I was thinking about cases in which X1 and X2 are non-linear functions of arrays of Monte Carlo samples generated from distributions of different types (e.g. loguniform and lognormal). To calculate E(X1), I can simply compute the mean of the elements of X1. I was looking for a similar simple formula to combine X1 and X2, without having to work with the original distributions used to compute X1 and X2.
A concrete simple example would be combining the following:
According to estimate 1, X is as likely to be 1, 3, 4, 6 or 8: X1 = [1, 2, 3, 4, 5].
According to estimate 2, X is as likely to be 2, 4, 6, 8 or 10: X2 = [2, 4, 6, 8, 10].
The generation mechanisms of estimates 1 and 2 are not known.
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).
I was thinking about cases in which X1 and X2 are non-linear functions of arrays of Monte Carlo samples generated from distributions of different types (e.g. loguniform and lognormal). To calculate E(X1), I can simply compute the mean of the elements of X1. I was looking for a similar simple formula to combine X1 and X2, without having to work with the original distributions used to compute X1 and X2.
A concrete simple example would be combining the following:
According to estimate 1, X is as likely to be 1, 3, 4, 6 or 8: X1 = [1, 2, 3, 4, 5].
According to estimate 2, X is as likely to be 2, 4, 6, 8 or 10: X2 = [2, 4, 6, 8, 10].
The generation mechanisms of estimates 1 and 2 are not known.
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).