Investigations into the Doomsday Argument

1996 (c) Nick Bostrom
Department of Philosophy, Yale University
Homepage www.nickbostrom.com

Note: For my more recent writings on this topic, please see www.anthropic-principle.com, especially the book Anthropic Bias: Observation Selection Effects in Science and Philosophy (Routledge, New York, April 2002)

 

The Doomsday argument in a nutshell

The Doomsday argument was conceived by the astrophysicist Brandon Carter some fifteen years ago, and it has since been developed in a Nature article by Richard Gott [1993], and in several papers by philosopher John Leslie and especially in his recent monograph The End of The World (Leslie [1996]). The core idea is this. Imagine that two big urns are put in front of you, and you know that one of them contains ten balls and the other a million, but you are ignorant as to which is which. You know the balls in each urn are numbered 1, 2, 3, 4 ... etc. Now you take a ball at random from the left urn, and it is number 7. Clearly, this is a strong indication that that urn contains only ten balls. If originally the odds were fifty-fifty, a swift application of Bayes' theorem gives you the posterior probability that the left urn is the one with only ten balls. (Pposterior (L=10) = 0.999990). But now consider the case where instead of the urns you have two possible human races, and instead of balls you have individuals, ranked according to birth order. As a matter of fact, you happen to find that your rank is about sixty billion. Now, say Carter and Leslie, we should reason in the same way as we did with the urns. That you should have a rank of sixty billion or so is much more likely if only 100 billion persons will ever have lived than if there will be many trillion persons. Therefore, by Bayes' theorem, you should update your beliefs about mankind's prospects and realise that an impending doomsday is much more probable than you have hitherto thought.

Consider the objection: "But isn't the probability that I will have any given rank always lower the more persons there will have been? I must be unusual in some respects, and any particular rank number would be highly improbable; but surely that cannot be used as an argument to show that there are probably only a few persons?"

In order for a probability shift to occur, you have to conditionalise on evidence that is more probable on one hypothesis than on the other. When you consider your rank in the DA, the only fact about that number that is relevant is that it is lower than the total number of individuals that would have existed in either hypothesis, while for all you knew, it could have turned out to be a number higher than the total number of people that would have lived on one of the hypothesis, thereby refuting that hypothesis. It makes no difference whether you perform the calculation with a specific rank or an interval within which the true rank lies. The Bayesian calculation turns out the same posterior probability. The fact that you discover that you have this particular rank value gives you information only because you didn't know that you wouldn't discover a rank value that would have been incompatible with the hypothesis that there would have existed but few individuals. It is presupposed that you knew what rank values were compatible with which hypothesis. It is true that for any particular rank number, finding that you have that rank number is an improbable event, but a probability shift occurs not because of its improbability per se, but because of the difference between its conditional probabilities relative to either of the two hypotheses.

There are numerous objections such as this one, objections that can easily be seen to be mistaken. When people first encounter the Doomsday argument (hereafter the DA), what happens is that most of them think that it is obviously false. Then any sign of consensus disappears when it comes to explaining what is wrong with it. Each comes up with his own objection, these objections tend to be incompatible with each other, and typically they rest on simple misunderstandings. This paper will zoom in on those features of the DA that are genuinely problematic.

A probability shift must be made

In the case of the two urns, or in any similar situation when we take a random sample, we should obviously draw a conclusion analogous to the one aimed at in the DA. This follows from indubitable Bayesian principles, and anyone denying this is invited to demonstrate his sincerity by betting according to his doctrine in an urn game with the author of this paper.

If we have agreed on that as a starting point then we can take the next step towards establishing the necessity of a probability shift by considering the following thought example (discussed in Leslie[1996]).

The amnesia chamber

Imagine you are in an isolation chamber and don't not know what birth rank you have, but let the two exclusive hypotheses MANY (= many people will have existed) and FEW (=few people will have existed) be equally probable relative to your present information set. Suppose you obtain a new piece of evidence: your rank is higher than the number of individuals in FEW. That conclusively proves that MANY is true. This implies that if you had instead found that you had a rank that was low enough to be compatible with both hypotheses, then that would have increased the probability of FEW; because if you thought that the new piece of evidence could lower but never raise the probability of FEW, then you would be inconsistent, as is easily shown by a standard Dutch Book argument, or more simply by the following little calculation.

Write "M" for "MANY", and "M" for "FEW"; and let e be the evidence that you are living "early", i.e. that you have a rank compatible with FEW. We can assume that . Then we have

and

.

Dividing these two equations and using , we get

.

So the quotients between the probabilities of MANY and FEW is less after e is known than before. In other words, learning e decreases the likelihood of MANY and increases the likelihood of FEW. Therefore, in the amnesia chamber experiment, a probability shift results from learning e.

 

The next step is to determine whether it is an illegitimate idealisation to assume, as we did, that you just got to know what your rank is. In reality, of course, there was never any doubt: you knew all along that you were living in the 1990s. But the crucial consideration is not when you obtained the evidence, but whether you have taken it fully into account. Presumably, when you pondered over the future of mankind before you first heard of the DA, you didn't take the datum that you are alive in 1997 fully into account, for you didn't then realise all its implications. Now, when an important new implication (i.e. that this fact can be used to argue for the likelihood of an impending extinction of mankind), has been pointed out to you, you should make up for your earlier neglect by updating your beliefs accordingly. It therefore seems that nothing essential in the thought experiment hinges on the assumption that you were until recently ignorant of your birth rank.

And this seems to be enough to show that a shift must be made. There are, however, a number of hidden assumptions that we have made, if we are thinking of the amnesia chamber experiment as establishing the DA, and these have to be brought out into the light. We will have to go over them carefully before we can assess the validity or otherwise of the DA. The chief ones I will discuss in this paper concern

  • The no-outsiders requirement
  • The reference class problem
  • The Self-Indication axiom

A whole lot of other objections have been raised as well, but I think they have been satisfactorily answered by Leslie. Rather than repeating the replies here, I refer the reader to Leslie's work.

 

The no-outsider requirement

Instead of skipping directly to my own view on the no-outsider requirement, we will warm up by conducting another thought experiment, which will also serve the didactic purpose of improving our intuitive grip on how DA-style reasoning works.

