I’m not sure I’m intuiting the transformation to and from log-normals. My intuition is that since the mean of a log-normal with location u scale s is e^(u+s^2/2) rather than e^u, when we end up with a mean log, transforming back into a mean tacks the s^2/2 back on (aka we’re back to value ~ X rather than value ~ 0). Maybe I’m missing something, I haven’t gone through to rederive your results, but even if everything’s right I think the math could be made more clear.
Great post! We need to see this kind of reasoning made explicit!
I’m not sure I’m intuiting the transformation to and from log-normals. My intuition is that since the mean of a log-normal with location u scale s is e^(u+s^2/2) rather than e^u, when we end up with a mean log, transforming back into a mean tacks the s^2/2 back on (aka we’re back to value ~ X rather than value ~ 0). Maybe I’m missing something, I haven’t gone through to rederive your results, but even if everything’s right I think the math could be made more clear.
Great post! We need to see this kind of reasoning made explicit!