Re: effective probability

From: Gale <wmgale.domain.name.hidden>
Date: Mon, 01 Feb 1999 11:47:05 -0500

Gilles HENRI wrote:
>[hal wrote:]
> >I thought that the explanation for why the doomsday argument fails in
> >the everything-exists case went like this:
> >
> >Either we are in a world where the human race lives for a very long
> >time but we are very early in its history, or we are in a world where
> >it lives for a short time and we are at a typical point in its history
> >(or something in between).

> >Hal
>
> I think it's true. Bayes theorem is correct but the prior probability is
> very strongly in favor of worlds with many humans.
> However we can do estimate something about the duration of the human race.
> For most of the distributions of births (including exponential growth of
> the population), the average (over all human beings) ratio of
> (tf-t)/(t-ti), the time left before doomsday over the time past since the
> beginning of mankind, is of the order of one. The best estimate is thus
> that mankind has to live around as much time it has already lived, between
> 10^5 to 10^7 years.

It is not correct to apply time as a measure for the human race, because
you and I are members thereof, and because the number of people per unit
of
time has not been uniform. John Leslie is correct in using the birth
order
of people as a measure of time passage. You and I are indeed not in
peculiar positions by this measure, while we are by direct time measure.

It is interesting to note that the "Copernican" estimate that you give
originally, to my knowledge from Gott, does seem to give plausible time
estimates, but has the following property that makes me very suspicious
of Copernican estimates. (First, the Copernican estimates as a whole
do not have a finite expectation, but the median estimate is for future
interval = observed past interval). Then for Copernican estimates on
humans (as if they were made by an alien). The time, as you indicate
is 10^5 to 10^7. The number of people is about 60 10^9. The pair of
forecasts can both be met if the total forecast is for a rapid decline
of current population, then a long period of very few members. This
is oddly reminiscent of a reversal of past history.

In fact, the following theorem is easy to see. Given a function
strictly montonically increasing on (-infinty, 0), then the
median Copernican forecast is the reversal about 0 of the function.

This may be the best that the Copernican principle can get you in
a single dimension in which you have only a one way view. Note that
it does put you at the most likely point (0). However, it is rather
clear that time reversal per se makes a pretty poor forecast for
anything about which one can have some notion that there is an
irreversibility.

> Note that this estimate is correct for most of your own remaining time of
> life (except the very few new born babies, dying old people and victims of
> mortal accidents). it is correct also to guess how long the Sun will
> bright, and so on...We may belong to the unfortunate part that will
> disappear, but despite the growth of population, we only represent a few
> percent of the 10^11 people that have probably already lived, so this
> estimate would have been correct for most of them.
>
> Gilles

I rather suspect that there could be a lot of empirical cases in which
the Copernican forecasts would be right, but it is hard to reconcile
that with the apparent basis in a time reversal.

Bill Gale
Received on Mon Feb 01 1999 - 08:49:26 PST