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Yes. As long as you think of some not-too-complicated scenario where the one would lead to the other, that's perfectly reasonable. For example, God might exist and decide to prove it to you by effecting that prediction. I certainly agree this has a probability of at least one in a billion squared. In fact, suppose you actually get heads the next 60 times you flip a coin, even though you are choosing different coins, it is on different days, and so on. By that point you will be quite convinced that the heads are not independent, and that there is quite a good chance that you will get 1000 heads in a row.
It would be different of course if you picked a random series of heads and tails: in that case you still might say that there is at least that probability that someone else will do it (because God might make that happen), but you surely cannot say that it had that probability before you picked the random series.
This is related to what I said in the torture discussion, namely that explicitly describing a scenario automatically makes it far more probable to actually happen than it was before you described it. So it isn't a problem if the probability of 1000 heads in a row is more likely than 1 in 2-to-1000. Any series you can mention would be more likely than that, once you have mentioned it.
Also, note that there isn't a problem if the 1000 heads in a row is lower than one in a billion, because when I made the general claim, I said "a claim that significant number of people accept as likely true," and no one expects to get the 1000 heads.
Probabilities should sum to 1. You're saying moreover that probabilities should not be lower that some threshhold. Can I can get you to admit that there's a math issue here that you can't wave away, without trying to fine-tune my examples? If you claim you can solve this math issue, great, but say so.
Edit: -1 because I'm being rude? Sorry if so, the tone does seem inappropriately punchy to me now. -1 because I'm being stupid? Tell me how!
I set a lower bound of one in a billion on the probability of "a natural language claim that a significant number of people accept as likely true". The number of such mutually exclusive claims is surely far less than a billion, so the math issue will resolve easily.
Yes, it is easy to find more than a billion claims, even ones that some people consider true, but they are not mutually exclusive claims. Likewise, it is easy to find more than a billion mutually exclusive claims, but they are not ones that people believe to be true, e.g. no one expects 1000 heads in a row, no one expects a sequence of five hundred successive heads-tails pairs, and so on.
I didn't downvote you.
Maybe I see. You are updating on the fact that many people believe something, and are saying that P(A|many people believe A) should not be too small. Do you agree with that characterization of your argument?
In that case, we will profitably distinguish between P(A|no information about how many people believe A) and P(A|many people believe A). Is there a compact way that I can communicate something like "Excepting/not updating on other people's beliefs, P(God exists) is very small"? If I said something like that would you still think I was being overconfident?
This is basically right, although in fact it is not very profitable to speak of what the probability would be if we didn't have some of the information that we actually have. For example, the probability of this sequence of ones and zeros -- 0101011011101110 0010110111101010 0100010001010110 1010110111001100 1110010101010000 -- being chosen randomly, before anyone has mentioned this particular sequence, is one out 2 to the 80. Yet I chose it randomly, using a random number generator (not a pseudo random number generator, either.) But I doubt that you will conclude that I am certainly lying, or that you are hallucinating. Rather, as Robin Hanson points out, extraordinary claims are extraordinary evidence. The very fact that I write down this improbable evidence is extremely extraordinary evidence that I have chosen it randomly, despite the huge improbability of that random choice. In a similar way, religious claims are extremely strong evidence in favor of what they claim; naturally, just as if I hadn't written the number, you would never believe that I might choose it randomly, in the same way, if people didn't make religious claims, you would rightly think them to be extremely improbable.
It is always profitable to give different concepts different names.
Let GM be the assertion that I'll one day play guitar on the moon. Your claim is that this ratio
P(GM|I raised GM as a possibility)/P(GM)
is enormous. Bayes theorem says that this is the same as
P(I raised GM as a possibility|GM)/P(I raised GM as a possibility)
so that this second ratio is also enormous. But it seems to me that both numerator and denominator in this second ratio are pretty medium-scale numbers--in particular the denominator is not miniscule. Doesn't this defeat your idea?
The evidence contained in your asserting GM would be much stronger than the evidence contained in your raising the possibility.
Still, there is a good deal of evidence contained in your raising the possibility. Consider the second ratio: the numerator is quite high, probably more than .5, since in order to play guitar on the moon, you would have to bring a guitar there, which means you'd probably be thinking about it.
The denominator is in fact quite small. If you randomly raise one outlandish possibility of performing some action in some place, each day for 50 years, and there are 10,000 different actions (I would say there are at least that many), and 100,000 different places, then the probability of raising the possibility will be 18,250/(10,000 x 100,000), which is 0.00001825, which is fairly small. The actual probability is likely to be even lower, since you may not be bringing up such possibilities every day for 50 years. Religious claims are typically even more complicated than the guitar claim, so the probability of raising their possibility is even lower.
--one more thing: I say that raising the possibility is strong evidence, not that the resulting probability is high: it may start out extremely low and end up still very, very low, going from say one in a google to one in a sextillion or so. It is when you actually assert that it's true that you raise the probability to something like one in a billion or even one in a million. Note however that you can't refute me by now going on to assert that you intend to play a guitar on the moon; if you read Hanson's article in my previous link, you'll see that he shows that assertions are weak evidence in particular cases, namely in ones in which people are especially likely to lie: and this would be one of them, since we're arguing about it. So in this particular case, if you asserted that you intended to do so, it would only raise the probability by a very small amount.
I understand that you think the lower bound on probabilities for things-that-are-believed is higher than the lower bound on probabilities for things-that-are-raised-as-possibilities. I am fairly confident that I can change your mind (that is, convince you not to impose lower bounds like this at all), and even more confident that I can convince you that imposing lower bounds like this is mathematically problematic (that is, there are bullets to be bitten) in ways that hadn't occurred to you a few days ago.
I do not see one of these bounds as more or less sound than the other, but am focusing on the things-that-are-raised-as-possibilities bound because I think the discussion will go faster there.
More soon, but tell me if you think I've misunderstood you, or if you think you can anticipate my arguments. I would also be grateful to hear from whoever is downvoting these comments.
Note that I said there should be a lower bound on the probability for things that people believe, and even made it specific: something on the order of one in a billion. But I don't recall saying (you can point it out if I'm wrong) that there is a lower bound on the probability of things that are raised as possibilities. Rather, I merely said that the probability is vastly increased.
To the comment here, I responded that raising the possibility raised the probability of the thing happening by orders of magnitude. But I didn't say that the resulting probability was high, in fact it remains very low. Since there is no lower bound on probabilities in general, there is still no lower bound on probabilities after raising them by orders of magnitude, which is what happens when you raise the possibility.
So if you take my position to imply such a lower bound, either I've misstated my position accidentally, or you have misunderstood it.
I did misunderstand you, and it might change things; I will have to think. But now your positions seem less coherent to me, and I no longer have a model of how you came to believe them. Tell me more:
Let CM(n) be the assertion "one day I'll play guitar on the moon, and then flip an n-sided coin and it will come up heads." The point being that P(CM(n)) is proportional to 1/n. Consider the following ratios:
How do you think these ratios change as n grows? Before I had assumed you thought that ratios 1. and 4. grew to infinity as n did. I still understand you to be saying that for 4. Are you now denying it for 1., or just saying that 1. grows more slowly? I can't guess what you believe about 2. and 3.