18 November 2005

Randomness vs. 'Random Sample'

Excuse me, everyone, for using this space for my own education and not for, well, whatever else it is supposed to be for. Señor gadfly writes in the comments:
what it [the doomdsday argument] tells you is that if there are two choices, ten time marbles or a hundred time marbles, you are (much) likelier to be in the former case.
Yes, but it only tells you that if I think that I have drawn myself from a random sample of all possible marbles. Before I draw myself, I have to assume, as a prior probability, that there is an equal chance of humans living for just a little bit of time, or for a very long bit of time. It's only after the sample is drawn (i.e. us) that we can revise to a posterior probability that doomsday comes early. And we cannot make that statistical leap because we are not a random sample.

Finally, and more importantly:

it's 2005 and I'm alive on the earth. What are the odds I'm American? Do the math and you'll come up with an answer that's around 1 in 24, or about a 4% chance. And you're wrong, the Odds Are One™, I am in fact American. Bayes says I should bet on me being not American, but if I were to do so, I'd lose. Don't get me wrong, I think the Doomsday Argument is totally, utterly invalid, but I don't think it lies in deciding whether you're a random human or not, because it seems very difficult to show that this isn't the case, just as it's difficult to show that it is. I think the answer is really more Platonic than that--you and I can't be defined as random human number 60-billion-and-some-odd, because we cannot be separated from our 20th century-ness, our American-ness, our iPod-having-ness.

Two things: 1) I'm totally with the Fly on the notion that we aren't random humans and we can't look back at prior events and say in a meaningful way 'what are the odds that would have happened' when, in fact, they have already happened. The odds are one (can I say that without getting sued? Any chance any lawyers read this blog?). 2) However. Before knowing who you are, you should bet on your not being 'merican. But once you 'sample yourself' as it were, wouldn't Bayes say you should revise your probabilities upward? But the Bayesian logic depends on the notion not that YOU are random, but that if we have chosen you, we have chosen you from a random sample of ALL humans. Bayes isn't even telling us about the odds that you are American. He's telling us, about the chances that there are only Americans in the world, with the prior possibility being X and the posterior possiblity (if you've picked an American) being a much larger X. That breaks out as follows:

A. If I say, I'm going to pick a human being (not me) at random from the billions of us out there, then the odds are 1 in 24 that I will pick an American.
B. If I say, what are the odds that I'm American, then 'the odds are one'.
C. If I then say, from out of the entirety of human history (forward in time until its end) what are the odds that I would pick someone who has lived prior to 2005, then the odds are slim. However, to then sample from the population alive today, I break the primary rule of statistics, in that we are not a random sample of all humans.

What I'm trying to say - if anyone still cares by now – is that point C and point B are very much distinct. The Odds are One (point B) is an argument about how we aren't random humans, and about how we can't look back in time and attribute statistical probabilities to things that have already happened. Point C is merely the assertation that we can't sample from the PRESENT and presume it is a random sample of ALL TIME, because we aren't living at the end of time.

On the other hand, maybe the Fly's point is simply beyond me, and involves something far past the simple Bayesian logic. But I still think that Bayes tells the Fly nothing about his being an American, until after he has found out his IS American, at which point Bayes and the Fly 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'.


rev. mr. t. bayes said...

Again, I agree. We people are being drawn sequentially (me in c. 1764, Dr. Chambers in c. 2005), not randomly. Hence, the Doomsday argument is a fundamental misapplication of my theory.

Perhaps another example will serve: a golf ball can take two possible trajectories. The first 20 feet of the trajectories are identical. Thereafter, there is a 90% chance that the ball takes trajectory 1 (sand trap) and a 10% chance that a butterfly in the amazon flaps its wings and the ball goes in the hole, trajectory 2. As we sit and watch the first 20 feet of the trajectory of the golf ball, the fact that it is still in the first 20 feet tells us nothing about which path it will take thereafter. This is where we are in the trajectory of the human race. It's a complete falacy to draw any inference about the future from that.

friedrich nietzsche said...

Yes, although Dr. Chambers is wrong about only being able to take a random sample "at the end of time." One can only take a random sample if one is outside of time and has the whole skein of the human race before one. This whole "Doomsday" nonesense simply ignores our historicity completely. Typical Anglo mumbo jumbo.

Transient Gadfly said...

This guy I used to work with, when you asked him what super power he wanted, would eschew such things as invisibility and the ability to fly for the power to have Sade's Smooth Operator start playing when he walked into a room. In that vein, New Rule: Every time I walk into a room, the riff from The Fly by U2 starts playing.

Anyway, I started to write the rest of this comment, but then it got too long, so it's going into my blog. Coming soon to a theater near you.