I think it makes sense to assess the annual risk of simulation shutdown based on the mean annual probability of simulation shutdown. However, I also believe the risk of simulation shutdown is much lower than the one guessed by the humble cosmologist.
The mean of a loguniform distribution ranging from a to 1 is −1/​ln(a). If a = 10^-100, the risk is 0.434 % (= −1/​ln(10^-100)). However, I assume there is no reason to set the minimum risk to 10^-100, so the cosmologist may actually have been overconfident. Since there is no obvious natural lower bound for the risk, because more or less by definition we do not have evidence about the simulators, I guess the lower bound can be arbitrarily close to 0. In this case, the mean of the loguniform distribution goes to 0 (= −1/​(ln(0))), so it looks like the humblest view corresponds to 0 risk of simulation shutdown.
In addition, the probability of surviving an annual risk of simulation shutdown of 0.434 % (= 10^-5.44) over the estimated age of the universe of 13.8 billion years is only 10^-75,072,000,000 (= (10^-5.44)^(13.8*10^9)), which is basically 0. So the universe would have needed to be super super lucky in order to have survived for so long with such high risk. One can try to counter this argument saying there are selection effects. However, it would be super strange to have an annual risk of simulation shutdown of 0.434 % without any partial shutdowns, given that tail risk usually follows something like a power law[1] without severe jumps in severity.
Hi titotal,
I think it makes sense to assess the annual risk of simulation shutdown based on the mean annual probability of simulation shutdown. However, I also believe the risk of simulation shutdown is much lower than the one guessed by the humble cosmologist.
The mean of a loguniform distribution ranging from a to 1 is −1/​ln(a). If a = 10^-100, the risk is 0.434 % (= −1/​ln(10^-100)). However, I assume there is no reason to set the minimum risk to 10^-100, so the cosmologist may actually have been overconfident. Since there is no obvious natural lower bound for the risk, because more or less by definition we do not have evidence about the simulators, I guess the lower bound can be arbitrarily close to 0. In this case, the mean of the loguniform distribution goes to 0 (= −1/​(ln(0))), so it looks like the humblest view corresponds to 0 risk of simulation shutdown.
In addition, the probability of surviving an annual risk of simulation shutdown of 0.434 % (= 10^-5.44) over the estimated age of the universe of 13.8 billion years is only 10^-75,072,000,000 (= (10^-5.44)^(13.8*10^9)), which is basically 0. So the universe would have needed to be super super lucky in order to have survived for so long with such high risk. One can try to counter this argument saying there are selection effects. However, it would be super strange to have an annual risk of simulation shutdown of 0.434 % without any partial shutdowns, given that tail risk usually follows something like a power law[1] without severe jumps in severity.
Although I think tail risk often decays faster than suggested by a power law.