The four-buildings experiment

"Imagine four tall buildings with 100 floors each. You are in a group of 303 people to take part in an experiment. You are told the following: All participants will be anaesthetised and randomly carried to windowless rooms (the lavatories) in the four buildings. In three buildings there will be one subject on each floor; in the forth, there will be one subject on each of the first three floors and the rest of the house will be empty. Then a tetragon die will be thrown to randomly select one building. All subjects in that building will be given a stimulant so that they wake up from their dreamless slumber. They might be offered bets of various kinds, which have been prepared by the experimenter in advance, before the die was thrown. Ready?

Consider first the case of an outside observer. She selects a house at random. If she looks at the 50th floor and there is someone there, then that proves, of course, that that building contains 100 persons. If there is nobody there, the building contains three persons. If she instead looks at the 3rd floor, she will get no information, since she already knows that in all houses there are persons there. Suppose now that she randomly chooses a name, Mr. N.N., from the participants list, and that the position of Mr. N.N. is then pointed out to her. If Mr. N.N. is somewhere between floor 4 and 100, then that house contains 100 persons. But what if Mr. N.N. is on floor 2? Again that would give the outsider no information: the odds that the building contains 100 persons would still be 3:1. In this case, the unlikelihood that Mr. N.N. would be on one of the first three floors given that he is in a 100-persons building is precisely balanced by the greater likelihood that Mr. N.N should be in such a building in the first place. (We shall come back to this consideration in a later section.) The overwhelming majority, after all, are in 100-persons buildings. Only if a building is selected, and a name is picked randomly from a list of the persons in that building, can we use the fact that the name denotes a person on the 2nd floor to infer that the probability that it is a 100-persons house is less than 75%. (In fact, it's 8%.) If this were the relevant analogy, then the Doomsday argument would be valid.

But now consider the case of an insider. You wake up in a lavatory and remember the experimental set-up. You begin to consider what odds you would take if offered a bet on whether you are in a 100-persons building. After a little reflection you decide the probability is 300/303=99%, rather than 75%. Why? Because, on average, 99% of all awake persons in experiments of this sort will find themselves in 100-person buildings, and you have no other relevant information than that you are an awake person in such an experiment. The very fact that you are awake indicates that it is a 100-person building that has had its inhabitants woken.

If you think otherwise, we may repeat the experiment a thousand times, and if you are willing to bet according to your estimations, and do not change your mind, you will lose money. If the experiment is repeated only 5 times, there is some chance that you will be lucky and win despite your mistaken estimation, and even more so if the experiment is performed only once; but it would still be rational to bet against you. However, no bet is offered.

After a while, the lavatory doors are unlocked and you enter a room where there is a window. You look out and are surprised to discover that you are on the second floor. The probability that you are in a 100-person building obviously decreases; the question is by how much. If this is a situation where a calculation based on a nave application of the DA is applicable, then the new probability that you are in a 100-person building should be 8%. But that is a loser's odds. The real probability is 75%. There is a note on the wall beside the window written by the experimenter (from Monte-Carlo, we may suppose). It says that if you sign it, you will be given $100 if you are in a 100-person building and have to pay $1000 if you aren't. Having been convinced by John Leslie’s book, you sign it. The next day the experimenter performs the experiment with another group of subjects. On average, he makes $8700 per four days of work.

If this is right then there is an essential difference between the outsider case and the insider case. In the former, there is a 8% chance that the person in question is in a 100-persons building; in the latter, the probability is 75%. The difference is that in the outsider case, the tendency for the sample person to be in a 100-persons building because most persons are in such buildings is offset by the postulate that the sample will be taken from inhabitants of a given house, and this house is selected without regard to how many persons there are in it. In the insider case, on the other hand, the sample and the building are selected independently. The coincidence that the building selected happens to contain the independently selected sample (i.e. you) indicates that the building was one with a large population of samples, i.e. many inhabitants. As we have seen, this difference results in different rational odds, the "rational odds" being defined as the best odds one could give without having a negative expected utility from a corresponding bet. So on which of the two cases, if any, should we model our situation when it comes to the prospect of Doomsday? Clearly we are insiders with respect to the human species.

To make the parallels explicit: The four buildings can be thought of as possible worlds, and the sleeping people as possible observers; the awakened observers are the real humans that will have existed; your discovery that you are on the second floor is the observation that relatively few humans have existed before you; the fact that non-dreaming sleeping subjects can't really think or make observations corresponds to the same fact about non-existing "possible" observers; the proportion between 100-person buildings and 3-persons buildings (chosen to be 3:1 just as an illustration) represents the prior probability we would assign to the proposition that mankind will not have become extinct by 2150, based on our direct estimates of the dangers of warfare, plagues, runaway global warming, high-energy experiments etc.; and so on."

 

Leslie seems to accept that the thought experiment describes a situation relevantly analogous to our own, though with one proviso. The experiment must be carried out only once. If the experiment were repeated a great number of times, the odds would asymptotically approach the one I gave above. But if the experiment is performed only once or a small number of times, then, according to Leslie, another principle applies and we should no longer be guided by estimates of expected utility when we determine what the probability for being in a 3-person building is.

This reply might sound extremely as hoc and bizarre, but in fact it isn't. To explain how one could be led to make this reply if one accepts Leslie’s position, we shall do a thought experiment. (I know it's an awful lot of thought experiments here, but for some reason they are very helpful when we are dealing with the DA.)

God's coin tosses

Suppose you know that you were created as the result of a divine coin toss; if the coin landed heads, ten humans were created; if tails, only one human was created. We assume that there are no humans other than those created as a result of this coin toss, and that the everybody kept in isolation so nobody can see anybody else. Now you, dear reader, are luckily alive; the question is whether this gives you any reason to believe that God’s coin landed heads, or whether you should rather believe that the chance is fifty-fifty. According to Leslie, you should believe the latter.

 

Here we see the reason why Leslie insists that my thought experiment should be carried out only once. If God had tossed His coin a great many times, and created either ten or one human for each toss depending on how the coin landed, then Leslie would agree that you should say that the probability that you were created as a result of a toss wherein God’s coin landed heads approaches 10:1. For in this case, we could contrast the size of the group of people created as a result of the heads outcome with that of the group of those created on a tails outcome. And these would all be actual, living humans. In the case where God’s coin is tossed only once, there would only be one group of humans. (In order not to complicate things with the possibility that you could have been God, let us assume that in this thought experiment "God" is a fanciful way of denoting an automatic randomised human breeder machine.) The other group would consist of merely "possible humans", and, of course, a merely possible human doesn’t actually think or "find herself in this or that situation". So in this case you gain no information from the fact that you find yourself alive that you could use to infer that God’s coin has probably landed heads; for how could you possibly have found yourself otherwise than alive?

