation is entirely different. At the end of the experiment 25% percent of participants will report that the room color was constant after each step, while 75% will report the color changing once. 

Now if you are a scientist who participates in this experiment you might try to tell for yourself whether your history of "red room, blue room" is deterministic or random. So you hypothesize that if you do the experiment many times you can do a statistical test to check whether the room color changes are consistent with a binomial distribution and if so with what p. Well if you enroll in the experiment many times and do the counting it is clear that the majority of scientist participants will find themselves with histories statistically consistent with a binomial distribution with p=0.5. So for the majority of scientists who are actually subjectively experiencing this cloning scenario, they will find that their experiences are described by a purely random probability distribution.

Of course, like any probability distribution (deterministic or not) there will be outliers. Some scientists will see X$\sigma$ fluctuations that later regress to the mean, and a very very few will see a deterministic world in which the room is always one color. But the treatment here is no different from any other statistical assessment (for example ordinary QM ala Copenhagen) -- some scientists will be very "lucky" but the vast majority will have experiences that are consistent with the "purely random" hypothesis. This is born out by simply counting of histories or knowing about binomial distributions.  

It would be helpful if you clarified where, exactly, in all of this, do you disagree? Do you disagree that a scientist enrolling in such an experiment will on average have a history that leads him to conclude the room changes color purely randomly with binomial p=0.5? Or do you have a philosophical objection that the clones are not "conscious"? You need to explain clearly exactly where you confusion lies, because currently you have not given me enough to be able to say.

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@user1247:

At the end of the experiment 25% percent of participants will report that the room color was constant after each step, while 75% will report the color changing once. 

Yes. A scientist observing the whole experiment will see the uniform distribution of the experimental result reproduced by the reports of most participants, so what? There is nothing subjective about it. If the experimenters would have chosen a different experimental distribution, this different distribution would have been observed, equally objective. The experimental setup completely determines the resulting most typical probabilities, and any set of probabilities can be obtained by picking appropriate rules for whom to clone when. This is what I meant in my answer with 

What is the probability distribution on the space of all worlds underlying Everett's interpretation? Without an assumed distribution, nothing definite follows.  

Note that you silently assumed the uniform distribution of the experiment, but you concluded the subjective 50% probability as a nontrivial result of your argumentation. It is this kind of fallacy that becomes apparent once one turns from mere philosophy to physics.

Note also that the cloning scenario is very different from the MWI setting, where only a single trajectory of cloned universes can be observed by mankind as a whole. I was talking about this single trajectory - it can have an arbitrary distribution completely unrelated to the distribution of the experiment, any distribution is as permissible as any other, and since one cannot observe any other trajectory, there is no way of assigning a meaningful experimental probability to which splits are chosen with which frequency by the assumed experimenter external to the universe. 

Taking your silently assumed uniform prior as an admissible a priori hypothesis, I should conclude that we live in an extraordinarily exceptional trajectory since rather than seeing everything happen at random we observe marvellous regularities that keeps busy generations of physicists. Knowing that our trajectory is so extraordinarily exceptional, it is clear that we cannot deduce the slightest thing from presumed indexical uncertainties, as these can only cover the most likely cases, and we have proof that our universe is far too regular for this.

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@Arnold, first of all, you continue to obfuscate the discussion by bringing in your objections to the MWI, which is putting the cart before the horse. Please stop until we clarify the more fundamental issue. You are already causing this discussion to nearly derail, as I am losing patience (you did not even answer my very direct questions).

Yes. A scientist observing the whole experiment will see the uniform distribution of the experimental result reproduced by the reports of most participants, so what? There is nothing subjective about it.

You are continuing to misunderstand. The point is not only that a scientist observing the whole experiment from the outside will see the binomial distribution of histories reported, but more to the point is that the experimental participants will report themselves experiencing those histories. What the scientist from the outside can conclude is that the participants, as reported by them, are subjectively experiencing pure randomness with statistics exactly consistent with a binomial distribution with p=0.5. The situation is no different, scientifically, from if those participants had instead measured whether a given radioactive atom had decayed during one interval of its half life, or any other quantum or assumed purely random process with p=0.5. In such a case they will also report experiences with the same statistical distribution. Therefore the two situations are empirically indistinguishable. 

