I find that the expected number of inhabitable planets is 50.9 billion, while my point-estimate approximation is just 619 million planets! Clearly when there are very high levels of uncertainty, point estimates perform poorly.
I think there is something misleading about this comparison.
Let’s first take a different example: assume we want to compute how much bread there is in the world (why not). You might model this number as (bread owned by people) + (bread in stores) + (bread in bakeries). Then derive from there that
Now you devise some probability distribution for each of those numbers and come up with your estimates. Question: how big will the difference be between the mean of the output distribution and the sum/products of the means? Can we predict in which direction the difference will go?
(Think about it, then hit the spoiler cache)
There will be no difference. This is because the mean of the product/sum of independent variables is the product/sum of the means.
The reason why you have a difference in your example is because mean(1/x)≠1/mean(x). This has little to do with how uncertain your estimates are.
I think there is something misleading about this comparison.
Let’s first take a different example: assume we want to compute how much bread there is in the world (why not). You might model this number as (bread owned by people) + (bread in stores) + (bread in bakeries). Then derive from there that
TotalWeightOfBread=NumberOfPeople∗avgBreadPerPeople+NumberOfStores∗avgBreadPerStore+NumberOfBakeries∗avgBreadPerBakery
Now you devise some probability distribution for each of those numbers and come up with your estimates. Question: how big will the difference be between the mean of the output distribution and the sum/products of the means? Can we predict in which direction the difference will go?
(Think about it, then hit the spoiler cache)
There will be no difference. This is because the mean of the product/sum of independent variables is the product/sum of the means.
The reason why you have a difference in your example is because mean(1/x)≠1/mean(x). This has little to do with how uncertain your estimates are.
Thanks for highlighting this! You have convinced me.
I’ve made a few changes to the point-estimate section.