Imagine you are on a ship that has been blown off-course by a storm near the equator. Without consulting the GPS it would be hard to tell which hemisphere you are in. In situations like this, we can assign a probability to this uncertainty. But here we are going to discuss self-locating probabilities of a different nature. In anthropic reasoning, self-locating probability refers to the probability distribution of an indexical in some proposed reference class. To illustrate this, consider the following thought experiment.
A Case of Fission
Imagine you are taking part in this experiment. Tonight during your sleep, an advanced alien Omega would make a clone of you down to the details at the atomic level. The process is highly accurate even memories are preserved. So the created clone would not be able to tell if he is physically new. The two resulting copies would be put into separate rooms. Omega would randomly pick a copy among the two and call him RED. The other one would be called BLUE. The entire procedure is disclosed to you. After waking up the next day, before knowing your label, consider the following questions:
1. What is the probability that RED is the original?
2. What is the probability that “I” am RED?
3. What is the probability that “I” am the original?
Base on my research, almost all camps would assign the same value to the three: 1/2. (The only exception I have found, which also questions the validity of self-locating probabilities, is from the debate of interpretations of quantum mechanics.) These questions are regarded as trivial, their answers obvious. However, I would argue each of them is logically distinct. Most notably, only Question 3 is a self-locating probability. I argue it is perspectively inconsistent, therefore invalid.
The following sections may seem like overanalyzing simple problems. Yet I think this is the best way to illustrate the importance of reasoning from a consistent perspective.
Question 1
“What is the probability that RED is the original?”
Question 1 is formulated as an objective question or formulated from an impartial observer’s perspective. It is understandable to anyone aware of the experiment. It is inherently indifferent to all observers/moments. We can test this by changing the experiment subject from “you” to someone/something else, say “Jerry”, and remove the time “after waking up and knowing the label” altogether. Even so, “What is the probability that RED is the original?” is still conceptually the same question.
Because of this indifference, the particular copy being discussed have to be specified by a sampling process. Selecting RED from both copies is this process. Both the original and the clone take part in this selection. Without any other information, a principle of indifference can be applied among the two. Therefore the probability of RED being the original ought to be half, 1/2.
We can get the same answer by adhering to a frequentist approach, i.e. by conducting the experiment a large number of times and check the relative frequency. Operationally there are many ways to do this. The experiments can be repeatedly performed on the same subject(s), or gather a large population to perform on each individual once, or any combination thereof. It does not matter because the question is indifferent to the subject/time used. It is easy to imagine as the number of experiments increases the relative frequency for RED being the original would approach 1/2.
Questions 2
“What is the probability that “I” am RED?”
For question 2, the topic is regarding the indexical “I”. It is formulated from the experiment subject’s first-person perspective. The problem is incomprehensible to anyone else. So to answer it we have to imagine being the subject. Taking this first-person perspective, after waking up, it is inherently clear who “I” refers to. There is no need to find any objective difference between “me” and “the other copy” to clarify the question because “I” am primitively identified by the perspective center. With this out of the way, the probability of “I” am RED is simple. “I” am in symmetrical positions in the labeling process as “the other copy”. Therefore the probability of “me” being RED ought to be 1/2.
The frequentist approach gives the same conclusion. Because the question is formulated from the first-person perspective of a subject we must also imagine repeating the experiment as a subject. As the subject himself, it is clear who “I” refers to after waking up from the first experiment. Hereon “I” can take part in another iteration of the experiment. After going through the fission process again and waking up the next day, “I” can take part in yet another experiment and so on. Just like in the first experiment where “I” might not refer to the same physical person who fell asleep the night before, here “I” might not refer to the same physical person throughout the repetitions. Nonetheless, it is not relevant to the question since who “I” am is always primitively clear to me. Because in every experiment my label is determined by a random and symmetrical process, as the number of repetitions increases “I” would experience being labeled RED about 1/2 of all times.
In the above analyses, “I” am not considered naturally indifferent to, or in the same reference class as, “the other copy”. “I” am inherently the focus of the analysis. The symmetry between the “I” and “the other copy” is specifically regarding the labeling process. It is worth noting the uncertainty questioned here is due to the labeling process only, not the cloning. It is similar to the storm-near-equator problem earlier. In both cases, uncertainties arise because of a random event, or the lack of information about that event. In these situations, it is uncontroversial to assign probabilities to such uncertainties. Which is different from self-locating probabilities in anthropics, such as question 3.
Question 3
“What is the probability that “I” am the original?”
Question 3 again uses the indexical “I”. We need to think from the first-person perspective of the subject to comprehend it. As before, from this perspective who “I” refers to is inherently clear. However, unlike the labeling event in Question 2, there is no process making “me” the clone or the original. In fact, there cannot be a logical explanation to it because perspective centers, as the starting point of reasoning, are primitively identified. So no symmetry, or any other relationships, can be claimed to assign a probability.
