How and how not to make predictions with temporal Copernicanism

Abstract

Gott (Nature 363:315–319, 1993) considers the problem of obtaining a probabilistic prediction for the duration of a process, given the observation that the process is currently underway and began a time t ago. He uses a temporal Copernican principle according to which the observation time can be treated as a random variable with uniform probability density. A simple rule follows: with a 95% probability,

$$39t > T - t > \frac{1}{39}t$$

where T is the unknown total duration of the process and hence T  −  t is its unknown future duration. Gott claims that this rule is of very general application. In response, I argue that we are usually only entitled to assume approximate temporal Copernicanism. That amounts to taking a probability distribution for the observation time that is, while not necessarily uniform, at least a smooth function. I work from that assumption to carry out Bayesian updating of the probability for process duration, as expressed by my Eq. 11. I find that for a wide range of conditions, processes that have already been underway a long time are likely to last a long time into the future—a qualitative conclusion that is intuitively plausible. Otherwise, however, too much depends on the specifics of various circumstances to permit any simple general rule. In particular, the simple rule proposed by Gott holds only under a very restricted set of conditions.