I'm sure that many of you here have read Quantum Computing Since Democritus. In the chapter on the anthropic principle the author presents the Dice Room scenario as a metaphor for human extinction. The Dice Room scenario is this:
1. You are in a world with a very, very large population (potentially unbounded.)
2. There is a madman who kidnaps 10 people and puts them in a room.
3. The madman rolls two dice. If they come up snake eyes (both ones) then he murders everyone.
4. Otherwise he releases everyone, then goes out and kidnaps 10 times as many people as before, and returns to step 3.
The question is this: if you are one of the people kidnapped at some point, what is your probability of dying? Assume you don't know how many rounds of kidnappings have preceded yours.
As a metaphor for human extinction, think of the population of this world as being all humans who ever have or ever may live, each batch of kidnap victims as a generation of humanity, and rolling snake eyes as an extinction event.
The book gives two arguments, which are both purported to be examples of Bayesian reasoning:
1. The "proximate risk" argument says that your probability of dying is just the prior probability that the madman rolls snake eyes for your batch of kidnap victims -- 1/36.
2. The "proportion murdered" argument says that about 9/10 of all people who ever go into the Dice Room die, so your probability of dying is about 9/10.
Obviously this is a problem. Different decompositions of a problem should give the same answer, as long as they're based on the same information.
I claim that the "proportion murdered" argument is wrong. Here's why. Let pi(t) be the prior probability that you are in batch t of kidnap victims. The proportion murdered argument relies on the property that pi(t) increases exponentially with t: pi(t+1) = 10 * pi(t). If the madman murders at step t, then your probability of being in batch t is
pi(t) / SUM(u: 1 <= u <= t: pi(u))
and, if pi(u+1) = 10 * pi(u) for all u < t, then this does indeed work out to about 9/10. But the values pi(t) must sum to 1; thus they cannot increase indefinitely, and in fact it must be that pi(t) -> 0 as t -> infinity. This is where the "proportion murdered" argument falls apart.
For a more detailed analysis, take a look at
http://bayesium.com/doomsday-and-the-dice-room-murders/
This forum has a lot of very smart people who would be well-qualified to comment on that analysis, and I would appreciate hearing your opinions.
Actually, no, improper priors such as you suggest are not part of the foundations of Bayesian probability theory. It's only legitimate to use an improper prior if the result you get is the limit of the results you get from a sequence of progressively more diffuse priors that tend to the improper prior in the limit. The Marginalization Paradox is an example where just plugging in an improper prior without considering the limiting process leads to an apparent contradiction. My analysis (http://ksvanhorn.com/bayes/Papers/mp.pdf) is that the problem there ultimately stems from non-uniform convergence.
I've had some email discussions with Scott Aaronson, and my conclusion is that the Dice Room scenario really isn't an appropriate metaphor for the question of human extinction. There are no anthropic considerations in the Dice Room, and the existence of a larger population from which the kidnap victims are taken introduces complications that have no counterpart when discussing the human extinction scenario.
You could formalize the human extinction scenario with unrealistic parameters for growth and generational risk as follows:
Let n be the number of generations for which humanity survives.
The population in each generation is 10 times as large as the previous generation.
There is a risk 1/36 of extinction in each generation. Hence, P(n=N+1 | n >= n) = 1/36.
You are a randomly chosen individual from the entirety of all humans who will ever exist. Specifically, P(you belong to generation g) = 10^g / N, where N is the sum of 10^t for 1 <= t <= n.
Analyzing this problem, I get
P(extinction occurs in generation t | extinction no earlier than generation t) = 1/36
P(extinction occurs in generation t | you are in generation t) = about 9/10
That's a vast difference depending on whether or not we take into account anthropic considerations.
The Dice Room analogy would be if the madman first rolled the dice until he got snake-eyes, then went out and kidnapped a bunch of people, randomly divided them into n batches, each 10 times larger than the previous, and murdered the last batch. This is a different process than what is described in the book, and results in different answers.
Thanks, interesting reading.
Fundamental or not I think my point still stands that "the prior is infinite so the whole thing's wrong" isn't quite enough of an argument, since you still seem to conclude that improper priors can be used if used carefully enough. A more satisfying argument would be to demonstrate that the 9/10 case can't be made without incorrect use of an improper prior. Though I guess it's still showing where the problem most likely is which is helpful.
As far as being part of the foundations goes, I was just going by the fact that ... (read more)