This is doubtless a frequentist approach, which perhaps isn’t allowed, but I asked myself, how should an observer update on the information that civilization has existed up until that time, if observers collectively wanted to minimize their total error.
Suppose that a civilization lasts T years, and I assume that observations are uniformly distributed over the T years. (This would mean, for example, that the civilization at each time point in the interval (0,T) makes a collective, single vote.) Given only the information that civilization has lasted x years (clearly, 0<=x<=T), the observer would guess that civilization will actually last some multiple of x years: cx. Should they choose a c that is low (close to 1) or high, etc?
The optimal value of c can be calculated as exactly sqrt(2).
By taking the minimum of the function that measures the total error, the integral of the error (error=|T-cx|) integrated from T to 0.
You would get a different c if you assume that growth is exponential and weight by the number of observers. It would be closer to 1.
Also, implicit is the assumption that T is uniformly distributed over an unknown range. Instead, T might be normally distributed with an unknown mean or extremely tight-tailed. These would also affect c, but by moving c further from 1 I think.
This is doubtless a frequentist approach, which perhaps isn’t allowed, but I asked myself, how should an observer update on the information that civilization has existed up until that time, if observers collectively wanted to minimize their total error.
Suppose that a civilization lasts T years, and I assume that observations are uniformly distributed over the T years. (This would mean, for example, that the civilization at each time point in the interval (0,T) makes a collective, single vote.) Given only the information that civilization has lasted x years (clearly, 0<=x<=T), the observer would guess that civilization will actually last some multiple of x years: cx. Should they choose a c that is low (close to 1) or high, etc?
The optimal value of c can be calculated as exactly sqrt(2).
By taking the minimum of the function that measures the total error, the integral of the error (error=|T-cx|) integrated from T to 0.
You would get a different c if you assume that growth is exponential and weight by the number of observers. It would be closer to 1.
Also, implicit is the assumption that T is uniformly distributed over an unknown range. Instead, T might be normally distributed with an unknown mean or extremely tight-tailed. These would also affect c, but by moving c further from 1 I think.
tldr;
If you observe that civilization is age X units, you should update that it will last another 0.4 X units.