Setting the latter claim (that finding yourself alive makes the larger group more likely, when the coin is tossed only once) to a side for the moment, we can at least agree that in the case when the coin is tossed many times, say a thousand times, it is more likely to be in the larger group. For the probability of belonging to a group of people is equal to the sum of the probabilities of being the particular individuals in that group. This is an immediate consequence of Kolmogorov's additivity axiom, since the alternatives are mutually exclusive: one cannot be more than one individual. Think of it like this: You know there are almost certainly about 5000 people. Hence there are about 5000 names of people. You know one of these names is yours, but you have absolutely no idea as to which one it is. By the principle of indifference, you should therefore believe that the probability that your name is among the names of persons who were created as a result of a heads toss is about 90%, for about ninety percent of all the names belong to such persons.

(If you are uncomfortable with this appeal to the principle of indifference, we can perhaps add further persuasion by considering a variant of the God's coin tosses experiment, where, instead of humans being created, the coin tosses result only in that either one or ten names of randomly chosen humans from an extant population get entered into columns of special list, either as "one-names" or as "ten-names". After a thousand tosses, you get to know that you are on the list and you are asked to guess in which column your name appears. (No other information is available, and you know that all people on the list are given the same instruction.) Here it would be easy to carry out a betting session in a scaled-down realisation of this experiment. Anybody who now thinks that the chances should be fifty-fifty is encouraged to contact the author in private. But nothing essential differs here from the original God's coin tosses with a thousand throws; the fact that it is your mode of creation in one case and the status of your name in the other makes no difference in this idealized context.)

So in order for the four-buildings example to model our true situation, the experiment has to be performed once only. (Forget about the possibility of extraterrestrials for the moment.) That is Leslie's position. However, upon reflection I have come to believe that not even when the experiment is only done once is the four-buildings model correct. For it is clearly supposed to be carried out in the real world, here on Earth, where there are about six billion outsiders; not in a possible world where the experimental subjects and the experimenter were the only beings that existed. And as we shall now see, that makes a crucial difference.

Consider a universe where there are a thousand humans. Then a coin is tossed and either one or ten more humans are created as a result. Now suppose that you discovered that you were in such a universe (after the coin was tossed). The most probable hypothesis is of course that you weren't created because of the coin toss but are one of the thousand that were there anyway. But it is still ten times more likely that you were created as a result of a heads-toss than a tails-toss. For consider the conditional probability Pr(I was created because of the coin toss. | There are 1010 humans.). This is clearly greater than Pr(I was created because of the coin toss. | There are 1001 humans.) If there are 1010 humans then about 1% of them will be coin-toss humans; if there are 1001, then the figure is about 0.1%. By Bayes' theorem, we conclude that after conditionalising on "I was created as a result of the coin toss." it is more likely (by about a factor 10) that the coin landed heads than that it landed tails. So in this case, where we have a thousand outsiders, if I know that I was created as a result of a coin toss, then that gives me strong reason to believe that that coin toss landed heads; in contrast to the case where there were no outsiders and the coin was only tossed once, where knowing that I was created as a result of the coin toss didn't modify the prior fifty-fifty probability distribution.

Thus we see that what is relevant is not how many times the experiments are repeated, but how many people are expected to exist independently of their outcome. Since this point isn't made in Leslie's work, it's worth epitomizing it by giving it a mathematical formulation:

 

The Generalized DA formula (GDA)

In the simplified case where mankind is the only intelligent species ever to exist (i.e. there are no outsiders) and considering only the two alternatives FEW and MANY, we can calculate the conditional probability Pr(h|e) of a long-lasting human race upon finding oneself alive at an early time by means of Bayes' theorem in the following form:

We have just seen that the DA in this form would be incorrect in case we couldn't be sure that there were no extraterrestrials. What we now want to do is to develop a way of calculating what predictions the DA would make in the more general case where there might be extraterrestrials. In order to do that, we shall consider again the model of God's coin tosses, but this time the coin can be tossed any given number of times and there can also be a group of outsiders that exist independently of the coin tosses.

We'll begin by making some definitions. Let "the group" consist of all individuals that are created as a result of the coin tosses, i.e. the insiders. Let "the reservoir" consist of all the other individuals, i.e. the outsiders; and let R be the number of these. Let G be the hypothesis that you are in the group. Let e be the evidence that you find yourself early, i.e. that within the batch in which you were created you have a rank that is compatible with that batch being one that contained only few individuals, i.e. being one that was created as a result of the coin landing tails. Let Hk be the hypothesis that the coin landed heads k times.

Then the probability that the coin (when all the tosses are completed) having landed heads k times is

(Where N is the number of times the coin is tossed.) The number of people in the group, given that k heads turned up, will then be

(Where m is the number of people that are created each time the coin lands heads, and f the number each time it lands.) The probability that you will be in the group, given that the coin landed heads k times is

From the general relationship

we get

             (#)

Likewise, we can write

since . Using and inserting into (#) and simplifying, we get

.

Rejecting the SIA (to be defined in a later section), at least for the sake of the present argument, the prior probability of M (you being born as a result of a toss where the coin landed heads) is expressed by the weighted sum over the possible fractions of all people that were created as a result of such coin tosses in the different possible scenarios:

.

Now we can define a new probability distribution, P', where the priors have changed to accommodate the added information that you are in the group (i.e. were created as a result of a coin toss):

             (*)

But from Bayes' theorem applied to this new distribution it now follows that

where all terms are known:

Inserting these values finally gives us

             (GDA)

which we can call the generalised Doomsday argument formula, the GDA. It should be straightforward, using standard probability theory, to extend this result to cover other cases where instead of a coin toss we have a process with more than two possible outcomes, each with a different probability, and so fourth. Thus we have given an exact expression of what happens on the DA point of view in the general case where an unknown number of aliens may exist.

 

The reference class problem

The DA is formulated in terms of the number of people that would have existed in different scenarios, but "people" is a fussy concept: there are in the early stages of our species a big grey zone where it is unclear what to count as human and what as ape, and it is not implausible that in the future, if we manage to survive, we will change into something else or perhaps replace ourselves with computers, our "mind children" (Moravec [1989]). In itself the fussiness may not be very harmful for the DA, but it raises the suspicion that it might be a symptom of some more fundamental affliction. In any case, the fussiness is so extensive, especially in the future direction, that unless we make more precise what we are supposed to include in the reference class, the conclusion of the DA, even if it followed from its premises, would be so vague as to be of strongly reduced practical import.

