Here, α is the observer-moment whose subjective probability function is . is the class of all possible observer-moments about whom h is true; is the class of all possible observer-moments about whom e is true; is the class of all observer-moments that places in the same reference class as herself; is the possible world in which is located; and γ is a normalization constant Pα Ωh Ωe Ωα α wα α ∑∈Ω Ω ∩Ω = e w P w σ σ σ α σ γ | ( ) | ( ) OE can be generalized to allow for different observer-moments within the reference class having different “weights”, an option that might be of relevance for instance in the context of the many-worlds version of quantum theory. 19 6 Redelmeier and Tibshirani (1999) • “Differential surveillance can occur because drivers look forwards rather than backwards, so vehicles that are overtaken become invisible very quickly, whereas vehicles that overtake the index driver remain conspicuous for much longer;” and • “Human psychology may make being overtaken (losing) seem more salient than the corresponding gains.” The authors recommend that drivers should be educated about these effects in order to reduce the temptation to switch lanes repeatedly. This would reduce the risk of accidents, which are often caused by poor lane changes. While all these psychological illusions might indeed occur, there is a more straightforward explanation for the drivers’ persistent suspicion that cars in the next lane are moving faster. Namely, that cars in the next lane actually do go faster! One frequent cause of why a lane (or a segment of a lane) is slow is that there are too many cars in it. Even if the ultimate cause is something else (for example, road work) there is nonetheless typically a negative correlation between the speed of a lane and how densely packed the vehicles driving in it are. This implies that a disproportionate fraction of the average driver’s time is spent in slow lanes. If you think of your present observation, when you are driving on the motorway, as a random sample from all observations made by drivers, then chances are that your observation will be made from the viewpoint that most such observer-moments have, which is the viewpoint of the slowmoving lane. In other words, appearances are faithful: more often than not, for most observer-moments, the “next” lane is faster. Even when two lanes have the same average speed, it can be advantageous to switch lanes. For what is relevant to a driver who wants to reach her destination as quickly as possible is not the average speed of the lane as a whole, but rather the speed of 20 some segment extending maybe a couple of miles forward from the driver’s current position. More often than not, the next lane has a higher average speed at this scale than does the driver’s present lane. On average, there is therefore a benefit to switching lanes (which of course has to be balanced against the costs of increased levels of effort and risk). Adopting a thermodynamics perspective, it is also easy to see that (at least in the ideal case) increasing the “diffusion rate” (that is, the probability of lane-switching) will speed the approach to “equilibrium” (where there are equal velocities in both lanes), thereby increasing the road’s throughput and the number of vehicles that reach their destinations per unit time. To summarize, in understanding this problem we must not ignore its inherent observation selection effect. This resides in the fact that if we randomly select an observer-moment of a driver and ask her whether she thinks the next lane is faster, more often than not we have selected an observer-moment of a driver who is in a lane which is in fact slower. When we realize this, we see that no case has been made for recommending that drivers change lanes less frequently.7 11. Observation selection theory (also known as anthropic reasoning), which aims to help us detect, diagnose, and cure the biases of observation selection effects, is a philosophical goldmine. Few branches of philosophy are so rich in empirical implications, touch on so many 7 The above reasoning applies to a driver who is currently on the road wondering why she is in the slow lane. When considering the problem retrospectively, that is, when you are sitting at home thinking back on your experiences on the road, the situation is more complicated and requires also taking into account differential recall (psychological factor may make you more likely to remember and bring to mind certain kinds of experiences) and the fact that 21 important scientific questions, pose such intricate paradoxes, and contain such generous quantities of conceptual and methodological confusion that need to be sorted out. Working in this area is a lot of intellectual fun. The mathematics used in this field, such as conditional probabilities and Bayes’s theorem, are covered by elementary arithmetic and probability theory. The topic of observation selection effects is extremely complex, yet the difficulty lies not in the math, but in grasping and analyzing the underlying principles. References Bartha, P. and C. Hitchcock, "No One Knows the Date or the Hour: An Unorthodox Application of Rev. Bayes's Theorem," Philosophy of Science (Proceedings) 66 (1999): S329-S53. Bartha, P. and C. Hitchcock, "The Shooting-Room Paradox and Conditionalizing on Measurably Challenged Sets," Synthese 108(3) (2000): 403-37. Bostrom, N., "Investigations into the Doomsday argument." Preprint (1997). Bostrom, N., "The Doomsday argument, Adam & Eve, UN++, and Quantum Joe." Synthese 127(3) (2001): 359-87. 22 while the slow lane contains more observer-moments, it may nevertheless be true that more drivers have passed through the fast lane. Bostrom, N., Anthropic Bias: Observation Selection Effects in Science and Philosophy (New York: Routledge, 2002a). Bostrom, N., "Self-Locating Belief in Big Worlds: Cosmology's Missing Link toObservation," Journal of Philosophy 99(12) (2002 b). Dieks, D., "Doomsday - Or: the Dangers of Statistics," Philosophical Quarterly 42(166) (1992): 78-84. Hall, N., "Correcting the Guide to Objective Chance," Mind 103(412) (1994): 505-17. Leslie, J., The End of the World: The Science and Ethics of Human Extinction (London: Routledge, 1996). Lewis, D., Philosophical Papers (New York: Oxford University Press, 1986). Lewis, D., "Humean Supervenience Debugged," Mind 103(412) (1994): 473-90. Oliver, J. and K. Korb, A Bayesian analysis of the Doomsday Argument,[is this a book or article…?] Department of Computer Science, Monash University, 1997. Olum, K., "The Doomsday Argument and the Number of Possible Observers," Philosophical 23 Quarterly 52(207) (2002): 164-84. Redelmeier, D. A. and R. J. Tibshirani, "Why cars in the other lane seem to go faster," Nature 401 (1999): 35. Smith, Q., "Anthropic Explanations in Cosmology," Australasian Journal of Philosophy 72(3) (1994): 371-82. Thau, M., "Undermining and Admissibility," Mind 103(412) (1994): 491-503.