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.
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.