Part of Leslie's answer (Leslie [1996], pp. 256-63) to this problem is to point out another feature of the analogy with the urn experiment that we used to introduce the DA in this paper. If the balls in the urn, instead of being numbered, had been of different colors and shades of colors, and we asked "What fraction of the balls in this urn is red?", then we would have been faced with the problem of what nuances of pink and purple to count as red. But here the answer would be that it would simply be up to us to decide. If we are interested in how many light-red-to-dark-purple balls there are, then what we should put into the Bayes’ formula is our estimate of the number of such balls; and when we draw the first ball, all we have to do is classify it either as light-red-to-dark-purple or as not-light-red-to-dark purple and then calculate the posterior probability in the same way as in the case with numbered balls.

This solves a part of the problem. If we are interested in how many blue-eyed humans there will exist, then we do the calculation with the parameters for blue-eyed humans; if we ask about the number of two-or-three-eyed humanoid descendants, then our beliefs and data about those are what should be fed to the equation. Bayes' formula is perfectly neutral as to how to define the hypotheses.

This response can only be part of the answer, however. As Leslie realises, for the purpose of the DA, it is necessary to place restrictions on which reference classes are permissible; otherwise they could be both too wide and too narrow. Let's begin by showing that they could be too wide.

Suppose we were interested in finding out how many oaks-or-humans there will have existed on Earth. (Set to a side for the moment the difficulty of exactly what to count as a human or a an oak tree.) Suppose we had reduced the quandary to two rival hypotheses: either (h1) there will have been 100 billion humans and 100 billion oaks, or else (h2) 100 billion humans and 1000 oaks. Now you find yourself alive at a time when there so far have been 50 billion humans and 50 billion oaks (say). In this case, it would obviously be wrong to reason that h1 is now much more likely than h2 because we have discovered that a random sample (you) turned out to be among the first 100 oaks-or-humans and that would be much more likely on h1 than on h2. The reason why it would obviously be wrong to reason thus is that you are clearly not a random sample from the population of oaks-or-humans: you couldn't possibly in any meaningful sense have turned out to be an oak. (We are assuming here that oaks don't have souls or such; which of course they haven't.)

This raises another question: "From what population am I a random sample for the purposes of the DA?". In my opinion, this turns out to be the problem in assessing the DA, and we will address it in a later section. For now, we can at least conclude that the reference class must be restricted in some way: it can't be made as inclusive as one pleases.

It can't be made arbitrarily restrictive either. For instance, we can't hope to derive any valid conclusions if we define the reference class to consist of all humans born the day you were born or later. Relative to that group you would indeed have had an exceptionally low rank, but there is nothing surprising about that and nothing follows from that fact about how long you should expect the human race to survive.

The reason why this is an impermissible choice of reference class is that it is defined with reference to yourself -- all humans born as late as or later than you were. Such a customized reference class fails because there is a systematic dependence between you and the reference class. It would definitely be wrong of you to regard yourself as a random sample from the population of all humans born on or after your date of birth.

It may be worth illustrating this randomness-proviso with a more mundane example. Suppose you are doing your laundry in a new washing machine which for all you know may stop with equal probability any time between the starting point and one hour later, but by then it must have stopped. You start it and go away to watch telly. After fifteen minutes you return to make an inspection and you find that it is still running. What is the probability that it will still be operating after another half hour?

Well, we can think of the stopping time as a uniformly distributed random variable which has been given an unknown value between 0 and 60 minutes. Checking the machine tells us that the value wasn't between 0 and 15. We can thus cut away that segment and renormalize the variable between 15 and 60. So the chance is one third that it is running at 45 minutes after time zero.

That is how we would normally reason. But now assume that the time of the inspection was neither chosen at random nor was determined by some other external factor but that it was instead fixed by randomly selecting a time point from all the time points at which the machine would be running. For example, if what determined when the machine is to stop is some kind of hidden timer, then one could have a computer read off that timer and then select a random number between zero and the preset timer value. Now if you decided to make the inspection at the time selected by this computer, then the inspection itself would give you no information, since you were guaranteed that the machine would still be running at that time; but the clock value when you made your inspection would contain information that you could use to calculate when the machine were likely to stop. For instance, if the inspection were made after only two minutes and five seconds, then that would give strong reason to believe that the machine would have ceased its toil within, say, ten more minutes.

It would evidently be wrong to reason that the machine was very likely to stop within the next thirty seconds because "if the machine were to continue to run for more than thirty seconds then the inspection call would have occurred extraordinarily early in the interval delta = (2 minutes, machine's stopping time). This would be faulty reasoning because the time of the inspection call is a random sample from delta-prime = (0, machine's stopping time), but not a random sample from delta. The faulty interval, or reference class, was defined by means of a reference to the time the sample was taken. The lesson is that, as a rule of thumb, one should not define the reference class in the DA by means of implicit or explicit reference to oneself.

This is, in my opinion, the right response from the advocate of the DA to problems stemming from an unduly narrowing of the reference class. Leslie, however, takes another route, which he says was suggested to him by Carter (Leslie [1996], p. 262). He thinks that defining the reference class as humans-born-as-late-as-you-or-later is fine and that ordinary inductive knowledge will make the priors so low that no absurd consequences will follow:

Imagine that you'd been born knowing all about Bayesian calculations and about human history. The prior probability of the human race ending in the very week you were born ought presumably to have struck you as extremely tiny. And that's quite enough to allow us to say the following: that although, if the human race had been going to last for another century, people born in the week in question would have been exceptionally early in the class of those-born-either-in-that-week-or-in-the-following-century, this would have been a poor reason for you to expect the race to end in that week, instead of lasting for another century. [My emphasis.]

I disagree witn the claim that the prior inductive improbability is enough to allow us to say this (given that the DA is correct). This insufficiency of the inductive improbability to compensate for the artificial reference class follows logically from the fact that the inductive evidence (for the hypothesis that mankind will end in a given time interval, say within a week from now) is constant no matter how we define the reference class, whereas the reference class can be defined in arbitrarily many ways, thereby varying the strength of the resulting probability shift. The inductive evidence could therefore compensate for at most one strength of the shift, whilst for all other choices of reference class, a different, incompatible, prediction about our future would result.

