Would you like to say a bit more about why that's not a valid argument. To be clear, I'm not saying it is (I don't know enough about the subject to do so) but it doesn't seem that far-fetched to me. Isn't similar probabilistic reasoning used to explain why evolution by natural selection gives rise to various complex life forms? If so, do you also think that that reasoning is shoddy?
Let N be the number of places life could arise, and p the probability that life arises in one of those places.
That argument is basically "there is a value of N such that for any p > 0, N p is much greater than 1."
But that's obviously wrong. For any N, there are values of p > 0 that make the product N p arbitrarily close to 0.
The dim intution behind the argument was that p can't be "too small". But given our current understanding of OoL, that's not a justified assumption. p could be exponentially small, if OoL requires some extremely unlikely step.
Natural selection is great once the system's reproductive fidelity is good enough to support it. The problem is bridging the gap from small molecules to that system. The smallest system we know of that can independently support Darwinian evolution has billions of atoms.
In this formulation, isn't p^N the probability that ALL places where life is possible, actually has life? It makes sense for that to approach zero.
What we want is the probability for at least one other place other than ours to have life. This would be 1 - (1-p)^N, which does tend to 1 as N gets arbitrarily large.
To get that formula: (1-p) is the probability that life does not exist in a place, so (1-p)^N is the probability that ALL places where life is possible, has no life. Therefore, 1-(1-p)^N is the probability of the opposite of that (where at least one place has life).
For a random variable X taking on non-negative integer values (here, the number of occurrences of life elsewhere in the universe), by Markov's inequality the probability that X = 0 is >= 1 - E[X]. Here, E[X] = Np, so if Np is very close to 0, the probability that X = 0 will be very close to 1.
That the probability goes to 1 as N goes to infinity FOR FIXED p is just another example of assuming p can't be "too small". The probability also goes to zero as p goes to zero. Why are you fixing p and not N? Why are you assuming p is large enough that N is in that asymptotic range where the probability has approached 1?
That seems right, but from a scientific point of view (as opposed to, say, a certain sort of theological view), two occurrences is not much more than one (even though one is so much more than zero.)
Two occurrences would actually be much more than one! Our own existence is useless due to observer selection, but discovery of even a single other independent OoL event nearby would allow us to infer OoL cannot be too uncommon.
Observer selection does not eliminate us as evidence for the proposition that life can exist. As for whether it is rare, you added the qualification 'nearby', and while it is true that it is most likely that any extraterrestrial life we detect will be nearby, the post I was replying to was arguing about the universal probability of life coming into existence, not about whether it will be discovered by us.
Furthermore, proponents of an extraterrestrial origin of life on Earth will doubtless argue that nearby life may have had a common origin.
Observer selection means p > 0 (ie the inequality is strict) but it can't tell us any more. Bayesian reasoning from our own solar system can put a reasonable upper limit on p but that isn't very helpful.
However, if we found life on Mars that same Bayesian reasoning would imply a meaningful lower limit on p as well, since life on Mars is independent of our existence to observe it.
If we found life on Mars that was independent of life on Earth it would imply a meaningful lower bound. Even finding a fundamentally different biosystem on Earth (life that didn't use nucleic acids, say) would be informative.
Just finding life on Mars that's the same kind of life as on Earth would not tell us much, as it could be explained by panspermia. There are Mars rocks on Earth, so transfer of life in those rocks should have happened constantly. If early Mars were habitable it almost certainly had life, due to this transfer.
This explains why it may not be a sound argument, not a demonstration of its invalidity. The distinction matters, because while invalid hypotheses can be summarily rejected, valid ones might turn out to be right.
Of course, if some people don't understand that this one is not an established fact, and that annoys you, I can't say you are wrong.
Yes. Of course, I was not arguing that life must be rare, I was arguing that the evidence we have does not compel one to believe life must exist elsewhere in the universe. The opposite of belief is not belief in the opposite.
There are rare instances where people say that life exists elsewhere, other just state that there is a possibility > 0.
I agree that it is arbitrary that the dimension of the exponent of n has to be larger than the negative one of p. That probably stems from the assumption that the universe is endless.
> p could be exponentially small, if OoL requires some extremely unlikely step.
