Bostrom's Incubator and the epistemic ensemble.

A lot of tricky problems involving probabilities, such as the famous Monty Hall problem that has lead astray even professional statisticians, can be solved quickly and painlessly with the ensemble method (see how). The question however is: is the ensemble method always sound, or does it work in some situations and fail in others? Here I want to do two things. First, I want to present a simple thought experiment that might, in some people's eyes, call into question the ensemble method. Second, I would like to explain a couple of routes for justifying the method.

The Incubator. 

This neat foundational thought experiment in anthropic reasoning is due to philosopher Nick Bostrom. I found a great summary of it along with an excellent discussion of surrounding issues:

Stage (a): In an otherwise empty world, a machine called “the incubator” kicks into action. It starts by tossing a fair coin. If the coin falls tails then it creates one room and a man with a black beard inside it. If the coin falls heads then it creates two rooms, one with a blackbearded man and one with a white-bearded man. As the rooms are completely dark, nobody knows his beard color. Everybody who’s been created is informed about all of the above. You find yourself in one of the rooms.

Question: What should be your credence that the coin fell tails?

Stage (b): A little later, the lights are switched on, and you discover that you have a black beard.

Question: What should your credence in Tails be now?

I will show you the simple calculation a little later, but for now will simply state how the ensemble method answers those questions. For stage (a), it says the credence for Tails should be 1/3. For stage (b), the credence for Tails should be 1/2. While for most "regular" problems, such as the Monty Hall problem, there is little disagreement about the right answer, a problem like the Incubator, which involves different numbers of observers for different possibilities, elicits substantial disagreements in the academic literature.

The answer the ensemble argument gives also follows from other approaches (such as that based on the Self Indication Assumption). That doesn't mean it's correct of course, other approaches (such as that based on the Self Sampling Assumption) give different answers. Briefly, in case you are not sure how different answers are possible, let me give you a quick bumper sticker version of two main positions on stage (a):

Bumper sticker for 1/2. Before the light turns on, the only relevant information you seem to have is that each scenario was chosen by a coin flip. Your experience (of being a person in a dark room) is perfectly consistent with either scenario, so you have no reason to deviate from your knowledge that a coin flip is 50-50.

Bumper sticker for 1/3. Whoever "you" are, you should think, absent other information, that you were twice as likely to be created if there are two people created than if there is only one. So you should believe it is more likely that the coin fell heads than tails, twice more likely.

The Ensemble Premise.

Let me now give you the principle that serves as the foundation for the ensemble method and then give some justifications for it. I suspect that some people, upon hearing the principle, will think it's completely obvious and feel that I am making a big deal out of nothing. But I know that others won't feel that way at all. The justifications are mostly for this second group. First, a quick and dirty version of the Ensemble Premise:

EP. We can assume that the experiment has been performed many times.

Let me clarify what this means. Let's use the Incubator as an example. The story says that the experiment (flipping the coin and creating one or two people) is performed "in an otherwise empty world". This might seem a little ambiguous: is it necessarily saying that in this thought experiment we assume nothing else exists?  Or maybe that there is some kind of isolated world where that experiment is performed but other, "parallel" worlds can exist too as long as they don't interact with the Incubator world? If you think "who cares? That doesn't affect the answer" then I agree and I think you will have no trouble agreeing with one of the justifications for EP I will give in a minute.

What EP is saying then is that we can, if it helps with the calculations, add to the problem the stipulation that there are many many other parallel worlds where the same experiment is independently performed (and this, like the other conditions of the setup, would be known to the created people). So there now would be G (gazillions) of such incubators and, if G is made very big, to a high degree of precision in half of them the coin would land heads and in half - tails. In total there would be:

• 0.5G black-bearded people in the Tails worlds 
• 0.5G black-bearded people in the Heads worlds
• 0.5G white-bearded people in the Heads worlds.

For stage (a), when the room is dark, all a person would know is that he is one of the 1.5G people, of whom 0.5G, one-third, are in the Tails worlds. Therefore he would assign a chance of 1/3 of him being one of the Tails people.

What if there's no coin?

