Reference Classes in the Doomsday Argument

The reference class determines for whom the bell tolls. If you're 100 billionth in h. sapiens, that means that, if given absolutely no other information, you should guess that h. sapiens will last about another 100 billion individuals. Although it is possible to choose fuzzy or ill-defined reference classes and get fuzzy or ill-defined answers, I wouldn't call that a "serious problem."

More serious is the fact that we have tons of extra information. For example, did you know that h. sapiens is a self-replicating life-form? The fact that we have trouble not knowing this (and more!) is probably where the conflict between our intuition and the doomsday argument comes from.

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.

The Doomsday Argument seems to be misnamed. It doesn't predict DOOM - merely the absence of births. For those who expect most future creatures to spread out by growing - rather than by reproducing - an absence of births would not be too surprising. Since that would happen in some of the very best futures, proclamations of DOOM seem to be rather unwarranted.

Bostrom agrees. From Chapter 11 of his book Anthropic Bias:

I wish to suggest that insensitivity (within limits) to the choice of reference class is exactly what makes the applications [of observation selection effects] just surveyed scientifically respectable. Such robustness is one hallmark of scientific objectivity.

[...]

It pays to contrast this list of scientific applications with the various paradoxical applications that we discussed in earlier chapters. Take the Doomsday argument. In order for it to work, one has to assume that the beings who will exist in the distant future if humankind avoids going extinct soon will contain lots of observer-moments that are in the same reference class as one’s current observer-moment. If one thinks that far-future humans or human descendants will have quite different beliefs than we have, that they will be concerned with very different questions, and that their minds might even be implemented on some rather different (perhaps technologically enhanced) neural or computational structures, then requiring that the observer-moments existing under such widely differing conditions are all in the same reference class is to make a very strong assumption. The same can be said about [other paradoxes described in the book]. These arguments will fail to persuade anybody who doesn’t use the particular kind of very inclusive reference class they rely on—indeed, reflecting on these arguments may well lead a reasonable person to adopt a more narrow reference class. Because they presuppose a very special shape of the indexical parts of one’s prior credence function, they are not scientifically rigorous. At best, they work as ad hominem arguments for those people who happen to accept the appropriate sort of reference class—but we are under no rational obligation to do so.

I think there might be a problem with assumptions for DA. In the cubicle scenario, the situation when the first person is placed is the same for when the 10th or 100th will be placed, while humanity has greatly changed since the first of any reasonable reference class, and will continue to do so (especially if at some point in the future we start colonizing planets).

The reference class is things you can be, i.e. things with subjective experience. This doesn't actually make answering the question any easier, though it does mean it has other implications. Also, a smaller reference class is more likely (hence, doomsday), so that suggests that fewer creatures are sentient, and if possible, they're sentient to a lesser degree, than we'd otherwise believe.