The Mysteries of Self-Locating Belief and Anthropic Reasoning Nick Bostrom Oxford University, Faculty of Philosophy 1. How big is the smallest fish in the pond? You take your wide-meshed fishing net and catch one hundred fishes, every one of which is greater than six inches long. Does this evidence support the hypothesis that no fish in the pond is much less than six inches long? Not if your wide-meshed net can’t actually catch smaller fish. The limitations of your data collection process affect the inferences you can draw from the data. In the case of the fish-size-estimation problem, a selection effect—the net’s being able to sample only the big fish—invalidates any attempt to extrapolate from the catch to the population remaining in the water. Had your net had a finer mesh, allowing it to sample randomly from all the fish, then finding a hundred fishes all greater than a foot long would have been good evidence that few if any fish remaining were much smaller. In the fish net example, a selection effect is introduced by the fact that the instrument you used to collect data sampled from only a subset of the target population. Analogously, there are selection effects that arise not from the limitations of the measuring device but from the fact that all observations require the existence of an appropriately positioned observer. These are known as observation selection effects. The study of observation selection effects is a relatively new discipline. In my recent book Anthropic Bias, I have attempted to develop the first mathematically explicit theory of observation selection effects. In this article, I will attempt to convey a flavor of some of the mysteries that such a theory must resolveThe theory of observation selection effects may have implications for a number of fields in both philosophy and science. One example is evolutionary biology, where observation selection effects must be taken into account when addressing questions such as the probability of intelligent life developing on any given earth-like planet. We know that intelligent life evolved on Earth. Naively, one might think that this piece of evidence suggests that life is likely to evolve on most Earth-like planets, but that would overlook an observation selection effect. No matter how small the proportion of all Earth-like planets that evolve intelligent life, we must be from a planet that did (or we must be able to trace our origin to a planet that did, if we were born in a space colony) in order to be an observer ourselves. Our evidence—that intelligent life arose on our planet—is therefore predicted equally well by the hypothesis that intelligent life is very improbable even on Earth-like planets, as it is by the hypothesis that intelligent life is highly probable on Earth-like planets. The evidence does not distinguish between the two hypotheses, provided that in both hypotheses intelligent life would very likely have evolved somewhere. 2. Another example comes from cosmology, where observation selection effects are crucial considerations in deriving empirical predictions from the currently popular so-called ‘multiverse theories’, according to which our universe is but one out of a vast ensemble of physically real universes out there. Some cases are relatively straightforward. Consider a simple theory that says that there are 100 universes, and that 90 of these are lifeless and 10 contain observers. What does such a theory predict that we should observe? Obviously not that we should observe a lifeless universe. Because lifeless universes contain no observers, an observation selection effect precludes them 1 from being observed. So although the theory says that the majority of universes are lifeless, it nevertheless predicts that we should observe one of the atypical ones that contain observers. Now let’s take on a slightly more complicated case. Suppose a theory says that there are 100 universes of the following description: 90 type-A universes; they are lifeless. 9 type-B universes; they contain one million observers each. 1 type-C universe; it contains one billion observers. What does this theory predict that we should observe? (We need to know that in order to determine whether it is confirmed or disconfirmed by our observations.) As before, an obvious observation selection effect precludes type-A universes from being observed, so the theory does not predict that we should observe one of those. But what about type-B and type-C universes? It is logically compatible with the theory that we should be observing a universe of either of these kinds. However, probabilistically, it is more likely, conditional on the theory, that we should observe the type-C universe, because that’s what the theory says that 99% of all observers observe. Couldn’t we hold instead that the theory predicts that we should observe a type-B universe? After all, it says that type-B universes are much more common than those of type-C. There are various arguments that show that this line of reasoning is untenable. We lack the space to review them all here, but we can hint at one of the underlying intuitions by considering an analogy. Suppose you wake up after having been sedated and find yourself blindfolded and with earplugs. Let’s say for some reason you come to consider two rival hypotheses about your location: that you are somewhere on the landmass of Earth, or that you at sea. You have no evidence in particular to suggest that you should be at sea, but you are aware that there are more square meters of sea than of land. Clearly, this does not give you ground for thinking you are at sea. For you know that the vast majority of observers are on land, and in the absence of more specific relevant evidence to the contrary, you should think that you probably are where the overwhelming majority of people like you are. In a similar vein, the cosmological theory that says that almost all people are in type-C universes predicts that you should find yourself in such a universe. Finding yourself in a type-C universe would in many cases tend to confirm such a theory, to at least some degree, compared to other theories that imply that most observers live in type-A or type-B universes. 3. Let us now look a little more systematically at the reasoning alluded to in the foregoing paragraphs. Consider the following thought experiment: Dungeon. The world consists of a dungeon that has one hundred cells. In each cell there is one prisoner. Ninety of the cells are painted blue on the outside and the other ten are painted red. Each prisoner is asked to guess whether he is in a blue or a red cell. (And everybody knows all this.) You find yourself in one of these cells. What color should you think it is? – Answer: Blue, with 90% probability. Since 90% of all observers are in blue cells, and you don’t have any other relevant information, it seems that you should set your credence (that is, your subjective probability, or your degree of belief) of being in a blue cell to 90%. Most people seem to agree that this is the correct answer. Since the example does not depend on the exact numbers involved, we have the more general principle that in cases like this, your credence of having property P should be equal to the fraction of observers who have P. You reason as if you were a randomly selected observer. This 3 4 principle is known as the Self-Sampling Assumption: 1 (SSA) One should reason as if one were a random sample from the set of all observers in one’s reference class. For the time being, we can assume that the reference class consists of all intelligent observers, although this is an assumption that needs to be revised, as we shall see later. While many accept without further argument that SSA is applicable to Dungeon, let’s briefly consider how one might seek to defend this view if challenged to do so. One argument one can adduce is the following. Suppose that everyone accepts SSA and everyone has to bet on whether they are in a blue or a red cell. Then 90% of the prisoners will win their bets; only 10% will lose. If, on the other hand, SSA is rejected and the prisoners think that one is no more likely to be in a blue cell than in a red cell, and they bet, for example, by flipping a coin, then on average merely 50% of them will win and 50% will lose. It seems better that SSA be accepted. What allows the people in Dungeon to do better than chance is that they have a relevant piece of empirical information regarding the distribution of observers over the two types of cells; they have been informed that 90% are in blue cells. It would be irrational not to take this information into account. We can imagine a series of thought experiments where an increasingly large fraction of observers are in blue cells—91%, 92%, …, 99%. As the situation gradually degenerates into the limiting 100%-case where they are simply told, “You are all in blue cells,” from which each prisoner can deductively infer that he is in a blue cell, it is plausible to require that the strength of prisoners’ beliefs about being in a blue cell should gradually approach probability one. SSA has this property. 1 For further explorations of this and related principles, see Bostrom (1997), (2001), and (2002b).