Note that you silently assumed the uniform distribution of the experiment, but you concluded the subjective 50% probability as a nontrivial result of your argumentation. It is this kind of fallacy that becomes apparent once one turns from mere philosophy to physics.

This is simply wrong. We apparently agree (see above) on what a scientist will see being reported to him. Such reports are indistinguishable from the uniform distribution you say I assumed. But this is where it would be helpful for you to directly answer my questions (such as "Do you disagree that a scientist enrolling in such an experiment will on average have a history that leads him to conclude the room changes color purely randomly with binomial p=0.5?").

Taking your silently assumed uniform prior as an admissible a priori hypothesis, I should conclude that we live in an extraordinarily exceptional trajectory since rather than seeing everything happen at random we observe marvellous regularities that keeps busy generations of physicists. Knowing that our trajectory is so extraordinarily exceptional, it is clear that we cannot deduce the slightest thing from presumed indexical uncertainties, as these can only cover the most likely cases, and we have proof that our universe is far too regular for this.

This is so wrong and confused and misinformed that I think it best to wait until clarifying the above.

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"Do you disagree that a scientist enrolling in such an experiment will on average have a history that leads him to conclude the room changes color purely randomly with binomial p=0.5?"

 Ah, the test persons are supposed to be the scientists. In this case, the (for definiteness male) scientist can perform an average only on the frequency of the colors he sees. Again, there is nothing subjective about it. The uncloned scientist will report only blue walls, and the scientists that have been subjected to k out of n cloning operations will report, objectively, a probability of $k/n$. If you ask only one scientist you'll get exactly one answer, which might be anything between 0 and 100%, but if n is odd, never exactly 50%.

This answer is an objective fact; in the scientist's report there is no subjective probabilities anywhere. Of course it depends on whom you cask what the resulting statistics will be. But this has nothing to do with subjective probability, it is rather standard conditional probability. Experiments under different conditions will give different results if the conditions affect the experiment.

You argue that it is most likely 50% since this is the peak of the binomial distribution with $p=0.5$ and a large number of trials. But this kind of likelihood has nothing to do with the probabilities the scientist can objectively record without having access to his clones.

These 50% are therefore only your subjecive probability that you assign knowing the cloning procedure. It is nothing the scientist can deduce from his color measurements. He measures a fact - an observed, objective probability.

Only you, who embed his observations into the cloning context, need subjective probabilities, since you argue from an assumed process to the outcome of a single trajectory of this process. And you want to argue from a single observed case for a probability law. But this is impossible to do either of the two in a rational manner, hence subjectivity enters. Rational probability statements never apply to a single case. The reason is that the meaning of probability is intrinsically tied to the possibility of frequently repeating the case under similar conditions.

Moreover, and that was my whole point, this number 50% completely depends on the details how the experiments are planned. If you modify the experiment so that the person in the room is cloned with a probability p only before being released, you don't get a symmetric binomial distribution, and the majority of scientists will report a probability different from 50%. or if you change the rule of how to color the walls. Thus which probability is most likely observed by a single scientist will depend completely upon the stochastic process used to generate the colors.

In your case, the cloning experiment is deterministic, it produces $2^n$ trajectories on equal footing. The randomness comes in through which person you ask. If you ask only one, you cannot attach rationally a probability to it. So you retreat to the uninformative prior, claiming that you chose this person uniformly from the set of all persons. A bold claim for a single choice, not supported by any underlying mathematics.

Let us return to the perspective of a single scientist. No matter which probability the scientist finds when he makes a statistics of the colors in his rooms it is an objective probability in the traditional sense of the word, number of good cases divided by number of all cases. Moreover, if as a scientist, he makes a theory about how the colors change in his room, the best theory he can actually test is to set up a stochastic or deterministic process for his changing walls. If he in addition speculates about any possible cloning that might have been going on, this is not science but pure speculation. For to make a prediction that explains his observed deterministic or stochastic trajectory (the only objective information about his current field of study), he must postulate [without the slightest evidence for it]

1. a mysterious cloning process that copies every detail about the past (in his memory, his notebook, his friends, etc.) except for changing the next moment.
2. a second deterministic or stochastic process for cloning - which will surely be at least as complex as the stochastic model he'd use to describe his single time series.
3. The assumption that his cloning observation is typical for cloning, i.e., that the most likely family of trajectories contains his own trajectory (whatever it is),
4. the mystery that all the generated alternatives in the cloning process are completely unobservable.

