In the case where there are zero observed successes (so 𝑆 = 0) in the last 𝑛 years, Gott’s formula
P(N≤Z)=∫N=ZN=nP(N|n)dN=Z−nZ
for the probability that the next success happens in the next 𝑚 = 𝑍 − 𝑛 years gives
mm+n=1−(1+mn)−1
which ends up being exactly the same as the time-invariant Laplace’s rule. The same happens if there was a success (𝑆 = 1) but we chose not to update on it because we chose to start the time period with it. So the time-invariant Laplace’s rule is a sort of generalization of Gott’s formula, which is neat.
Yes, this is true. We note in a footnote that performing an anthropic update is similar to assuming an extra (virtual) success in the observation period, so you can indeed justify our advice of introducing such a success on anthropic grounds.
In the case where there are zero observed successes (so 𝑆 = 0) in the last 𝑛 years, Gott’s formula
P(N≤Z)=∫N=ZN=nP(N|n)dN=Z−nZ
for the probability that the next success happens in the next 𝑚 = 𝑍 − 𝑛 years gives
mm+n=1−(1+mn)−1
which ends up being exactly the same as the time-invariant Laplace’s rule. The same happens if there was a success (𝑆 = 1) but we chose not to update on it because we chose to start the time period with it. So the time-invariant Laplace’s rule is a sort of generalization of Gott’s formula, which is neat.
Yes, this is true. We note in a footnote that performing an anthropic update is similar to assuming an extra (virtual) success in the observation period, so you can indeed justify our advice of introducing such a success on anthropic grounds.