It is true that we can always choose the values we substitute for the variables representing the priors in the Bayesian formula in such a way that we reproduce the posterior probability resulting from the correct choice of reference class and priors; but these values which we substitute will then no longer be the prior probabilities but mere ad hoc numbers chosen because, inserted into Bayes' formula, they happen to give the output we had decided in advance we wanted to get. Instead of Bayes' formula we could have used any function onto [0,1], or simply not bothered at all.

I therefore conclude that, contrary to what Leslie and Carter assert, it is essential that the reference class not be improperly restricted by means of overt or covert reference to the subject applying the DA.

We have also seen that the reference class must not be too wide: for instance, it must not include noncognitive beings. Further, we saw that there was an element of choice: we can device different versions of the DA, depending on what feature of the future we want to predict; provided, of course, that we don't make the reference class too wide or confine it in an illegitimate way. These specifications are not sufficient, however, to completely disambiguate the DA. In particular, we need to know what things, if any, must be excluded from the reference class except noncognitive beings, and what probability metric to use over the reference class we choose. We shall address these issues in a later section.

 

The Self-Indication Axiom

What I call the self-indication axiom is defined as follows:

(SIA) The fact that one is an observer gives one some reason to believe that the world contains many observers.

"Observer" should here be taken to denote those entities which can be justifiably included in the reference class of the DA. For example, a noncognitive thing is not an observer, but exactly what types of cognitive things are observers has not been definitely settled by our discussion so far. "the world contains" should be understood in the atemporal sense.

The motivation for defining this axiom is that if we accept it then that buys us a very neat "solution" to the DA problem, i.e. a way of giving all the arguments advanced by Carter and Leslie their due without changing our minds one bit about the future of our species and its robotic descendants. The idea is that we could grant that a shift must be made (a conclusion we seem to have established with the "Amnesia Chamber"-argument above) while insisting that this shift is counterbalanced by the greater a priori likelihood of there being many observers. Intuitively, if a world contains more observers, then there are in some sense more "slots to be born into", and hence a greater probability that you should find yourself in such a world rather than in a world that only contains a few observers. Prima facie, the SIA is admittedly dubious, and perhaps it is no less dubious ultimo facie, but it deserves serious consideration as it might be the only consistent way of resisting the conclusion of the DA. We postpone the evaluation to the next section and concentrate here on how it works to achieve this result.

The easiest way to explain this is to take again the example of God's coin tosses. Assume it is tossed only once in an otherwise empty universe, creating either ten or one individual (and neglect the existence of God Himself; it is better to assume that a mindless machine was responsible for creating the humans). According to the SIA, before you know anything else than that you were created as a result of that coin toss, you should believe that the probability of heads is greater than that of tails. On the simplest version, the chances of heads might be 10:1, for if it did then there would be ten times more "opportunities for you to exist" than if tails, and since you know you exist, this should increase the likelihood of the hypothesis which made that more probable, i.e. (assuming the SIA) the hypothesis that the coin landed heads.

So if we assume the SIA, then we have two effects to consider. First, the SIA says that finding yourself alive indicates that there are many observers in the world. Second, the DA says that finding yourself to be in the early group (i.e. finding that you have a rank that would have existed whether FEW or MANY were true), indicates that there are only few observers in the world. The neat thing is that, as first noted by Dieks [1992] and shown by Kopf et al. [1994], these two effects cancel each other out precisely!

Even were we convinced that the SIA is true, we would still have to be careful when we advocated it or we would risk creating more confusion than we would sort out. For the SIA would only be to be recommended with the proviso that when we apply it to practical cases, we must not forget to take the second step in our calculations if the total available information allows us to do so. Otherwise the probability estimates we come up with will not reflect the true implications of the data we are analysing. This always true when we are estimating probabilities, but it is especially pertinent to point it out in this case because in practice we will always have more information available than goes into the SIA. Moreover, if we reject an inference of the DA type, as many of us are inclined to do, then we must also abstain from applying the SIA: we should either apply both considerations or none. From a pedagogical point of view it could even be argued that it would advisable to regard the SIA as unsound, for people will tend not to make the DA probability shift in any case, so one would hardly do them a favour by insisting that they use the SIA, unless one went to great length to explain the whole situation. There is nothing deep or very subtle about this. Either you cross the street or you stay on the pavement, and in both cases you will be safe; but if you stop midway you’ll get run over by a bus.

If the SIA is accepted then the DA is obliterated, without residue. But is the SIA true? Leslie does not think so, of course, and Dieks did not seriously attempt to give independent motivation for what I have called the SIA. So how are we to decide?

In general one would expect that most arguments working against the DA would also be arguments working in favour of the SIA, and vice versa, simply because the SIA is such a plausible way for the DA to fail, if indeed it does. In the remainder of this section we will focus on four arguments that are more directly aimed at the SIA: (1) the argument involving IPC considerations (to be explained in a moment), (2) the argument from expected utility, (3) the argument from infinity, and (4) the argument from no coincidence.

1. IPC considerations

One argument, presented in the following paragraph (and which is subsequently to be shown to be flawed), attempts to show that the is what I shall call interpersonal congruence violating (IPC violating). An alleged principle of rationality is IPC violating iff it violates

(IPC) In any idealised betting context, if a group of persons have exactly the same priors, equivalent information sets (up to "indexical facts" of the type "I am Mr. B"), the same preferences, and accept the same canons of rationality, then no bet is possible between them from which they should all expect to benefit.

The meaning of the IPC condition will be made clearer as we proceed.

One (fallacious) argument that the negation of SIA is IPC violating

"Consider again the thought experiment with God’s coin tosses. If we modify the experiment to include one additional person, the owner of a nearby betting shop, then this wouldn't change anything essential. (We can see this if we transform the experiment so that instead of one or ten persons being created, either one billion or ten billion persons are created; the bookie can be made an arbitrarily small part of the scheme.) But if one refused to accept the SIA then one would have no good reason to decline a bet on whether God’s coin had landed heads or tails that gave us a 50:50 + e odds for tails. The bookie, on the other hand, would have a very good reason to offer everybody such a bet: he would buy a 50% chance of winning 10 * s for the prize of a 50% risk of losing 1 * s * (1 + e ). Surely he would be irrational not to offer you that bet in this idealised situation. But then, how could it be rational of you to accept it? For you have exactly the same information as the bookie, and we can also assume that you accept the same canons of rationality and make the same estimates of all the prior probabilities.