And as the scientific fruit of these assumptions he gets an ''explanation'' for his stochastic trajectory that caries no additional information beyond the stochastic model used for explaining his single time series without cloning. Thus he will conclude that it is most rational to drop the mysteries invoking cloning, following Ockham's famous principle.

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If you want specific questions answered, please post each question separately as a comment (preferably one by one, so that iknow which one is the most important one for you). In my previous (not the last) mail I had answered to your question

It would be helpful if you clarified where, exactly, in all of this, do you disagree?

because I thought this was the most basic one, and then you complained that I had not answered the other ones.

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@arnold

Rational probability statements never apply to a single case. The reason is that the meaning of probability is intrinsically tied to the possibility of frequently repeating the case under similar conditions.

You seem to be continuing to just completely misunderstand the point of the thought experiment. We are not talking about a single case. The experiment can be repeated as many times as a scientist/participant want to, it is exactly the same in every way as if the scientist were doing experiments on a quantum observable. If 100 scientists measure whether an atom has decayed during its halflife, some scientists will find that it didn't decay, some will find that it did, and if you take a census of the scientists, the distribution will be binomial with p=0.5. How is this any different from the example with the clones? All you have to do is work out an example. Do it for yourself on paper. It. Is. Exactly. The. Same.

Moreover, and that was my whole point, this number 50% completely depends on the details how the experiments are planned. If you modify the experiment so that the person in the room is cloned with a probability p only before being released, you don't get a symmetric binomial distribution, and the majority of scientists will report a probability different from 50%. or if you change the rule of how to color the walls. Thus which probability is most likely observed by a single scientist will depend completely upon the stochastic process used to generate the colors.

yes... what is your point? Of course p=0.5 was just an example. If you clone the person three times into a red blue and green room, then p=1/3, etc etc. 

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In your case, the cloning experiment is deterministic, it produces 2n trajectories on equal footing. The randomness comes in through which person you ask. If you ask only one, you cannot attach rationally a probability to it. So you retreat to the uninformative prior, claiming that you chose this person uniformly from the set of all persons. A bold claim for a single choice, not supported by any underlying mathematics.

Again, your position can immediately seen to be wrong, since you can apply the exact same logic to taking a census of scientists doing any quantum experiment in the real world. Please think about this carefully.

Let us return to the perspective of a single scientist. No matter which probability the scientist finds when he makes a statistics of the colors in his rooms it is an objective probability in the traditional sense of the word, number of good cases divided by number of all cases. Moreover, if as a scientist, he makes a theory about how the colors change in his room, the best theory he can actually test is to set up a stochastic or deterministic process for his changing walls. If he in addition speculates about any possible cloning that might have been going on, this is not science but pure speculation. For to make a prediction that explains his observed deterministic or stochastic trajectory (the only objective information about his current field of study), he must postulate [without the slightest evidence for it]

The fact that I am not at all understanding you continues to indicate that you simply do not understand the thought experiment itself. Maybe I have not explained it well enough, but there is not much to explain. One thing in the above you seem confused about, and which I thought I made clear given the wording I used, is that the scientist is well aware that he is being cloned. There is no reason whatsoever for him to be in the dark about this. It doesn't matter to the thought experiment.

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If you want specific questions answered, please post each question separately as a comment (preferably one by one, so that iknow which one is the most important one for you). In my previous (not the last) mail I had answered to your question

No you did not. I had asked:

It would be helpful if you clarified where, exactly, in all of this, do you disagree? Do you disagree that a scientist enrolling in such an experiment will on average have a history that leads him to conclude the room changes color purely randomly with binomial p=0.5? Or do you have a philosophical objection that the clones are not "conscious"? You need to explain clearly exactly where you confusion lies, because currently you have not given me enough to be able to say

You did not quote any of the above or respond to any of it. You ignored it completely.