It looks like that time-dependence in Laplace’s rule appears because of the “+1” part which is not measured in time units.
The “+1” appeared in the original Laplace’s rule exactly because it was derived for predicting discrete events with known periodicity. That is, if the sun has risen today, we know that it will not rise next 24 hours, and the question is will it appear after that on the 25th hour. This is not true for earthquakes which could happen at any moment.
This problem has been addressed in Gott’s equation which looks exactly like Laplace’s rule of succession but without “+1” and “+2″ parts, and is used to predict the future duration of continuous processes, like life expectancy. Gott’s equation gives the same result no matter if days or years are used.
If S=1 your equation does look like Gott’s equation.
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.
It looks like that time-dependence in Laplace’s rule appears because of the “+1” part which is not measured in time units.
The “+1” appeared in the original Laplace’s rule exactly because it was derived for predicting discrete events with known periodicity. That is, if the sun has risen today, we know that it will not rise next 24 hours, and the question is will it appear after that on the 25th hour. This is not true for earthquakes which could happen at any moment.
This problem has been addressed in Gott’s equation which looks exactly like Laplace’s rule of succession but without “+1” and “+2″ parts, and is used to predict the future duration of continuous processes, like life expectancy. Gott’s equation gives the same result no matter if days or years are used.
If S=1 your equation does look like Gott’s equation.
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.