Both you and the bookie know that God has made his toss of a fair coin and created either ten or one human as a result. You both know that you, the bet-taker, are alive; that you were created because of the coin toss. Neither of you has seen how the coin landed. What other relevant discrepancy between your information sets is there to warrant the divergence of your probability estimates?

There is none; your sets of relevant information are the same, and therefore your rational probability estimates must coincide. But since the bookie is obviously rational to offer you the bet, there must be something wrong with the principle that leads you to accept it and expect to benefit from it. This principle is the principle that your existence does not give you any reason whatever to believe that many observers will ever exist, not even "other things equal" as in this idealised case. Thus we should reject this principle. If we also reject the implausible principle that your existence indicates that few observers will ever live (an argument for rejecting that principle could be construed along the same lines), then we are logically forced to adopt principle SIA, which is, in effect, no more than the negation of both of these two rejected principles."

 

In response to this, the advocate of the DA could choose to sacrifice the IPC and admit that it may be violated in the contexts where the DA applies. I think this would be pretty damning for the DA. It would then begin to seem more plausible to assume the SIA, if we could thereby avoid IPC violation.

There is, however, a definite flaw in the argument above. It was claimed that by introducing a bookmaker in the God’s coin toss example, we didn’t change anything essential. But that's not the case. Whereas it is true that by scaling up the numbers of people that are created as a result of the coin toss we can make the change in these people’s rational probability estimates due to the introduction of the bookie arbitrarily small, the same does not hold for the bookie’s rational probability estimates! The more people that are created as a result of the coin toss whichever way it landed, the more surprising it should be for the bookie to find himself being the bookie rather than one of the people in the experiment. One could think that since the potential gain from offering bets that the coin landed heads are much greater for the bookie than the potential loss, it would be rational for him to offer bets at a price at which it would be advantageous for the subjects in the group to accept it (in conflict with the IPC). This reasoning, however, overlooks the fact that it would be unlikely in the first place that the bookie would be the bookie if there were many other roles he could have been in instead. So from observing that he himself is the bookie, he obtains evidence that the coin probably landed tails. It turns out that this effect exactly counterbalances the factor defining the quotients of the potential gains to the potential loss.

Since it is not trivial to see that the compensation is precise, we have to perform a calculation. We want to show that the fair odds, y, for the bookie are the same as the fair odds, x, for the subjects in the group. Since everybody have the same information sets etc., this is a necessary requirement if we are to avoid IPC violation. In order to prevent ambiguity, we attach the subscript "y" to all the probabilities that are associated with the bookie, and the subscript "x" to all those associated with the subjects in the group. We can then write the conditions for fairness of the odds as

.

and

We have already derived . We can reduce to known quantities using Bayes’ theorem:

If we solve the two formulae representing the fairness conditions, solve them for for x and y respectively, and then set x = y, we get

                     ($)

Thus ($) is the equality that must be satisfied if the IPC condition is not to be violated. Some algebraic manipulation can take ($) into an equivalent expression that is more useful if we want to check whether it is satisfied by specific numbers

                 ($$)

where we have introduced the abbreviations

In the special case we discussed above, the parameter values were {R=1, f=1, m=10, N=1}, and it turns out that they satisfy ($). Numerical calculations indicate that this holds for all parameter values. Thus is not violated in these (quite general) situations. Note that this has as a consequence, however, that, provided the subjects in the group assign the same priors as the bookmaker, the condition ($) is always satisfied. This means that the SIA is not IPC violating either! Thus we find that the IPC requirement cannot be used to arbiter between SIA and.

2. Expected utility

Another argument for the SIA might be derived from the fact that acceptance of the SIA will maximize a quantity that we can call epistemic utility. In the simplest case the idea is that when a coin is tossed once and either one or ten humans are created, and there are no outsiders, then if the SIA is generally adopted as a principle of rationality, so that the people created will believe that the coin landed heads, then there would either be ten persons who were right or there would be one person who was wrong. If, on the other hand, the negation of the SIA were generally adopted, so that people made their guesses at random, then on average there would be either half a person who was right or five persons who were right; and that outcome seems inferior to the outcome if the SIA were generally accepted.

So if the maximizing epistemic utility is what we are trying to do then we should adopt the SIA. This would then have the consequence that the shift we have to make after finding that we have a low birth rank would be exactly cancelled by the initial higher probability that many observers exist; as we saw in a previous section. Thus the DA would fail.

It is, however, far from clear that maximising epistemic utility is our concern here. If I am trying to decide how likely it is that mankind will go extinct soon, then the probabilities I am interested in are the probabilities which it would be most rational for me to accept. How many other persons will be right or wrong seems entirely irrelevant.

Consider a case where you are being closely monitored by a sexist king who has the power to instil beliefs in a large portion of the population, though propaganda etc. Suppose you have good reasons to believe that if the monarch would find that you believe that his first child will be a son then he will be in a good humour and instil true beliefs in his people; and that if you believe that his first child will be a daughter then the opposite will happen. Would this make it overwhelmingly likely that the queen will give birth to a boy?

It seems that there is a sense of "rationality" according to which it would be irrational for you to believe that the probability that first child will be a boy is anything other than about 50% (depending on your earlier experience). In this sense of rationality it would, analogously, be wrong to regard the greater expected epistemic utility of the SIA as a reason for its rationality.

Given some more facts about the situation and the likely outcome of your actions, it may of course be instrumental for you to adopt the belief that there is a 99% chance that the king's first child will be a son. (Perhaps you have to drink some magical brew to be able to believe this). If this were so, then there would also be a sense of rationality according to which it would be rational for you to believe (or to adopt the belief) that the chance of a son is 99%. This is the sense of rationality according to which an option (e.g. changing one of your beliefs) is rational only if there is no other option that has a greater expected utility. However, if we take rationality in this sense, then the conclusion does not carry over to the DA. For we have not seen any reason to believe that adopting the SIA would maximize expected utility.

The calculation above indicated that acceptance of the SIA would maximize expected epistemic utility, in a certain rather contrived sense. But that is something completely different from expected utility simpliciter. There is no obvious reason to believe that the world would be more likely to become a better place if the SIA is accepted than if its negation is. Perhaps it could even be argued that the important thing, if we want to maximise the chances of making the world a better place, is that as large a fraction as possible of the people who exist, regardless of how many they are or will have been, have views that are correct. (Because that determines what democratic decisions are made.) In that case the argument would rather favour the . (This could be true even if people having true beliefs were the sole good. It would still not follow that it would be irrational in this sense not to accept SIA. For whereas the SIA would score better than its negation in the context of the DA, there are many other areas of knowledge that would presumably be best served by whatever principle it is (SIA or) that, if adopted, would be most likely to lead as large a fraction of existing people to be right; the absolute numbers seem less important. Only if the good equals the number of people who hold correct views about when mankind will become extinct would an expected utility argument show that it would be rational to favour the general acceptance of the SIA, in the instrumental sense of rationality.)

