Saturday, August 30, 2008

Speed and Likelihood

So I'm reading Carl Sagan's posthumous The Varieties of Scientific Experience, and overall it's pretty good. But Sagan does say something about the likelihood of the origin of life that I found nonsensical. He's talking about the origin of life on Earth and how it happened within a span of 500 million years, from the time conditions were probably suitable for the origin of life to the first actual evidence we have for life appearing. And then he says:

It must have happened very quickly. A process that happens quickly is a process that in some sense is likely. The faster it happens, the more likely it is.


He then gives lip service to the problem of extrapolating from a single instance of an event, but that's entirely the point. We have very poor understanding of how life originated, so we can't say how likely it was. And the speed with which a process happens does not correlate with the likelihood of that event happening. If somebody does have evidence or an argument to the contrary, I'd like to hear it.

Think of it this way: Let's say you roll 100 dice, and you want to know how likely it is that all of them turn up as sixes. Now, you're not measuring the likelihood of it recurring, just happening once. This could happen on the very first roll, or it could happen after 40 million rolls. Do either of those situations (it happening quickly vs. after a very long time) convey any particular information about the likelihood of the event?


Ian said...

I'd have to find my probability and statistics textbooks to come up with a good answer, but I'll describe what I think Sagan's description means.

If you rolled those 100 dice only 10 times, and you had _no_ other information about them other than an assumption of randomness (such as, a die has 6 sides, it's fair/uniform, etc), you can assume that the observed results will cluster around the mean. It's possible that you would see incredible outliers (for instance, all 100 dice came up with sixes), but that is statistically unlikely. It's more likely you'd see 16 dice with one pip showing, 17 with two pips, 15 with three and so on. With no other information, you could reasonably assume that the dice follow a random distribution, the mean is "each number 1-6 should show on approx 16 dice", etc.

I think origin-of-life events occurring over a specific time span could be modeled with a Poisson distribution; I haven't calculated the likelihood from the values we have (3 habitable zone planets in our solar system, 500 million year time span, 1 observed event), but it could be done.

Basically, I think Sagan makes two assumptions: one, our world is _not_ due to a series of incredibly lucky and special events (reflected in most of his works); and two, origin-of-life events follow a random distribution and would cluster around the mean (ie, given some planet, when the conditions are right and 500 million years pass, an origin-of-life event has a non-zero, non-trivial probability).

Whether those assumptions are valid or not, only time (and more data) will tell... :)

Philip said...

Now, you're not measuring the likelihood of it recurring, just happening once.

If they're independent events, there's no difference between those.

I don't see why this is confusing. Give that you have no other information, if it takes 3 rolls to get all sixes, this event is would be considered more likely to happen than if it took 300 rolls.

It seems silly to me to boil it down to a single event though. Everything we know about life is that it's a stepwise process. At any point, it is very unlikely to have occurred from the beginning, but it is very likely to occur give the previous step.

Think of it this way - given the primordial soup, what is the likelihood of a Tiger evolving? Infinitesimally small. What is the likelihood of predators evolving - much higher.

So, the probability of life as we know it evolving is almost certainly impossibly small. But the possibility of something like life might be higher. The problem is we don't know what that is.

Derek James said...

Maybe my sense of probability isn't working all that great in this instance, or maybe I'm actually right.

Let's try another example, the good old "socks in a drawer". Let's say there are billions of socks in a big drawer, and there are two colors, blue and red. You have no idea what the ratio is.

Let's say you pull out a red sock on the very first trial. Does this tell you anything about the ratio of red socks to blue socks? It could very well be that there were 40 billion blue socks and only one red sock, and the sample of one was extremely skewed, right?

In the absence of more information, I don't think it's reasonable to conclude anything about the possible ratio of red to blue socks. We just don't have enough information about the system.

We're pretty sure life arose only once on this planet, because the evidence points to shared ancestry between all living things. If conditions were ripe for life on earth, why didn't it arise multiple times independently? Doesn't this actually suggest that the likelihood of life arising, given favorable conditions, should be estimated as very small, rather than the other way around?

Ian said...

I think it's a matter of personal choice of a priori probabilities of Earth's "rareness". Your perspective is that the Earth is a (potentially) rare occurrence; the fact that a "red" was chosen implies nothing about the likelihood of a "red" chosen on the next selection.

Sagan's perspective is that Earth is not especially rare; i.e., the "sock drawer" is mostly equal in amounts of blue and red. A "red" being selected early implies "red"'s are plentiful. In essence, Sagan's argument is: If the Earth isn't rare, the speed of life's emergence implies origin-of-life events should be likely. The argument only works if the Earth isn't rare.

You're absolutely right that Sagan hand-waves away this initial presupposition. Without more data, there is no real traction on the problem.

(Hopefully the rest of the hurricanes will miss your neck of the woods...)