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I'm not sure I can agree with this analysis. Consider two hypothetical universes. In one, p = 10^-1000, but by a statistical miracle, here we are. We would cer

I'm not sure I can agree with this analysis. Consider two hypothetical universes. In one, p = 10^-1000, but by a statistical miracle, here we are. We would certainly be alone in this universe, because the number of stars in the visible universe is inconsequential compared to this number. In the second one, the universe is the same size, but p is such that there are 100 planets with intelligent life like ours. If you cannot say "anything at all about P" from our existence, then we should be indifferent between these two hypotheses, on the basis of the evidence of our existence. However, this is not the case. The evidence supports the second hypothesis much better than the first. If X occurred, and is more likely under hypothesis B than A, then X is evidence for hypothesis B, otherwise you don't have a consistent system of probability. Now you can further imagine that God comes down from heaven and tells you that in fact both of these two universes actually exist, and they are the only two universes that exist, so you are definitely in one of the two, but doesn't tell you which one. Is your posterior probability now 50% between the two? "God" here represents the anthropic principle leaning over and saying "psst, don't worry about the 10^-1000, because, well, here you are". But you still have to worry about the difference between 1 and 100. If N beings in universe A and 100 * N beings in universe B all face this same question, your only reasonable conclusion as one of them is that the odds are 100-to-1 in favor of you being in universe B. So it's not the case that our existence tells us nothing about p. Of course, p is what it is, and only physics and chemistry and biology can actually reveal it, but the fact that we are here means that theories that make our existence much less likely are themselves much less likely.

anthuswilliams 5y ago
This is a very thoughtful rejoinder, thanks. > If X occurred, and is more likely under hypothesis B than A, then X is evidence for hypothesis B, otherwise you don't have a consistent system of probability. The point is, this idea breaks down when neither hypothesis would exist if X had not occurred. This differs from other questions of probability because the conditional probability of an event is predicated on the ability of an observer to ask about it. We should indeed be indifferent between the two hypotheses, if our own existence is all we have to go on. (This might seem strange, if you think of hypothesis B as having 100x the prior probability, but the point here is that we really don't have a basis for a prior at all.) > Now you can further imagine that God comes down from heaven and tells you that in fact both of these two universes actually exist, and they are the only two universes that exist, so you are definitely in one of the two, but doesn't tell you which one. Is your posterior probability now 50% between the two? No, of course I would assume I was in universe B. This is a very different question. Instead of "what is the nature of our universe" the question is "of the two existing universes God has described, which do I occupy?"
- inimino 5y ago
  Thanks for the reply. The two scenarios don't seem all that different to me, because "there are two possible universes, and you're in one, but you don't know which" is how I think about probability and belief in general. Here's a third scenario: Instead of God and two universes, let's use an unknowable physical constant and one universe. In this scenario, we have solved physics, except for one fundamental physical constant that we know for a fact can take one of exactly two values, but for some reason we know that experimental evidence will never be able to distinguish between the two. From the physics side, it will always be a toss-up between values A and B. But, as before, we also know for some reason (maybe it has to do with conditions in the early universe that can never be reproduced) that if the value is A, then p is 10^-1000, and for B etc. as before. What would you say in this case? In this case, all that's changed is that we don't say that both universes exist. Instead only one exists, but we don't know which one it is. We also said that from the physics side, A and B are equally likely, which means our state of knowledge about A and B are meant to be the same as in scenario two. Would you now say that we should be indifferent between A and B in this case? A fourth scenario: There's a multiverse, and universes exist with every possible value of p. The distribution over values of p is unknown to us. If it's uniform, then almost all the people asking themselves this question are in universes where p is high, because those universes are teeming with life. None of the people are in universes where p is 0, which is just the anthropic principle again. Where p is very low, most universes are completely unpopulated. Chemistry and biology may give us arguments to say that p is low, so our prior over p certainly shouldn't be uniform. But whether you look at it as all the universes existing, or whether you look at it as a state of uncertainty about the universe, either way, as far as I can tell, the math has to work out the same way. If most reasoning beings are in the universes where reasoning beings are more likely, then we are justified in saying it is more likely that that's the kind of universe we're in. > This differs from other questions of probability because the conditional probability of an event is predicated on the ability of an observer to ask about it. I think this is the part of your argument that I don't follow. The physicists in the third scenario will report that either the physical constant is B, and we can start looking for the 100 other civilizations, or it is A, and a miracle occurred. Observers exist in both cases. Wouldn't we say that the case that requires a miracle to have occurred is less well supported by the evidence?

anthuswilliams 5y ago

This is a very thoughtful rejoinder, thanks. > If X occurred, and is more likely under hypothesis B than A, then X is evidence for hypothesis B, otherwise you don't have a consistent system of probability. The point is, this idea breaks down when neither hypothesis would exist if X had not occurred. This differs from other questions of probability because the conditional probability of an event is predicated on the ability of an observer to ask about it. We should indeed be indifferent between the two hypotheses, if our own existence is all we have to go on. (This might seem strange, if you think of hypothesis B as having 100x the prior probability, but the point here is that we really don't have a basis for a prior at all.) > Now you can further imagine that God comes down from heaven and tells you that in fact both of these two universes actually exist, and they are the only two universes that exist, so you are definitely in one of the two, but doesn't tell you which one. Is your posterior probability now 50% between the two? No, of course I would assume I was in universe B. This is a very different question. Instead of "what is the nature of our universe" the question is "of the two existing universes God has described, which do I occupy?"

inimino 5y ago
Thanks for the reply. The two scenarios don't seem all that different to me, because "there are two possible universes, and you're in one, but you don't know which" is how I think about probability and belief in general. Here's a third scenario: Instead of God and two universes, let's use an unknowable physical constant and one universe. In this scenario, we have solved physics, except for one fundamental physical constant that we know for a fact can take one of exactly two values, but for some reason we know that experimental evidence will never be able to distinguish between the two. From the physics side, it will always be a toss-up between values A and B. But, as before, we also know for some reason (maybe it has to do with conditions in the early universe that can never be reproduced) that if the value is A, then p is 10^-1000, and for B etc. as before. What would you say in this case? In this case, all that's changed is that we don't say that both universes exist. Instead only one exists, but we don't know which one it is. We also said that from the physics side, A and B are equally likely, which means our state of knowledge about A and B are meant to be the same as in scenario two. Would you now say that we should be indifferent between A and B in this case? A fourth scenario: There's a multiverse, and universes exist with every possible value of p. The distribution over values of p is unknown to us. If it's uniform, then almost all the people asking themselves this question are in universes where p is high, because those universes are teeming with life. None of the people are in universes where p is 0, which is just the anthropic principle again. Where p is very low, most universes are completely unpopulated. Chemistry and biology may give us arguments to say that p is low, so our prior over p certainly shouldn't be uniform. But whether you look at it as all the universes existing, or whether you look at it as a state of uncertainty about the universe, either way, as far as I can tell, the math has to work out the same way. If most reasoning beings are in the universes where reasoning beings are more likely, then we are justified in saying it is more likely that that's the kind of universe we're in. > This differs from other questions of probability because the conditional probability of an event is predicated on the ability of an observer to ask about it. I think this is the part of your argument that I don't follow. The physicists in the third scenario will report that either the physical constant is B, and we can start looking for the 100 other civilizations, or it is A, and a miracle occurred. Observers exist in both cases. Wouldn't we say that the case that requires a miracle to have occurred is less well supported by the evidence?