There is one serious flaw with the SDO paper. The issue is that for N = “# detectable civilizations in the Milky Way”, they interpret Pr(N<1) as “the probability that there is no other detectable civilization in the Milky Way”. But this interpretation is mistaken, since Pr(N<1) doesn’t take into account the fact that we (our civilization) exist, i.e. that the number of civilizations in the Milky Way, N, is at least 1.
In fact, doing so would change their conclusion. Since small values for N would make our existence unlikely, our existence makes such small values unlikely. (It’s a theorem of probability theory that Pr(A|B)<Pr(A) implies Pr(B|A)<Pr(B).)
Edit: What they’d want for “the probability that there is no other detectable civilization in the Milky Way” would be something like Pr(N=1|N≥1).
The milky way was “choosen” because it happens to be the galaxy we are in, so we shouldn’t update on that as if if had first chosen a galaxy to study and then found life in it. Similarly, we can’t even be sure we can update on the fact that there is life at least one place in the universe, because we don’t know how many empty multiverses are out there. The only thing we can conclude from us self existing is that life is possible in the universe.
The milky way was “choosen” because it happens to be the galaxy we are in, so we shouldn’t update on that as if if had first chosen a galaxy to study and then found life in it.
This doesn’t follow; one way or the other we have to update on the fact that our civilization exists. The order is not relevant.
Similarly, we can’t even be sure we can update on the fact that there is life at least one place in the universe, because we don’t know how many empty multiverses are out there. The only thing we can conclude from us self existing is that life is possible in the universe.
Anthropic reasoning (see SIA, Sleeping beauty paradox) suggests otherwise. That’s what I was talking about above: If hypothesis B makes evidence A unlikely, then observing evidence A makes hypothesis B unlikely.
Yes, the ordering does matter. Compare two hypotheses, one, H1, says that one average there will be 1 civilisation in each galaxy. The other, Hϵ says that on average there will be 10−8 civilisations in each galaxy. Suppose the second hypothesis is true.
If you now do the experiment of choosing a random galaxy, and counting the number of civilisations in that galaxy, you will probably not find any civilisation, which correctly supports Hϵ.
If you do the second experiment of first finding yourself in some galaxy and then counting the number of civilisation in the galaxy, you will at least find your own civilisation and probably not any other. If you don’t correct for the fact that this experiment is different, you would update strongly in the direction of H1 even when Hϵ is true and the evidence of the experiment is as favourably towards Hϵ as possible. This cannot be the correct reasoning, since correct reasoning should not consistently lead you to wrong conclusions.
You might argue that there is another possible state which is even more favourable evidence towards Hϵ: that you do not exist. However, in a universe with 1011 galaxies, the probability of this is 10−434.
The crucial point is that on H1 our existence (say, E) is much more likely than on Hϵ. That is, P(E|H1)>P(E|Hϵ). If our prior is P(H1)=P(Hϵ), then P(H1|E)>P(Hϵ|E). That is, our existence implies that H1 is more likely.
As a sanity check, this also doesn’t reliably lead us to wrong conclusions: If we have two possible universes, one where H1 is true and one where Hϵ is true, and in both possible universes the civilizations reason as above and regard H1 as more likely, much more civilizations will reach the true rather than the false conclusion, since the possible universe where H1 is true has much more civilizations (or a much higher density etc).
Now I see, yes you are right. If you want the beliefs to be accurate at the civilisation level, that is the correct way of looking at it. This corresponds to the 1⁄3 conclusion in the sleeping beauty problem.
I was thinking of it on the universe level, were we are a way for the universe to understand itself. If we want the universe to form accurate beliefs about itself, then we should not include our own civilisation when counting the number of the civilisations in the galaxy. However, when deciding if we should be surprised that we don’t see other civilisations, you are right that should include ourselves in the statistics.
There is one serious flaw with the SDO paper. The issue is that for N = “# detectable civilizations in the Milky Way”, they interpret Pr(N<1) as “the probability that there is no other detectable civilization in the Milky Way”. But this interpretation is mistaken, since Pr(N<1) doesn’t take into account the fact that we (our civilization) exist, i.e. that the number of civilizations in the Milky Way, N, is at least 1.
Essentially the same was already pointed out by Alexey Turchin at the time. SDO should have updated their probability distribution not just on the Fermi observation, but also on the fact that we exist.
In fact, doing so would change their conclusion. Since small values for N would make our existence unlikely, our existence makes such small values unlikely. (It’s a theorem of probability theory that Pr(A|B)<Pr(A) implies Pr(B|A)<Pr(B).)
Edit: What they’d want for “the probability that there is no other detectable civilization in the Milky Way” would be something like Pr(N=1|N≥1).
The milky way was “choosen” because it happens to be the galaxy we are in, so we shouldn’t update on that as if if had first chosen a galaxy to study and then found life in it. Similarly, we can’t even be sure we can update on the fact that there is life at least one place in the universe, because we don’t know how many empty multiverses are out there. The only thing we can conclude from us self existing is that life is possible in the universe.
This doesn’t follow; one way or the other we have to update on the fact that our civilization exists. The order is not relevant.
Anthropic reasoning (see SIA, Sleeping beauty paradox) suggests otherwise. That’s what I was talking about above: If hypothesis B makes evidence A unlikely, then observing evidence A makes hypothesis B unlikely.
Yes, the ordering does matter. Compare two hypotheses, one, H1, says that one average there will be 1 civilisation in each galaxy. The other, Hϵ says that on average there will be 10−8 civilisations in each galaxy. Suppose the second hypothesis is true.
If you now do the experiment of choosing a random galaxy, and counting the number of civilisations in that galaxy, you will probably not find any civilisation, which correctly supports Hϵ.
If you do the second experiment of first finding yourself in some galaxy and then counting the number of civilisation in the galaxy, you will at least find your own civilisation and probably not any other. If you don’t correct for the fact that this experiment is different, you would update strongly in the direction of H1 even when Hϵ is true and the evidence of the experiment is as favourably towards Hϵ as possible. This cannot be the correct reasoning, since correct reasoning should not consistently lead you to wrong conclusions.
You might argue that there is another possible state which is even more favourable evidence towards Hϵ: that you do not exist. However, in a universe with 1011 galaxies, the probability of this is 10−434.
The crucial point is that on H1 our existence (say, E) is much more likely than on Hϵ. That is, P(E|H1)>P(E|Hϵ). If our prior is P(H1)=P(Hϵ), then P(H1|E)>P(Hϵ|E). That is, our existence implies that H1 is more likely.
As a sanity check, this also doesn’t reliably lead us to wrong conclusions: If we have two possible universes, one where H1 is true and one where Hϵ is true, and in both possible universes the civilizations reason as above and regard H1 as more likely, much more civilizations will reach the true rather than the false conclusion, since the possible universe where H1 is true has much more civilizations (or a much higher density etc).
Now I see, yes you are right. If you want the beliefs to be accurate at the civilisation level, that is the correct way of looking at it. This corresponds to the 1⁄3 conclusion in the sleeping beauty problem.
I was thinking of it on the universe level, were we are a way for the universe to understand itself. If we want the universe to form accurate beliefs about itself, then we should not include our own civilisation when counting the number of the civilisations in the galaxy. However, when deciding if we should be surprised that we don’t see other civilisations, you are right that should include ourselves in the statistics.