However, Bayes says that if we assign greater than 10^-8 prior probability to “strange” explanations
Well, don’t do that then. Does 10^-8, besides being the chances of a ticket winning a typical big lottery, carry in addition the implied meaning “unimaginably small”, “so small that one must consider all manner of weird other possibilities that in fact we have no way of assessing the probability of, but 10^-8 is so extraordinarily small that surely they must be considered alongside the simple explanation that my ticket won”? “How could we ever be 10^-8 sure of anything?”
Because I would dispute that. Consider someone who has a lottery ticket in their hand, for a draw about to be announced, with 1 chance in 100,000,000 of having the winning numbers. If their numbers are drawn, they must overcome 80dB of prior improbability to be persuaded of that. (It does not matter whether they know that is what they are doing: they are nonetheless doing that.) An impossible task? No, almost all jackpots in the Euromillions lottery (probability 1⁄76275360) are claimed. Ordinary people, successfully comparing two strings of seven numbers and getting the right answer. It is news when a Euromillions jackpot goes unclaimed for as little as one week.
One of the alternative hypotheses that one must consider, of course, is the mundane “I am mistaken: this is not a winning ticket, despite the fact that I have stared at the two sets of numbers and the date over and over and they still appear to be identical.” I don’t know how many false positives the claims line gets. But the jackpot is awarded at least every few weeks, and every time it is claimed by people who were not mistaken.
There are two questions we must consider, according to Bayes: What is the prior probability of living in a simulation, and given we live in a simulation, what is the probability of winning the lottery?
We can invoke your argument at either point, and I’m not sure which you intended.
-- Is 10^-8 enough evidence to overcome the prior improbability? In this case, “prior” means just before we bought the ticket; so we have a lifetime of evidence to help us decide whether we live in a simulation. (Determining this may be difficult, of course, but the lottery argument presumes we can get some distinguishing evidence in various ways!)
—Is 10^-8 actually so much lower than the probability of winning the lotto in a simulation? It could even be higher, depending on what we think is likely!
Other commentors pointed out the second possibility, but I dodn’t think of the first until your post. We might accept the idea that winning the lotto is rather more probable in a simulation, and yet reject the idea that we should believe we’re in a simulation if we win, simply because the simulation hypothesis is much more complex than the regular-world hypothesis. We are then “safe” unless we win the lotto twice. :)
Well, don’t do that then. Does 10^-8, besides being the chances of a ticket winning a typical big lottery, carry in addition the implied meaning “unimaginably small”, “so small that one must consider all manner of weird other possibilities that in fact we have no way of assessing the probability of, but 10^-8 is so extraordinarily small that surely they must be considered alongside the simple explanation that my ticket won”? “How could we ever be 10^-8 sure of anything?”
Because I would dispute that. Consider someone who has a lottery ticket in their hand, for a draw about to be announced, with 1 chance in 100,000,000 of having the winning numbers. If their numbers are drawn, they must overcome 80dB of prior improbability to be persuaded of that. (It does not matter whether they know that is what they are doing: they are nonetheless doing that.) An impossible task? No, almost all jackpots in the Euromillions lottery (probability 1⁄76275360) are claimed. Ordinary people, successfully comparing two strings of seven numbers and getting the right answer. It is news when a Euromillions jackpot goes unclaimed for as little as one week.
One of the alternative hypotheses that one must consider, of course, is the mundane “I am mistaken: this is not a winning ticket, despite the fact that I have stared at the two sets of numbers and the date over and over and they still appear to be identical.” I don’t know how many false positives the claims line gets. But the jackpot is awarded at least every few weeks, and every time it is claimed by people who were not mistaken.
There is no such thing as a small number.
There are two questions we must consider, according to Bayes: What is the prior probability of living in a simulation, and given we live in a simulation, what is the probability of winning the lottery?
We can invoke your argument at either point, and I’m not sure which you intended.
-- Is 10^-8 enough evidence to overcome the prior improbability? In this case, “prior” means just before we bought the ticket; so we have a lifetime of evidence to help us decide whether we live in a simulation. (Determining this may be difficult, of course, but the lottery argument presumes we can get some distinguishing evidence in various ways!) —Is 10^-8 actually so much lower than the probability of winning the lotto in a simulation? It could even be higher, depending on what we think is likely!
Other commentors pointed out the second possibility, but I dodn’t think of the first until your post. We might accept the idea that winning the lotto is rather more probable in a simulation, and yet reject the idea that we should believe we’re in a simulation if we win, simply because the simulation hypothesis is much more complex than the regular-world hypothesis. We are then “safe” unless we win the lotto twice. :)