# Copernican probability estimates

Benjamin Kay writes,

The article Survival Imperative for Space Colonization applies what they call Copernican probability estimates. Essentially, this is a way of getting a probability estimate of the life span being in an interval by assuming their is nothing special about it being alive today. Is this like choosing a very particular prior such that the posterior distribution is uniform, and essentially updating it with a single observation? I’m just wondering what this is exactly, and I’d be interested in seeing a Statistical Modeling, Causal Inference, and Social Science post on this subject if you were looking for something to write about.

Is statistical inference or really more of a parlor trick?

My reply: I think this is the same as the “doomsday argument,” and I think it’s wrong. To quote myself from a couple years ago:

The argument is that there is there is a high probability that humanity will be extinct (or drastically reduce in population) soon, because if this were not true–if, for example, humanity were to continue with 10 billion people or so for the next few thousand years–then each of us would be among the first people to exist, and that’s highly unlikely.

Anyway, the (sociologically) interesting thing about this argument is that it’s been presented as Bayesian (see here, for example) but it’s actually not a Bayesian analysis at all! The “doomsday argument” is actually a classical frequentist confidence interval. Averaging over all members of the group under consideration, 95% of these confidence intervals will contain the true value. Thus, if we go back and apply the doomsday argument to thousands of past data sets, its 95% intervals should indeed have 95% coverage. If you look carefully at classical statistical theory, you’ll see that it makes claims about averages, not about particular cases.

However, this does not mean that there is a 95% chance that any particular interval will contain the true value. Especially not in this situation, where we have additional subject-matter knowledge. That’s where Bayesian statistics (or, short of that, some humility about applying frequentist inferences to particular cases) comes in. The doomsday argument is pretty silly (and also, it’s not Bayesian). Although maybe it’s a good thing that Bayesian inference has such high prestige now that it’s being misapplied in silly ways. That’s a true sign of acceptance of a scientific method.

P.S. I also noticed this discussion. Unfortunately all the commenters treat this as a Bayesian analysis. Again, this could be considered a good sign, that frequentist inference is now so obscure that nobody recognizes it when it is not labeled as such.

## 5 thoughts on “Copernican probability estimates”

1. I think that many people have decided that they can turn a frequentist analysis into a Bayesian one simply by saying "we use a uniform prior", eg in the Doomsday argument case assuming that the observer was uniformly sampled out of all people. The uniform prior is of course ignorant so this way we get a Bayesian inference from the data alone without making any subjective assumptions (that's a joke, at least I think it is, but this attitude is common enough in the literature).

Since basically everyone (at least in my field of climate science) has spent so long misinterpreting frequentist confidence intervals as Bayesian credible intervals in this way, they don't even realise that there might be some problem with what they are doing. Indeed this "theory" has been explicitly proposed as a new approach to probability by some prominent climate scientists…

2. The Doomsday argument does have a Bayesian form (associated with other people than Gott), but it doesn't work unless 1) you accept the "self-sampling assumption" that says you can reason as if you were randomly selected from the set of all observers and 2) you do not accept the "self-indication assumption" that says possible universes containing more observers should have a proportionally higher prior. It's philosophically very tricky. I recommend Nick Bostrom's book and other work on this: http://www.anthropic-principle.com/book/

Here's a paper that claims to translate Gott's argument into a Bayesian one (albeit one with a specific prior that I don't think is right for any of the cases it's being used): http://spot.colorado.edu/~monton/BradleyMonton/Ar

James, in the Doomsday argument case, isn't the relevant prior the one over total observer numbers? That's the one that gets turned into a posterior (which you could then translate into a credible interval, which coincides with the frequentist confidence interval if you chose the right prior). Am I confused here?

3. It seems to me that there are certain philosophical problems with reasoning about the probability of the current state of affairs. Since it's already happened, any calculations involving its probability are moot.

Everybody who disagrees is invited to argue over the probability that I will press the Post button.

4. Maybe interesting in this context:
The biologist Richard Lewontin wrote, in the context of explaining why evolution does not result in a general increase of the fitness of life to the external world, the following in a book review:
"Judging from the fossil record a typical mammalian species lasts roughly ten million years, so we [humans] might expect to last another nine million unless, as a consequence of our immense ability to manipulate the physical world, we either extinguish ourselves a good deal sooner or invent some extraordinary way to significantly postpone the inevitable." (p.52)
The wars over evolution. New York Review of Books, Oct.20 2005