An alternative approach might be to suggest the two copies are inherently indifferent so the probability must be 1/2. However, to be indifferent towards the two requires an impartial outsider’s perspective, e.g. reasoning from a God’s eye view. From such perspectives, a copy needs to be specified by sampling. There is no reason to focus on “I”. In fact, from outsiders’ perspectives, “I” do not refer to a particular subject. So the question won’t be comprehensible. We can treat the two copies indifferently or we can treat one as primitively unique, but not both in the same analysis. As a result, the probability cannot be formulated from any perspective.
The probabilities’ invalidity can also be shown with a frequentist model, or rather, the lack thereof. “I” can repeat the experiment the same way as in Question 2. Imagine yourself as a subject, waking up from the previous experiment, and then take part in the next. It may seem doing this repeatedly and counting the times where “I” am the original in each respective iterations would eventually give the probability. Yet there is no reason for the relative frequency of “me” being the original to converge on any particular value. Because the perspective center is primitively identified. It cannot be explained as an experiment outcome. Consequently, there is no valid experiment to use in a frequentist analysis. Surely, half of all copies are the originals. If a random copy is selected each time (e.g. RED) the relative frequency of it being the original would approach 1/2. However, that frequency reflects a problem formulated from an outsider’s perspective, e.g. Question 1, not the self-locating probability here.
Comparison of the Questions
The most important difference between Question 1&2 and Question 3 is perspective consistency. The probabilities being asked in Question 1&2 can be formulated from one consistent perspective, respectively. While the self-locating probability in Question 3 cannot. It needs both an outsider’s view and the subject’s first-person view to make sense.
It is also worth emphasizing that apart from the labeling process, the experiment is completely deterministic and known to the subject(s). Unlike Question 1&2, Question 3 is specifically asking about this deterministic and known process. The fact that “I” am, say the clone, means the agent at the perspective center is the newly created physical person. The experiment does not have an alternative outcome where “I” am the original. Therefore from the subject’s first-person perspective, the clone and original cannot be interpreted as two complementary events.
Some Anticipated Counter Arguments
It is understandable that many would not like my position about self-locating probability. Here I would provide brief counters to some foreseeable arguments.
It may feel natural to reinterpret the self-locating probability in the following way: first, come up with a description of “I” or “now” so that it can be used from an outsider’s perspective. Then reason from the outsider’s perspective, using said description and implied sampling process to give a probability. Then claim that is the self-locating probability. For example, say after waking up and learning that “I” am labeled RED. One might suggest the probability of “me” being the original is the same as the probability of RED being the original. However, regarding the first-person center as the outcome of a sampling process is switching perspectives. Comparing Question 3 and Question 1, they are obviously not equivalent. (Using arbitrary descriptions to imply different sampling processes is a more general problem in probability calculations. It is not limited to anthropic reasoning. See my discussion about the Full Non-Indexical Conditioning for more examples. ）
Another argument is by using decision-making outcomes. For example, if 90 people were created in blue cells and 10 in red cells, given I am one of these 100 people then the probability of “me” in a blue cell should be 90%. Because if everyone uses this probability to guess they are in a blue cell then 90% of all would be correct (this example is due to Nick Bostrom here). However, in this argument, the underlying objective is to maximize the total number of correct guesses for all 100 people combined. This is very peculiar. Because the self-locating probability is specifically about the indexical “I”, the corresponding objective should be maximizing “my” correct guesses, i.e. a simple selfish objective. Changing the objective to include everyone’s decision treats each individual indifferently. So the aggregated fraction only justifies probabilities formulated from an outsider’s perspective when the sampling is uninformative, e.g. the probability that a randomly selected person being from a blue cell is 90%. It does not justify the self-locating probability.
Self-Locating Probabilities in Paradoxes
Self-locating probabilities are used in almost all anthropic paradoxes. It seems to be a common trend to assign a probability for indexicals in a proposed reference class. Examples include the following.
In the Sleeping Beauty Problem: the probability that “today” is Monday/Tuesday, or the probability that “this awakening” is the first/second awakening, are self-locating probabilities.
In the doomsday argument: the prior probability distribution of “my birth rank” among all humans, past, present, or future, sometimes this is expressed with an extension of the first person by using “our birth rank” or “the current generation”, nonetheless, such concepts are still defined by the first-person center, they are all self-locating probabilities.
In presumptuous philosopher, the prior probability that “I”, as a potentially existing human/observer, actually exist is a self-locating probability.
In the simulation argument, the probability that “I” am a simulated observer or the probability that “this world” is simulated, are both self-locating probabilities.
In Dr.Evil and Dub, the probability of “me” being Dr.Evil or Dub is a self-locating probability.
All these probabilities are invalid.
See Part 4 for why perspective disagreement can occur in anthropic reasoning.
For the topic relating self-locating probability with the Many Worlds Interpretation, see Section 8 and Section 9.