I hope the above illustrates how EP works and how it lets us calculate probabilities. But there's one important consideration that the Incubator doesn't illustrate. Basically, in real life sometimes there's no coin. What am I talking about? Many situations in real life are actually more analogous to a different version of the Incubator: suppose everything is as before, except each created person is not told the precise probabilities for creating a one-person world vs a two-person world. Suppose instead he just has some vague information about who or what was in charge of that decision. What should he do then if he still needs to make his best guess about being in a one or two-person room? 

In that case, the only thing to do is to use his best judgment about whatever vague information is available and translate that into an educated estimate of the probabilities for the "creator" to choose which world to create. These probabilities are called the priors. For example, suppose, as far as the person knows, the "creator" had no significant obstacles for creating either world and is not aware of any other considerations why the "creator" would choose one over the other. Then it seems sensible to not give any preference to either scenario and guess a prior probability of 50-50 for the "creator" to make either one. In that case, he can treat the situation as if a coin was flipped. 

If, on the other hand, the person has some reason to believe the "creator" was more likely to make one world over the other then again he has to use his best judgment to translate those considerations into an estimate of prior probabilities. There is obviously no straightforward algorithm for this, it just comes down to the person's best understanding of what's going on - a situation we all encounter many times a day. Suppose, for example, the person assigns a 70%-30% prior probability for the "creator" to make one person vs two. In that case, he can treat the situation as if the "creator" generated a random number between 1 and 100, and if it turned out to be in the range 1-70 created one person, otherwise two people. 

Only after this preliminary step is complete he can apply EP and imagine that this whole experiment has been performed many times. For example, if the priors are 70%-30%, out of G total worlds 0.7G would be one-person worlds, and 0.3G would be two-person worlds.

Precise statement of EP.

We are now ready to formulate EP more precisely.

1. Preliminary step: assign priors to epistemic uncertainties and replace them with decisions based on a random number generator in a way that, to your best judgment, wouldn't change the answer.
2. EP proper: the answer wouldn't change if it's assumed that the situation in question has been repeated independently many times.

This collection of independent experiments, which I call the epistemic ensemble, will then contain many versions of the agent trying to assign probabilities to various hypotheses. Just like in the Incubator example above, the desired probabilities are obtained by using the fact that, as far as such an agent can tell, they are equally likely to be any of these versions.

Justifications of EP.

Note that the first part, the preliminary step, doesn't change the answer by construction. Only the second part, adding many repetitions of the situation, could conceivably change the answer, and one could rightfully ask for an explanation why it doesn't. I think for some people this would seem so obvious that they would be puzzled by anybody making a big deal out of it. But, as the Incubator example shows, EP implies a certain answer to a class of questions for which there is currently no consensus among philosophers. That alone creates the need to provide justifications for EP. 

Note however that we shouldn't expect to be able to prove EP in the same sense as proving some mathematical theorem. There are very few of such proofs in philosophy. Actually, even mathematical proofs rely on fundamental assumptions, axioms, that are themselves granted without proof, so the best we should expect to do in justifying EP (or some competing principle) is to derive it from some other assumptions that would themselves be granted without proof. The reason one would then accept EP is because one would accept those other assumptions.

This opens one legitimate avenue for accepting EP: take it as one of those basic assumptions that you accept without reducing it to further assumptions, until and unless there is a defeater for it. While that would be a perfectly kosher position if you find EP itself sufficiently plausible, let's look at a couple of arguments in support of EP.

Justification based on a lack of causal influence.

We can put it in this compact form:

1. Rational credences about properties of a given situation can only be affected by information about things that can causally affect those properties.
2. The stipulation about many independent repetitions of the situation is, by construction, a stipulation about parts of reality causally disconnected from one another.
3. Therefore, adding such a stipulation should not affect an agent's credences about properties of his situation.

The last point follows from the first two and is equivalent to EP proper. Given that the second point is true by definition, the argument stands or falls with the first point. But the first point seems to simply express an essential feature of what it means for credences to be rational.

Justification based on the concept of credences.

What exactly is actually meant by a credence of, let's say, 70% that a certain situation S has property A? There are different accounts of the definition of credence, but one reasonable answer might be: this just means that if the situation S were to repeat, independently, many times we expect that 70% of the time it will have property A. If that's a definition that you accept then you accept EP by definition.