The conclusion is that whether we are talking about maximising your immediate likelihood of being right or about maximising any plausible form of expected utility then the above argument fails to support the claim for superiority of the SIA to its negation.

3. The problem of infinity

One of Leslie's favourite objections against the SIA is that it would lead to the conclusion that it is certain that the world contains infinitely many observers. For so long as our prior probability of the world containing infinitely many observers is greater than zero then any plausible formalization of the SIA would seem to imply that the posterior (before the DA is applied) is infinitely probable, i.e. certain.

One could think that this might not matter, if the application of the DA were then to restore the status quo. But imagine a situation (the amnesia chamber) where you happened not to know your birth rank: there you would now have to say that with probability one the world contains infinitely many individuals, despite the knowledge you might possess that the observers in the universe were caused by a random process which was extremely unlikely to generate infinitely many of them. This would clearly be a highly counterintuitive consequence.

I think infinity problem is very serious for the proponent of the SIA, though as it stands I don't think it amounts to a conclusive refutation. What are the loopholes? One could perhaps hope to formalize some variant of the SIA that would somehow discount very large shifts, or make some kind of renormalization, but I don't see how this could be done without violating basic Bayesian principles. Alternatively, one might bite the bullet and accept that had we not known our birth rank then we should indeed assign probability one to the propositions that the universe contains infinitely many observers. Perhaps some would try to motivate this move by saying that, although counterintuitive, that wouldn't matter for practical purposes since we always know our approximate birth rank. This motivation is wrong though, because we are not looking for a rule of thumb but for a way to determine whether the DA is correct or not. Yet another escape route would be to say that infinities cause lots of problems in decision theory and probability theory anyway (consider Pascal’s wager for example), so it's not particularly blemishing for the SIA that they cause problems for the SIA too. Another route still would be to argue that for some deep reason, we should really have expected all along that it would be certain that the world contains infinitely many observers. For instance, one could follow David Lewis and believe that all possible worlds exist.

4. No coincidence, no surprise

A further blow is dealt the SIA by the following consideration. A slightly nave-sounding but useful way of putting it is in terms of coincidences (following Derek Parfit [1997] and personal communication). A coincidence is something that requires two independent things that coincide; one thing makes no coincidence. Now, if I am in the amnesia chamber and learn that my birth rank is very low then there is a coincidence: the coincidence between me (the one that is here and now, in the amnesia chamber) and a certain (very low) birth rank. These two things are independent, at least as far as I know in the amnesia chamber.

But what are the two independent things in the case of the shift that the SIA postulates that I should make after "finding myself alive"? One thing is me (the guy that is here and now). But the other --me being alive -- could hardly be said to be independent of the first thing. The only reason why me was picked out as the first thing was that there happened to be a real person (me) who chose to take himself as a random sample. It is not surprising that a non-existing, merely possible human, was not picked out as the first thing; -- for there would be nobody to pick him. Thus the second thing (me being alive), far from being independent of the first thing, was in fact a necessary prerequisite for the choice of the first thing.

If we wanted to modify this second case so that we would have something where there was a coincidence, what we would have to do would be to create a big urn with paper slips containing descriptions of all possible humans or observers. Then we would draw a paper slip randomly from that urn, and then send somebody out in the world to check if the possible observer described on that slip is actual. If he or she is, then this would give us increased reason to believe the world contains many observers. But failing a procedure such as this, there is no coincidence; and hence no shift should be made; and hence the SIA is false.

This argument holds some persuasive and explanatory power, at least for me. But formulated as it is in rather loose terms it cannot be regarded as a conclusive refutation of the SIA.

A summary of the reasons for and against the SIA

One of the central issues involved in the assessment of the DA is the self-indication axiom, the SIA. In this section we have discussed four main arguments that are directly related to the SIA. The first was an argument stemming from considerations about IPC violation. It was hoped that such considerations would show that the were IPC violating, thereby giving us strong reason to adopt the SIA and thereby to reject the DA. It was found that neither the nor the SIA is IPC violating. The second argument tried to support the SIA by means of calculating "expected epistemic utilities". This argument was shown to be wrong. The third argument noted that adoption of the SIA seems to lead to the prediction that with probability one the universe contains infinitely many observers. This was assessed as a weighty problem with the SIA, though not one that has (yet) conclusively shown the SIA to be false. The last argument pointed out that there is a sharp disanalogy between shift postulated by the DA and the shift resulting from starting to adopt the SIA: the former shift is motivated by the existence of a significant coincidence, the latter shift lacks such motivation.

It is fair to say that the outcome of this evaluation is in favour of the. The only ground for still preferring the SIA would be if one had overriding reasons to believe that the DA is false and that accepting the SIA is the only way to maintain this conviction. If that is what one wants to do then one certainly owes the doomsdayer a couple of good replies to the third and the fourth argument.

You as a random sample

Let's now make a new stab at the problem of the reference class.

We have already seen that though your choice of a reference class contains an element of arbitrariness, it is also restricted by the requirement that non-cognitive things must be excluded and that it must not make explicit or implicit reference to yourself. The problem is now to determine what cognitive things can be included.

The question you must ask yourself is this: "What are the probability distributions from which, according to the rival hypotheses, respectively, you should regard your birth rank as a random sample?". Given these distributions then you can straightforwardly apply Bayes' rule to derive the new probabilities of the hypotheses.

To my knowledge, nobody has been able to answer this question convincingly. Let us list a number of possible answers with some degree of prima facie plausibility:

The individuals to which the distributions assign non-zero probabilities (roughly: individuals which you could have turned out to have been, had you suffered amnesia and were subsequently to rediscover your identity) are all:

  • Individuals who would actually have thought about the DA, given that the hypothesis (on which they exist) were true.
  • Individuals who could have understood about the DA, i.e. who are intelligent enough to understand the argument were it to be explained to them.
  • Individuals who possess a conception of their own approximate place in human history.
  • Individuals who have self-awareness.
  • Individuals who are conscious.

All of these possible maximal reference classes seem to use criteria that involve matters of degrees. They are therefore in need of further specification.

