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.
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 it's in Jaynes, but you clearly know a lot more about this topic than I do. I would be interested to know your answer to the following questions though: "Can a state of ignorance be described without the use of improper priors (or something mathematically equivalent)?", and "Can Bayesian probability be used as the foundation of rational thought without describing states of ignorance?".
On the Doomsday argument, I would only take the Dice Room as a metaphor not a proof of anything, but it does help me realise a couple of things. One is that the setup you describe of a potentially endlessly exponentially growing population is not a reasonable model of reality (irrespective of the parameters themselves). The growth has to stop, or at least converge, at some point, even without a catastrophe.
It's interesting that the answer changes if he rolls the dice first. I think ultimately the different answers to the Dice Room correspond to different ways of handling the infinite population correctly - i.e. taking limits of finite populations. For any finite population there needs to be an answer to "what does he do if he doesn't roll snake-eyes in time?" and different choices, for all that you might expect them to disappear in the limit, lead to different answers.
If the dice having already being rolled is the best analogy for the Doomsday argument then it's making quite particular statements about causality and free will.