For what it’s worth, Bostrom seems to favor reference classes that neutralize the DA. He considers a similar thought experiment in his paper “The Mysteries of Self-Locating Belief and Anthropic Reasoning”. If you apply his reasoning to the DA, it amounts to saying that, when you learn that you are the nth human, you should update on this information under the assumption that your reference class contains only those observer-moments who know that they are the nth human.
The thought experiment that Bostrom considers in the paper is called “Incubator”:
Incubator. Stage (a): The world consists of a dungeon with one hundred cells. The cells are numbered on the outside consecutively from 1 to 100. The numbers cannot be seen from inside the cells. There is also a mechanism called “the incubator”. The incubator first creates one observer in cell #1. It then flips a coin. If the coin lands tails, the incubator does nothing more. If the coin lands heads, the incubator creates one observer in each of the remaining ninety-nine cells as well. It is now a time well after the coin was tossed, and everyone knows all the above.
Stage (b): A little later, you are allowed to see the number on your cell door, and you find that you are in cell #1.
Question: What credence should you give to tails at stages (a) and (b)? We shall consider three different models for how to reason, each giving a different answer. These three models may appear to exhaust the range of plausible solutions, although we shall later outline a fourth model which is the one that in fact I think points to the way forward.
Bostrom then describes and rejects three models. Model 3 says that you should assign credence 1⁄2 to heads at both stages. Model 3 accomplishes this by just refusing to update after you learn your cell-number. But this leads to Bayesian incoherence. Later, he introduces his own “model 4”:
Before, we rejected model 3 because it seemed to imply that the reasoner should be incoherent. But we can now construct a new model, model 4, which agrees with the answers that model 3 gave, that is, a credence of 1⁄2 of heads at both stage (a) and stage (b), but which modifies the reasoning that led to these answers in a such a way as to avoid incoherency.
Bostrom avoids incoherence by changing reference classes in stage (b). In stage (b), when you know that you are in cell #1, your reference class includes only those observer-moments who know that they are in cell #1. Bostrom tries to justify this in the paper. I won’t try to give his justification. My point is just that Incubator and the DA are the same in all the respects that are relevant to how you should apply anthropic reasoning. The elements of the paradoxes map to each other as follows:
The occurrence of heads in Incubator is analogous to there being no early doomsday in the DA. Learning that your cell-number is #1 is analogous to learning that you are the nth human. Continuing to assign credence 1⁄2 to heads, even after you learn that you are in cell #1, is analogous to continuing to go with the empirical prior for the lifespan of the human race, even after you learn that you are the nth human. So, in practice, Bostrom’s model lets you use the empirical prior, without indexical updating, both before and after you learn that you are the nth human.
It is possible I have missed an important point you are making, but here is how I interpret what you wrote:
Observer #1 in this scenario is a special class because he knows that he was created before Doomsday. Because of this, he knows that his retrospective probability for heads is 50% because he was created regardless of the coin’s outcome.
In our world, the timing of Doomsday is not so well-defined that we can say whether we’re in the same position as Observer #1 or not. Maybe we have lived past the most likely Doomsday scenario, and maybe we haven’t.
For what it’s worth, Bostrom seems to favor reference classes that neutralize the DA. He considers a similar thought experiment in his paper “The Mysteries of Self-Locating Belief and Anthropic Reasoning”. If you apply his reasoning to the DA, it amounts to saying that, when you learn that you are the nth human, you should update on this information under the assumption that your reference class contains only those observer-moments who know that they are the nth human.
The thought experiment that Bostrom considers in the paper is called “Incubator”:
Bostrom then describes and rejects three models. Model 3 says that you should assign credence 1⁄2 to heads at both stages. Model 3 accomplishes this by just refusing to update after you learn your cell-number. But this leads to Bayesian incoherence. Later, he introduces his own “model 4”:
Bostrom avoids incoherence by changing reference classes in stage (b). In stage (b), when you know that you are in cell #1, your reference class includes only those observer-moments who know that they are in cell #1. Bostrom tries to justify this in the paper. I won’t try to give his justification. My point is just that Incubator and the DA are the same in all the respects that are relevant to how you should apply anthropic reasoning. The elements of the paradoxes map to each other as follows:
The occurrence of heads in Incubator is analogous to there being no early doomsday in the DA. Learning that your cell-number is #1 is analogous to learning that you are the nth human. Continuing to assign credence 1⁄2 to heads, even after you learn that you are in cell #1, is analogous to continuing to go with the empirical prior for the lifespan of the human race, even after you learn that you are the nth human. So, in practice, Bostrom’s model lets you use the empirical prior, without indexical updating, both before and after you learn that you are the nth human.
It is possible I have missed an important point you are making, but here is how I interpret what you wrote:
Observer #1 in this scenario is a special class because he knows that he was created before Doomsday. Because of this, he knows that his retrospective probability for heads is 50% because he was created regardless of the coin’s outcome.
In our world, the timing of Doomsday is not so well-defined that we can say whether we’re in the same position as Observer #1 or not. Maybe we have lived past the most likely Doomsday scenario, and maybe we haven’t.
Edit: grammar