Sample from parameterized distribution rather than estimate parameter

Interfaces brms
21

Sorry to keep drawing this out. I’m actually not so sure that something like this should work. I think that you want a posterior that is proportional to the sum of the posterior PDFs that correspond to each value of \theta (or equivalently, proportional to the arithmetic mean of the PDFs). But the approach you’ve suggested gives PDF that is proportional to the geometric mean of the PDFs, not the arithmetic mean. Thus, I think that rather than combining the lpdfs by summing and dividing by N, you’d need to combine the lpdfs with a log_sum_exp. But I don’t have time to work through carefully and make sure I’m not leading you astray.

What definitely works is brute-force multiple imputation :)

22

I think you’re right, if we’re thinking about an analog to integration, then it would make sense to average (or even simpler, sum as you suggest) the density function, not calculate its geometric mean. I also agree that the brute force approach would work and serve as a good check.

Fortunately, I think I figured out what the issue that caused the aberrant behavior I was trying to understand and that sent me down this road. Nevertheless, this was a useful discussion and suggests an alternative to the approach you initially suggested.

1 Like
23

Sorry for the late reply. I had a long day. I also believe this is the right answer.