As a first improvement, we should substitute "episodes" for "individuals". That way we will take care of the ambiguity of how to weigh individuals who spend different amounts of time thinking about the DA; who live differently long lives; who are awake varying amounts of time; etc. For example, according to (1) then each moment of thinking about the DA, for each individual, counts as an entity in the reference class over which the probability distribution is non-zero.

This goes some way towards disambiguation, but there are still problematic grey-zones left. For instance, an episode of thinking about the DA can come in different degrees of clarity, intensity and focus. Assuming (1), should we say that if there are equally many deep, clear as muddled, superficial thinkers about the DA (which is certainly not the case) then one should expect to find oneself as one of the clear thinkers? Or assuming (5), are highly intense consciousness-moments to be given more weight than torpid ones?

The question about the probability distribution (i.e. which one of (1)-(5) -- or some variant of these -- to adopt and how to disambiguate the chosen criterion) is an awkward one for the proponent of the DA, for two reasons.

First, the conclusion of the DA might vary drastically depending on how we choose the distributions. For example, if we choose (1) then the preferred conclusion might be that the people of the future will simply not think about the DA anymore. There could be many explanations of why that might be. This alternative would not be open to us if we chose any of (2)-(5). Even if we accept the basic soundness of the DA style reasoning, it is difficult to know what conclusion to draw until we have settled the question about the reference class and the distribution over it.

Second, pondering over the reference class question, one may come to wonder whether at the bottom there really is any fact of the matter at all. If there is no fact of the matter then the DA proponent would have to defend the implausible-seeming position that it is a fact of the matter that one of (1)-(5) is right, but there is no fact as to which one. He would have to argue that our notion of rationality hasn't got sufficiently high resolution to determine which one of (1)-(5) (or some variant) is the right one, only that the reference class must not include non-cognitive beings, that it must not be defined in an ad hoc manner with respect to the subject who is the sample, and that it must satisfy some other rather general criteria.

Conclusion: There are genuine problems remaining about the reference class. They make it difficult to say exactly what the DA would show, and they also cast some doubt on its validity.

 

So what are we to believe?

The DA has so far withstood all attempts at refutation. The main reason for still doubting it is its extraordinary nature (anthropic reasoning) and its shocking conclusion. But how could it be consistently disbelieved? We don’t want to give up the basics of Bayesianism, and if we don’t do that then there seems to be no way to avoid making a shift. Our discussion about changing the priors showed that that course of action didn’t look promising on closer inspection. Keeping the priors and making the shift, we inevitably end up with a DA-like conclusion. Exactly, or even approximately, what this conclusion is, we cannot tell, because that depends on how we define the reference class, and no one knows why we should do that one way rather than another (which raises an additional suspicion about the DA’s validity).

It should be mentioned that Leslie believes that the conclusion of the DA is much weakened by the indeterminism of the world, made plausible by quantum physics. I decided not to treat that aspect in this paper.

The most important further illumination that we now need on the DA is in my opinion on how to find a way to settle the question about the reference class. If the DA is wrong then in trying to solve the reference class problem we may hope to find out why it is wrong. If the DA is right then after the reference class problem is solved, we will have better grounds for believing it, and we would know what the DA really argues for. Considering the extremely important beliefs (concerning the very survival of humankind) that are a stake, it should be a priority to try to resolve these perplexities as soon as possible.

 

Acknowledgements

I am grateful to Nigel Armstrong, Wei Dai, Jean-Paul Delahaye, Jean Delhotel, J. F. G. Eastmond, Hal Finney, Mark Gubrud, Robin Hanson, Colin Howson, Thomas Kopf, Kevin Korb, John Leslie, Jonathan Oliver, Derek Parfit, David Pearce, Sherri Roush, Anders Sandberg and Damien Sullivan for valuable comments on some of these thoughts.

Back to anthropic-principle.com

 

NOTES    [The numbers referring to these notes are unfortunately missing in the web-version of this document]

See also the articles listed in the reference section, including the papers by Eckhardt, Dieks, Oliver & Korb, Kopf, T.& Krtous, P.& Page, D. N., T nnsj , T., and Delahaye, and Leslie’s responses to some of these, as well as the discussion in Nature that followed Gott’s article.

See e.g. Howson & Urbach [1993], pp. 403-11, or Leslie [1996], pp. 218-20.

Leslie [1996] covers most of the material previously published in his journal articles. For his replies to objections, see especially chapters five and six.

Personal communication. See also Leslie [1996], p. 228.

This thought experiment is also used by Leslie [1996], p. 227 ff.

This argument supports that in the present context we can make what Eckhardt has called the Human Randomness Assumption. Cf. Eckhardt [1997], p. 248.

Leslie [1996], p. 262

It is possible and might be advisable to substitute the following weaker formulation:

(SIA-) The world is a priori more likely to contain many observers than to contain few.

This achieves the same effect as the (SIA) while being weaker in that it does not commit us to a specific view on what a priori grounds it is that make the world likely to contain many observers.

The calculation is trivial. Let be the naive prior for the hypothesis that in total i observers will have existed, and assume that for i greater than some finite N. (This restriction allows us to disregard the problem of infinities, which we will deal with in a later section.) Then we can formalize the SIA as saying that

where is a normalization constant. Let r(x) be the rank of x, and let "I" denote a random sample from a uniform probability distribution over the set of all observers. We then have

We can assume that . (If not, then the example simplifies to the trivial case where the hypothesis is conclusively refuted regardless of whether the SIA is accepted.) Using Bayes’s formula, we can expand the quotient between two possible hypotheses:

and we see that after we have applied both the SIA and the DA, we are back to the naive probabilities that we started from.

For Leslie's view on the SIA, see [1996], pp. 224-8.

One could think that talking of episodes instead of individuals could screw up the way the no-outsider requirement works. Suppose there is just one outsider and either one or ten million insiders. Then, according to the formula we called GDA, this one outsider won’t make much of a difference to the rational probability estimates of an insider, relative to the case where no outsider exists. But the life of this outsider might typically contain many millions of episodes. So now the number of entities in the outsider class will be counted in millions. However, this won’t change the probability estimates, provided that each of the lives of the insiders contains as many episodes as the life of the outsider. This is easy to see if we examine GDA. The variables that would change value as a result of substituting "episodes" for "individuals" occur only in quotients, in such a way that it doesn’t matter, for instance, into how many episodes we divide one hour of thinking about the DA; the multipliers cancel out

 

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