Given 2 distributions $X_1$ and $X_2$ which are independent estimates of the distribution $X$, this be estimated with the inverse-variance method from:

• $X=\frac{\frac{X_1}{V\left(X_1\right)}+\frac{X_2}{V\left(X_2\right)}}{\frac{1}{V\left(X_1\right)}+\frac{1}{V\left(X_2\right)}}$.

Under which conditions is this a good aproach? For example, for which types of distributions? These questions might be relevant for determining:

• A posterior distribution based on distributions for the prior and estimate.
• A distribution which combines estimates of different theories.

Some notes:

• The inverse-variance method minimises the variance of a weighted mean of $X_1$ and $X_2$.
• Calculating $E\left(X\right)$ and $V\left(X\right)$ according to the above formula would result in a mean and variance equal to those derived in this analysis from Dario Amodei, which explains how to combine $X_1$ and $X_2$ following a Bayesian approach if these follow normal distributions.