This post started as a comment on Sam's last post on the ongoing dialog about the Doomsday Argument (see below), but then it became hopelessly too long for a comment. If you are coming late to this argument...um, move on to something else more interesting (This must be what it is like if you're, e.g. dailykos versus drudge report, or whatever. Endless co-mingled threads that eventually manage to lose absolutely everybody. What fun).
Sam: I still think that Bayes tells (you) nothing about (you) being an American, until after he has found out his IS American, at which point Bayes and (you) agree, in that the latter says 'the odds are one' and the former says, 'we do not have a random sample any longer, so my logic does not apply'. Also, by the way, SEGA! iPOD NANO! RAGE!
The Stoat: Indeed, and perhaps I have previously said Bayes, author of the probability theory, where I meant Leslie, who is the author of the Doomsday Argument. Sorry about that.
Sam: I see. So we agree that the Doomsday Argument is wrong, and that it's because Bayes' Theory does not apply. Then what is your quibble?
The Stoat:My quibble is perhaps really only semantic, but it's that trying to attack the Doomsday Argument by figuring out whether you are a random sample of all of humanity is the path of madness. It seems hung up in problems of temporality, which Leslie tries to overcome with this idea of "Doom Early" (a bag with a few marbles) versus "Doom Late" (a bag with many, many marbles). Much argumentative reasoning ensues. My point is that it is for naught. Leslie is unlikely to be convinced, and you are unlikely to be convinced by Leslie, and I think it's because randomness doesn't enter into it. My argument is that deciding whether you are a random human is logically equivalent to asking "What are the odds that I would be born at the time I was actually born?" The question has meaning, but it is not about statistics or randomness, because you were actually born at that time, it has already happened. It's an Odds Are One™ question (I'm totally going to trademark this and then the residuals are going to start pouring in).
Sam: What does this have to do with Bayes' Theorem?
The Stoat: Nothing. My argument is that not that it doesn't apply because you're not a random human, my argument is that probability does not apply at all because you are already you, stating the argument.
Sam: I have completely, totally and utterly lost you.
The Stoat: Here is the argument I thought of on the walk to work this morning. Either this will help, or I should abandon this line of reasoning forever. The argument begins now. *Ahem*. One of the (many) things we gloss over in the Doomsday Argument is the idea that there is a discrete first human (or for that matter a discrete last one). Unless we are Creationists, we wouldn't argue this, but we are assuming that it doesn't matter too much. Anyway, assume you believe in evolution.
Sam: You may make that assumption.
The Stoat: Then, you, Sam, for the sake of argument the current person considering the Doomsday Argument, cannot trace your lineage back to a distinct first human. Nor can anyone. The so called "bag" containing all of humanity, even as a species, cannot be traced back to a distinct Homo Sapien #1.
Sam: Yes, I agree. But I don't see how it's a problem that the line is a little bit fuzzy between Homo Sapien and Homo Erectus, or whomever.
The Stoat: So if it's not, is it okay if our "bag" contains some Homo Erecti, or some thousands to million of years of transitional species, just to make sure we get everybody who might possibly be considered Homo Sapien #1?
Sam: No, no, I see where you're going with this. I'm going to get on a slippery slope wherein we wind up having to trace the human lineage back to amoebas or something when in fact I could never have been born as an amoeba.
The Stoat: Indeed, the odds of Sam being born an amoeba are zero, For You Are Sam.
Sam: For I Am Sam.
The Stoat: Anyway, that's not quite where I'm going with this. My proposal is that including subjects in your metaphysical grab bag, such as homo erecti or amoebas as whom you have no chance of being born invalidates the terms of the experiment.
Sam: I want to believe you, but I don't quite see that as a problem. Nor do I see that it can't be solved by starting a couple thousand years into the advent of Homo Sapiens and numbering one of them human number one, just to be safe.
The Stoat: I'll grant you the latter thing, because I don't actually need it to make my argument. What I need is the former thing. Recall why the Doomsday Argument actually seems to work: You have a bag labeled "All Humans For All Time," and you know that this bag has two possible identities: Doom Early, meaning that there are 70 billion total humans in the bag, and Doom Late, in which there are many trillion humans in the bag. You reach into the all-humans-for-all-time grab bag and pull out a human--it happens to be you, and you have a number affixed to you that's your birth order, and it is under 70 billion, you apply Bayes theorem, blah blah blah.
Sam: (makes the "blah blah blah" hand motion)
The Stoat: There's all that confusion about whether you being born is equivalent to somebody outside of time and space reaching into the bag a picking out a random person. In order for this to be valid, not only does the probability of you being selected have to be as likely as selecting anyone else (you are truly random), but the entity you select out of the bag has to be as likely to be you as anyone else (your probability space is truly random).
Sam: Um...no? Why?
The Stoat: Those are the terms of the experiment. To be a random sample you'd have to be able to pop out at any point in history (that is, with any particular "number" affixed to you).
Sam: Ah, right. Isn't this what I was arguing?
The Stoat: Perhaps. Anyway, it's obvious that you have no chance of being an amoeba or a homo eretus. Now, if we solve this problem by removing the transitional species as you suggest above, and then reach into the bag and select...Njorl Hroffssen, fierce Norse warrior living in 800 C.E., what are the odds that this person is Sam?
Sam: Oh no....
The Stoat: ZERO! There are no odds! That person is not Sam. You reach into the bag and pull out Zarf VIII, Galactic Neural Coupling Plumbing Engineer, living in the 389th Solar Mega-Cycle after the dawn of the Total Information Era, what are the odds that this person is Sam?
Sam: I beg you to stop now....
The Stoat: None! Nada! Zilch! That person is not Sam. Sam does not and cannot exist outside the historical and societal context in which Sam exists.
Sam: Yeah. I made this exact same argument, only I used fewer words, and it made more sense.
The Stoat: You almost made this argument. My argument is that the Doomsday Argument might work fine in the abstract, with the idea of looking outside of time and space and reaching into the bag and pulling out a human and looking at his or her birth order. But the moment you apply the problem to yourself, a person alive right now, and this is the only way the argument could work, you fix yourself as you, Sam, and suddenly you cannot be a human pulled out of a bag at random, you can only be Sam. Sam does not exist at other times and places in the history of time. He only exists now. Bayes theorem does not apply not because you're un-randomly selected, but because probability does not apply to this question. The odds that you are you are one. The odds that you are not you are zero.
The Stoat: Yes?
Sam: I hate you.
The Stoat: I know you do.