Combining forecasts or distributions to form a more accurate one Suppose you are interested in getting as good an estimate as possible for a random variable $X$, this could be for example a stock price in the future. You go to see $N$ experts, each gives you a distribution for the value (for simplicity, assume Gaussian so you get the mean $\mu_i$ and standard deviation $\sigma_i$, $i = 1, ..., N$).
How do you combine these estimates to form a better one? I believe one can't just average the means, because the accuracy of each estimate is related to the standard deviation.
 A: Combinations of distributions are called mixtures. You typically use a weighted combination. In the specific case of combining normal distributions, the results are gaussian-mixtures.
The mean of the mixture is just the weighted average of the component means, so you can use this as a point forecast.
Of course, you can try to find "good" weights for your mixture. As whuber notes, I would not necessarily weight a very confident expert (low variance) highly. If at all, I would recommend you collect a track record of your experts' density forecasts, then evaluate them using proper scoring-rules and see whether low-variance experts are truly well calibrated, or whether they are just overconfident.
However, we find surprisingly often that simple equally weighted combinations perform better than trying to find "optimal" weights. This has been called the "forecast combination puzzle". One possible explanation is that estimating weights adds another source of variance, which may dominate and reduction in bias (Claeskens et al., 2016, IJF).
