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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

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
  The third scenario here is equivalent to the "God tells us we're in one of two universes" experiment. The only difference between the two scenarios is that, in the latter, the two universes "exist", but if God had said "there were two universes and I destroyed one; you inhabit the one which remains; which is it?" our answer wouldn't change. The fourth scenario is more interesting because we've admitted the distribution of p is unknown to us. You note that if p is uniform, all intelligent lifeforms should conclude they live in a universe where p is high, but I'm not certain of that. It seems to me that this logic only works if you conclude there is _one_ universe for each possible value of p. But if, for each possible value of p, the multiverse contains universes for each possible outcome of a Bernoulli trial with probability p, then we can't conclude anything at all. To see this, notice that the number of Lucky universes (where p is high) is uncountably infinite, just as the number of Unlucky-Lucky universes (where p is low but a miracle has occurred) is uncountably infinite. I haven't thought this next part through as deeply, but I believe this problem compounds given the unknown distribution of p. This is because subjectivist models of probability (e.g. Bayes' Theorem) are heavily influenced by the distribution of the prior when we have only observed one trial. And in cases like this one, where the only trials that _could_ be observed are successes, I believe it's all but entirely driven by the distribution of the prior. (You could imagine a hyperprior - a distribution of distributions of p, but I don't think we have any good reason for deciding what that would be either). Anyway, thanks for this. I love geeking out about epistemology.
  - inimino 5y ago
    I agree that scenarios two and three are equivalent. But then in three, the fact that we exist is allowed to influence our belief in what the value of a physical constant is. From this contrived example, can't we get to the argument being made in the original post? The fourth scenario, to me, is the closest to our actual situation. The many universes correspond to our uncertainty about p. It doesn't matter whether they are actual universes and we are in one, or whether they are possible universes and we are in one. I wasn't sure why you had the number of universes uncountably infinite. Is it because p is a real number? To make this argument, wouldn't you need to show that not only are they both uncountably infinite, but that they have equal measure? We can construct two sets that are both uncountably infinite, but one has measure 1 and the other measure 0. If the Lucky and Unlucky-Lucky universes have equal measure, there's still more people in the Lucky ones. Only if the measure of Unlucky ones is much higher, then there would be more people in the Unlucky-Lucky universes. We said we don't know anything about the distribution. But! In my opinion, we do have some idea about the distribution. It's just our prior, from whatever we know of chemistry, biology, etc, about p. We have some reasons to believe it's low. (The arguments the intelligent design folks make.) Then we update that with the observation that we exist. In other words, we start with a distribution over possible universes based on what we know about physics, and the complexity that's necessary for life, and we say that it's remarkable that we are here. But then we update by saying that, of all beings capable of reasoning through this, more of them would be in universes where it's less remarkable. So it doesn't tell us what p is, but it should shift our distribution over values of p to the right. Eventually, we may get such a clear understanding of how life must have formed, that we could say exactly what steps had to happen and in what order, and how probable they all were, and could determine p exactly, and even simulate everything. Then this argument wouldn't buy us anything. Because the physics is what it is, and p is what it is. It's only because our prior over p is currently so vague, that the argument is interesting. Anyway, it's an interesting discussion and you may be completely right. I'm certainly not certain of anything here :-)

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
The third scenario here is equivalent to the "God tells us we're in one of two universes" experiment. The only difference between the two scenarios is that, in the latter, the two universes "exist", but if God had said "there were two universes and I destroyed one; you inhabit the one which remains; which is it?" our answer wouldn't change. The fourth scenario is more interesting because we've admitted the distribution of p is unknown to us. You note that if p is uniform, all intelligent lifeforms should conclude they live in a universe where p is high, but I'm not certain of that. It seems to me that this logic only works if you conclude there is _one_ universe for each possible value of p. But if, for each possible value of p, the multiverse contains universes for each possible outcome of a Bernoulli trial with probability p, then we can't conclude anything at all. To see this, notice that the number of Lucky universes (where p is high) is uncountably infinite, just as the number of Unlucky-Lucky universes (where p is low but a miracle has occurred) is uncountably infinite. I haven't thought this next part through as deeply, but I believe this problem compounds given the unknown distribution of p. This is because subjectivist models of probability (e.g. Bayes' Theorem) are heavily influenced by the distribution of the prior when we have only observed one trial. And in cases like this one, where the only trials that _could_ be observed are successes, I believe it's all but entirely driven by the distribution of the prior. (You could imagine a hyperprior - a distribution of distributions of p, but I don't think we have any good reason for deciding what that would be either). Anyway, thanks for this. I love geeking out about epistemology.
- inimino 5y ago
  I agree that scenarios two and three are equivalent. But then in three, the fact that we exist is allowed to influence our belief in what the value of a physical constant is. From this contrived example, can't we get to the argument being made in the original post? The fourth scenario, to me, is the closest to our actual situation. The many universes correspond to our uncertainty about p. It doesn't matter whether they are actual universes and we are in one, or whether they are possible universes and we are in one. I wasn't sure why you had the number of universes uncountably infinite. Is it because p is a real number? To make this argument, wouldn't you need to show that not only are they both uncountably infinite, but that they have equal measure? We can construct two sets that are both uncountably infinite, but one has measure 1 and the other measure 0. If the Lucky and Unlucky-Lucky universes have equal measure, there's still more people in the Lucky ones. Only if the measure of Unlucky ones is much higher, then there would be more people in the Unlucky-Lucky universes. We said we don't know anything about the distribution. But! In my opinion, we do have some idea about the distribution. It's just our prior, from whatever we know of chemistry, biology, etc, about p. We have some reasons to believe it's low. (The arguments the intelligent design folks make.) Then we update that with the observation that we exist. In other words, we start with a distribution over possible universes based on what we know about physics, and the complexity that's necessary for life, and we say that it's remarkable that we are here. But then we update by saying that, of all beings capable of reasoning through this, more of them would be in universes where it's less remarkable. So it doesn't tell us what p is, but it should shift our distribution over values of p to the right. Eventually, we may get such a clear understanding of how life must have formed, that we could say exactly what steps had to happen and in what order, and how probable they all were, and could determine p exactly, and even simulate everything. Then this argument wouldn't buy us anything. Because the physics is what it is, and p is what it is. It's only because our prior over p is currently so vague, that the argument is interesting. Anyway, it's an interesting discussion and you may be completely right. I'm certainly not certain of anything here :-)