John Tierney today writes about Richard Gott’s Copernican principle. He has a little more on his blog, along with some useful discussion from Bradley Monton. The principle in question says that you should treat the time of your observation of some event as being a random point in its duration. Slightly more formally, quoting Gott via a paper Monton wrote with Brian Kierland,
Assuming that whatever we are measuring can be observed only in the interval between times tbegin and tend, if there is nothing special about tnow, we expect tnow to be located randomly in this interval.
As Monton and Kierland note, we can use this to argue that the probability that
tfuture is between a and b times tpast
is 1/ ( a + 1) – 1 / ( b + 1), where tpast is the past duration of the event in question, and tfuture is its future duration. Most discussion of this has focussed on the case where a = b = 39. But I think the more interesting, or at least easy to interpret, case is where a = 0 and b = 1. In this case we get the result that the probability of the entity in question lasting longer into the future than its current life-span is 1/2.
As a rule I tend to be very hostile to these attempts to get precise probabilities from very little data. I have a short argument against Gott’s Copernican formula below. (Against the general version, not for any particular values of a and b.) But first I want to try a little mockery. I’d like to know anyone who would like to take any of the following bets.
Wikipedia’s History of the Internet dates the founding of the World Wide Web to around the early 1990s, so it is 15 or so years old. Gott’s formula would say that it less than 50/50 that it will survive until around 2025. I’ll take that bet if anyone is offering.
The iPhone has been around for about 3 weeks at this time of writing. Again, Gott’s formula would suggest that it is 50/50 that it will last for more than 3 weeks from now. Again, I’ll take that bet!
Finally, it has been about 100 years since there were over 4,000,000 people on the Australian continent. I’m unlikely to be around long enough to see whether there still will be more than 4,000,000 in 100 years time, but I’m a lot more than 50/50 confident that there will be. I will most likely be around in 10 years to see whether there are more than 4,000,000 people there in 11 years time. Gott’s formula says that the probability of that is around 0.9. I’m a little more optimistic than that, to say the least.
Anyway, enough mockery, here is the argument. Consider any two plays, A and B, that have been running for x and y weeks respectively, with x > y. And consider the following three events.
E1 = Play A is running
E2 = Play B is running
E3 = Plays A and B are both running
Note that E3 has been ongoing for y, just like E2. The Copernican formula tells us that at some time z in the future, the probabilities of these three events are
Pr(E1 at z) = x / (x + z)
Pr(E2 at z) = y / (y + z)
Pr(E3 at z) = y / (y + z)
Now let’s try and work out the conditional probability that A will still be running at z, given that B is running at z. That is, Pr(E1 at z | E2 at z). It is
Pr(E1 at z & E2 at z) / Pr(E2 at z)
= Pr(E3 at z) / Pr(E2 at z)
= (y / (y + z)) / (y / (y + z))
So using the Copernican formula, we can deduce that the conditional probability of A still running at z given that B is still running at z is 1. And that’s given only the information that z is in the future, and that A has been running at B. That is, to say the least, an absurd result. So I’m sure there is something deeply mistaken with the Copernican formula.