"exponentially" is not a measure of size, nor is it a measure of relative size. If you think this anything base on "exponentially small" is a valid argument, go look in a mirror and slap yourself.
"Exponentially small" here means "the probability could be ~ e^-n" where n is a number proportional to the complexity of the minimal evolving system. This would happen if there's some gap that has to be bridged by random chance before we get a system capable of sustaining natural selection.
The point here is that this could easily be vastly smaller than 1/N, where N is (say) the number of atoms in the universe x age of the universe x rate at which atoms might interact to form such systems.
I think you could have easily understood this point if you had made an effort to do so, without me having to spoonfeed it to you here.
The problem there is we don't know the "world" of possibilities from which our existence was drawn. It might be the universe (which I read "observable universe"), or it might be out of a large number of causally disconnected universes, or even other branches of a universal wave function (in a Many Worlds interpretation). The "N" there is not the same as the "N" of "our universe".
We know approximately the lower bound of N, which is the approximate number of stars in the observable universe multiplied by an informed estimate of the expected number of planets within the goldilocks zone. That's usually what people mean when they discuss N. N could be that, or it could be much much larger, but I think it's fine to limit the discussion to the lower bound, we still have a huge N to work with.
Also I think you missed my point which is about Bayesian estimation of p, not of N.
I ignored the comment about Bayesian estimation because I couldn't turn that comment into something that made any sense. Perhaps you could explain in detail what you meant?
Your statements in this thread have assumed we have no info to work with (as far as estimating p goes) because we have no understanding of the mechanisms behind how life came to be. But this ignores the evidence that we are here, which is info that can be used in a Bayesian framework to estimate p. The fact we exist, as well as information about how many billions of years it took for us to evolve, contains significant information about p.
I still don't understand. We have no useful lower bound on the probability that life arises, so how does Bayesian reasoning bootstrap to any meaningful lower bound?
Who said anything about a lower bound of p? I was talking about a lower bound on N, not a lower bound on p.
Bayesian reasoning (by using the fact that we exist rather than don't exist, as well as other info about our existence, such as how long it took us to evolve) helps us estimate a probability distribution of p, as well as a central tendency estimate.
But selection can happen with autocatalysts as well. I agree that you can't say life /has/ to exist elsewhere, but I think the trend in research has shown that life seems likelier and likelier to arise the more it is studied.
"Trend in research"? How could that possibly work? Research will tend to clear the low hanging fruit early, which means the easy steps. This tells us nothing about how difficult the difficult steps (if any) might be.
The analogy I like here is those "collect the letters" games you see at fast food outlets and grocery stores. Buy a Happy Meal, get a scratch off ticket. If you collect all the letters in some phrase you win $N million. When you start the game, the trend is great. Letters are arriving and the phrase is filling in. But try as you might, that last letter never shows up. The game ends and you've won nothing. Of course, the way the game was designed was that last letter controls how many winners there could be. All the rest were distractions.
It does however tell us that the "easy" steps are easy, which was never a foregone conclusion. The other steps will remain what they are. It doesn't mean the trend will continue.
I find it weird to use a deliberately rigged game as an example. If one of the previous letters was wrong, the last letter being right means you don't win either.
It's like saying the difficult steps are going to be extra difficult because other steps were found easier than expected.
The point is that if you have N independent boolean random variables X1 ... Xn, establishing a lower bound on the probability that some proper subset of the Xi are true doesn't provide any useful lower bound on the probability they all are true.
Sure, my point was only that if the lower bound on the subset is higher than anyone expected, that will increase the probability of them all being true compared to your prior belief. And it will also increase the probability that life is more common.
You could argue that the priors were garbage I suppose. I'm not arguing for any particular probability.
The McDonalds example does not have independent variables as X1..Xn-1 are deliberately increased as Xn is decreased.
I'd also argue that origin of life doesn't have independent variables. If chemistry turns out to be more or less powerful in one setting, it should do something for our assessment of other settings, especially when it's similar processes.
It's not up to me to show that, since I'm not claiming life is rare. It's up to the person making the strong statement that life is (not just could be) common to convince me that there is no sufficiently difficult step. All I need to do is plausibly argue there could be a difficult step. Pointing out the complexity of all known self contained systems capable of Darwinian evolution is sufficient for that.
