Teacakes, Trains, Taxicabs and Toxins: A Bayesian Account of Predicting the Future
Teacakes, Trains, Taxicabs and Toxins: A Bayesian Account of Predicting the Future Thomas L. Griffiths & Joshua B. Tenenbaum Department of Psychology Stanford University Stanford, CA 94305-2130 USA {gruffydd ,jbt}@psych. stanford. edu Abstract This paper explores how people make predictions about the future. Statistical approaches to predicting the fu- ture are discussed, focussing on the method for predict- ing the future suggested by J. R. Gott (1993). A gener- alized Bayesian form of Gott’s method is presented, and a speciﬁc psychological model suggested. Three experi- ments show that the predictions people make about the future are consistent with a Bayesian approach. Despite the difficulty of predicting the future, people happily do it every day. We are conﬁdent about being able to predict the durations of events, how much time we will need to get home after work, and how long it will take to ﬁnish the shopping. In many cases we have a great deal of information guiding our judgments. How- ever, sometimes we have to make predictions based upon much less evidence. When faced with new situations our decisions about how much longer we can expect events to last are based on whatever evidence is available. When the only information we possess concerns how long a par- ticular event has lasted until now, predicting the future becomes a task of induction. In this paper we explore the question of how people predict the future when told only about the past. We examine a simple statistical method of predicting the future, and consider how such a method could be made sufficiently ﬂexible to be useful in everyday situations. The resulting Bayesian model makes strong predictions about the effects of providing further information, the symmetry of this form of reasoning, and how it should be affected by prior knowledge. We test these predictions empirically. The Copernican Anthropic Principle A simple solution to the problem of predicting the fu- ture was recently proposed by the cosmologist J. Richard Gott III (1993). Gott’s method is founded upon what he calls the “Copernican anthropic principle”, which holds that ...the location of your birth in space and time in the Universe is priveleged (or special) only to the extent implied by the fact that you are an intelli- gent observer, that your location among intelligent observers is not special but rather picked at random (1993, p. 316) Gott extends this principle to reasoning about our po- sition in time — given no evidence to the contrary, we should not assume that we are in a “special” place in time. This means that the time at which an observer encounters a phenomenon should be randomly located in the total duration of that phenomenon. Denoting the time since the start of a phenomenon tpast, and its total duration tt0m;, Gott forms what he terms the “delta t argument”. Deﬁne the ratio t T : past ttotal and assume that this is a random number between 0 and 1. It is possible to form probabilistic predictions about the value of 7“. For example, 7“ will be between 0.025 and 0.975 with a probability P : 0.95, meaning that Etpast l tfuture l 39tpast with 95% conﬁdence, where tfutwe = tmm; —tpa3t. Sim- ilarly, 7“ will be less than 0.5 with probability P : 0.5, so tpast l tfutwe with 50% conﬁdence. This method of reasoning has been used to predict a wide range of phenomena. Gott (1993) tells of his visit to the Berlin Wall in 1969 (tpast : 8 years). Assuming that his visit was randomly located in the period of the wall’s existence, the 95% conﬁdence interval for tfum,.e would be 2.46 months to 312 years. The wall fell 20 years later, consistent with these predictions. Gott made similar cal- culations of tfutwe for Stonehenge, the journal Nature, the U.S.S.R., and even the human race. Subsequent tar- gets of the principle have included Broadway musicals and the Conservative government in Britain (Landsberg, Dewynne, & Please, 1993). What’s Bayes got to do with it? Gott’s (1993) method for predicting the future yields interesting predictions in a wide range of situations. It is simple, but could prove useful in forming effective plans and expectations about future events. On this basis, it would be plausible for people to apply similar principles when making judgments concerning time. Despite the attractiveness of this claim, there may be good reasons why Gott’s (1993) method would not be- long in our cognitive armory. One reason could be the restrictive assumptions of such an inference. In many