Nick Bostrom and I have had a long conversation here about the Doomsday Argument, and I think I need to stop now. So let me now summarize my position.
Our standard approach to analyzing the chances for doom soon is to imagine various sorts of descructive processes, and then choose a probability distrubution over the frequency of such events. In doing so one trivally and usually implicitly conditions on the fact that we have made it so far without doom. And one typically does not infer that doom is likely soon.
There are two doomsday arguments I can see. The first one notes that for exponential growth processes which end suddenly, most population members appear near the end. Thus estimating that doom is not soon implies that we are not typical/random population members.
Since people often like to think they are special, the fact that an inference suggests we are special does suggest we may have fooled ourselves somehow. But while this may motivate skeptics to seek weak points in standard assumptions, it does not by itself seem to me much of a reason to much modify standard analyses. We *are* special on some dimensions, and this could be one of them.
The second doomsday argument invites us to imagine that we had amnesia regarding which person in history we are, and that in this situation our probability distribution over doom was that suggested by standard analyses. The argument then correctly notes that learning that we appear early in history must make us expect that doom occurs earlier in history. It then concludes that we should expect doom earlier than standard analyses would suggest.
The weak point of this argument is the assumption that we should invoke our standard prior probabilities over doom in the amnesia context. The usual place to invoke priors over doom is in the context where we don't know whether humanity lasted long enough for us to be born. In this context, learning that one exists, but has amnesia, gives one a lot of information, and in particular suggests that doom occurs later than suggested by the priors over doom.
Learning that one is alive with amnesia makes one more optimistic, and then learning that one was born early makes one more pesimisti. If you started with standard priors, you end up with standard expectations about how soon doom may be. Let me illustrate with some examples.
Imagine that you found yourself hung over in a hotel room, and couldn't remember who you were, other than that you are a musician on tour. You wonder: am I a one-hit-wonder, or do/will I have a lasting music career? If on average only 25% of musicians have a lasting career, but musicians with lasting careers spend three times as much time in hotel rooms on tour, then you should estimate a 50% chance you have a lasting career. This is because only half of the total musician hotel room-days are filled with one-hit-wonders. Amensia implies optimism.
Similarly, if you can't recall how old you are, you should expect to be older than the average person. Why? Because older people have more hotel-days in their lifetime. Similarly, it seems to me that if you can't recall who you are in humanity, you should become more optimistic about humanity's chances.
The amnesia situation does not seem to me to be a good place for imposing priors, unless one takes into account the fact that amnesia suggests optimism. The fact that amnesiacs who learn they are early in history should become more pessimistic does not seem a reason for me to be pessimistic. After all, I don't have amnesia.