I upvoted this and I think you proved SIA in a very clever way, but I still don't quite understand why SIA counters the Doomsday argument.
Imagine two universes identical to our own up to the present day. One universe is destined to end in 2010 after a hundred billion humans have existed, the other in 3010 after a hundred trillion humans have existed. I agree that knowing nothing, we would expect a random observer to have a thousand times greater chance of living in the long-lasting universe.
But given that we know this particular random observer is alive in 2009, I would think there's an equal chance of them being in both universes, because both universes contain an equal number of people living in 2009. So my knowledge that I'm living in 2009 screens off any information I should be able to get from the SIA about whether the universe ends in 2010 or 3010. Why can you still use the SIA to prevent Doomsday?
[analogy: you have two sets of numbered balls. One is green and numbered from 1 to 10. The other is red and numbered from 1 to 1000. Both sets are mixed together. What's the probability a randomly chosen ball is red? 1000/1010. Now I tell you the ball has number "6" on it. What's the probability it's red? 1/2. In this case, Doomsday argument still applies (any red or green ball will correctly give information about the number of red or green balls) but SIA doesn't (any red or green ball, given that it's a number shared by both red and green, gives no information on whether red or green is larger.)]
I think I've got a proof somewhere that SIA (combined with the Self Sampling Assumption, ie the general assumption behind the doomsday argument) has no consequences on future events at all.
(Apart from future events that are really about the past; ie "will tomorrow's astonomers discover we live in a large universe rather than a small one").
EDIT: This post has been superceeded by this one.
The doomsday argument, in its simplest form, claims that since 2/3 of all humans will be in the final 2/3 of all humans, we should conclude it is more likely we are in the final two thirds of all humans who’ve ever lived, than in the first third. In our current state of quasi-exponential population growth, this would mean that we are likely very close to the final end of humanity. The argument gets somewhat more sophisticated than that, but that's it in a nutshell.
There are many immediate rebuttals that spring to mind - there is something about the doomsday argument that brings out the certainty in most people that it must be wrong. But nearly all those supposed rebuttals are erroneous (see Nick Bostrom's book Anthropic Bias: Observation Selection Effects in Science and Philosophy). Essentially the only consistent low-level rebuttal to the doomsday argument is to use the self indication assumption (SIA).
The non-intuitive form of SIA simply says that since you exist, it is more likely that your universe contains many observers, rather than few; the more intuitive formulation is that you should consider yourself as a random observer drawn from the space of possible observers (weighted according to the probability of that observer existing).
Even in that form, it may seem counter-intuitive; but I came up with a series of small steps leading from a generally accepted result straight to the SIA. This clinched the argument for me. The starting point is:
A - A hundred people are created in a hundred rooms. Room 1 has a red door (on the outside), the outsides of all other doors are blue. You wake up in a room, fully aware of these facts; what probability should you put on being inside a room with a blue door?
Here, the probability is certainly 99%. But now consider the situation:
B - same as before, but an hour after you wake up, it is announced that a coin will be flipped, and if it comes up heads, the guy behind the red door will be killed, and if it comes up tails, everyone behind a blue door will be killed. A few minutes later, it is announced that whoever was to be killed has been killed. What are your odds of being blue-doored now?
There should be no difference from A; since your odds of dying are exactly fifty-fifty whether you are blue-doored or red-doored, your probability estimate should not change upon being updated. The further modifications are then:
C - same as B, except the coin is flipped before you are created (the killing still happens later).
D - same as C, except that you are only made aware of the rules of the set-up after the people to be killed have already been killed.
E - same as C, except the people to be killed are killed before awakening.
F - same as C, except the people to be killed are simply not created in the first place.
I see no justification for changing your odds as you move from A to F; but 99% odd of being blue-doored at F is precisely the SIA: you are saying that a universe with 99 people in it is 99 times more probable than a universe with a single person in it.
If you can't see any flaw in the chain either, then you can rest easy, knowing the human race is no more likely to vanish than objective factors indicate (ok, maybe you won't rest that easy, in fact...)
(Apologies if this post is preaching to the choir of flogged dead horses along well beaten tracks: I was unable to keep up with Less Wrong these past few months, so may be going over points already dealt with!)
EDIT: Corrected the language in the presentation of the SIA, after SilasBarta's comments.
EDIT2: There are some objections to the transfer from D to C. Thus I suggest sliding in C' and C'' between them; C' is the same as D, execpt those due to die have the situation explained to them before being killed; C'' is the same as C' except those due to die are told "you will be killed" before having the situation explained to them (and then being killed).