How are both X1 and X2 estimates of X when they are different distributions? At this point I am out of my depth so I do not have an informative answer for you.
X1 could be a distribution fitted to 3 quantiles predicted for X by forecaster A (as in Metaculus’ questions which do not involve forecasting probabilities).
X2 could be a distribution fitted to 3 quantiles predicted for X by forecaster B.
Meanwhile, I have realised the inverse-variance method minimises the variance of a weighted mean of X1 and X2 (and have updated the question above to reflect this).
How are both X1 and X2 estimates of X when they are different distributions? At this point I am out of my depth so I do not have an informative answer for you.
I will try to illustrate what I mean with an example:
X could be the total number of confirmed and suspected monkeypox cases in Europe as of July 1, 2022.
X1 could be a distribution fitted to 3 quantiles predicted for X by forecaster A (as in Metaculus’ questions which do not involve forecasting probabilities).
X2 could be a distribution fitted to 3 quantiles predicted for X by forecaster B.
Meanwhile, I have realised the inverse-variance method minimises the variance of a weighted mean of X1 and X2 (and have updated the question above to reflect this).