I'm doing atmospheric retrievals using non-linear optimal estimation, and I would like to calculate the associated uncertainty on the mean of two atmospheric profiles each starting from a different prior solution, but I'm not entirely sure how to do this in the most mathematically rigorous way possible.
My known variables are:
- $P(x_1|y)$, which is given by the covariance matrix $\mathbf{\hat{S}_1}$
- $P(x_2|y)$, which is given by the covariance matrix $\mathbf{\hat{S}_2}$
- $P(x_1)$, which is given by the prior covariance matrix $\mathbf{\hat{S}_{a1}}$
- $P(x_2)$, which is given by the prior covariance matrix $\mathbf{\hat{S}_{a2}}$
and I want to calculate: $P(\frac{x_1 + x_2}{2} | y)$