As a general rule if you have a domain like this that extends indefinitely in one direction, the correct prior is one that diminishes as you move further away in that direction, rather than picking a somewhat arbitrary end point and using a uniform prior on that.
Just a quick thought on this issue: Using Laplace’s rule of succession (or any other similar prior) also requires picking a somewhat arbitrary start point. You suggest 200000BC as a start point, but one could of course pick earlier or later years and get out different numbers. So the uniform prior’s sensitivity to decisions about how to truncate the relevant time interval isn’t a special weakness; it doesn’t seem to provide grounds for prefering the Laplacian prior.
I think that for some notion of an “arbitrary superlative,” a uniform prior also makes a lot more intuitive sense than a Laplacian prior. The Laplacian prior would give very strange results, for example, if you tried to use it to estimate the hottest day on Earth, the year with the highest portion of Americans named Zach, or the year with the most supernovas.
Moreover in your case in particular, there are also good reasons to suspect that the chance of a century being the most influential should diminish over time.
I agree with this intuition, but I suppose see it as a reason to shift away from a uniform prior rather than to begin from something as lopsided as a Laplacian. I think that this intuition is also partially (but far from entirely) counterbalanced by the countervailing intuitions Will lists for expecting influence to increase over time.
Just a quick thought on this issue: Using Laplace’s rule of succession (or any other similar prior) also requires picking a somewhat arbitrary start point.
Doesn’t the uniform prior require picking an arbitrary start point and end point? If so, switching to a prior that only requires an arbitrary start point seems like an improvement, all else equal. (Though maybe still worth pointing out that all arbitrariness has not been eliminated, as you’ve done here.)
You are right that having a fuzzy starting point for when we started drawing from the urn causes problems for Laplace’s Law of Succession, making it less appropriate without modification. However, note that in terms of people who have ever lived, there isn’t that much variation as populations were so low for so long, compared to now.
I see your point re ‘arbitrary superlatives’, but am not sure it goes through technically. If I could choose a prior over the relative timescale of beginning to the final year of humanity, I would intuitively have peaks at both ends. But denominated in years, we don’t know where the final year is and have a distribution over this that smears that second peak out over a long time. This often leaves us just with the initial peak and a monotonic decline (though not necessarily of the functional form of LLS). That said, this interacts with your first point, as the beginning of humanity is also vague, smearing that peak out somewhat too.
Just a quick thought on this issue: Using Laplace’s rule of succession (or any other similar prior) also requires picking a somewhat arbitrary start point. You suggest 200000BC as a start point, but one could of course pick earlier or later years and get out different numbers. So the uniform prior’s sensitivity to decisions about how to truncate the relevant time interval isn’t a special weakness; it doesn’t seem to provide grounds for prefering the Laplacian prior.
I think that for some notion of an “arbitrary superlative,” a uniform prior also makes a lot more intuitive sense than a Laplacian prior. The Laplacian prior would give very strange results, for example, if you tried to use it to estimate the hottest day on Earth, the year with the highest portion of Americans named Zach, or the year with the most supernovas.
I agree with this intuition, but I suppose see it as a reason to shift away from a uniform prior rather than to begin from something as lopsided as a Laplacian. I think that this intuition is also partially (but far from entirely) counterbalanced by the countervailing intuitions Will lists for expecting influence to increase over time.
Doesn’t the uniform prior require picking an arbitrary start point and end point? If so, switching to a prior that only requires an arbitrary start point seems like an improvement, all else equal. (Though maybe still worth pointing out that all arbitrariness has not been eliminated, as you’ve done here.)
You are right that having a fuzzy starting point for when we started drawing from the urn causes problems for Laplace’s Law of Succession, making it less appropriate without modification. However, note that in terms of people who have ever lived, there isn’t that much variation as populations were so low for so long, compared to now.
I see your point re ‘arbitrary superlatives’, but am not sure it goes through technically. If I could choose a prior over the relative timescale of beginning to the final year of humanity, I would intuitively have peaks at both ends. But denominated in years, we don’t know where the final year is and have a distribution over this that smears that second peak out over a long time. This often leaves us just with the initial peak and a monotonic decline (though not necessarily of the functional form of LLS). That said, this interacts with your first point, as the beginning of humanity is also vague, smearing that peak out somewhat too.