I asked this on math.stackexchange with no response, I'm hoping someone here might have something.

Suppose I want to estimate $x \in \mathbb{R}^n$ from two signals with zero mean, normally distributed noise: $y_1 = x + \epsilon_1$ and $y_2 = x + \epsilon_2$, where $\epsilon_1 \sim N(0, \Sigma_1)$, $\epsilon_2 \sim N(0, \Sigma_2)$. Using Bayesian estimation, the posterior mean is $\hat{x} = (\Sigma_1^{-1} + \Sigma_2^{-1})^{-1} (\Sigma_1^{-1} y_1 + \Sigma_2^{-1} y_2) $, which is a "convex combination" of the data points $y_1$ and $y_2$.

In the univariate case, the posterior mean is a convex combination of the data points in the usual sense. In the multivariate case, the "weights" are matrices that add up to $I$. What can we say about the posterior mean (e.g. is the set of possible $\hat{x}$ convex and in what sense, what are the extreme points, how does it vary with the parameters, etc)? A search for "matrix convex combination" gives this result: http://www.math.uni-sb.de/ag/wittstock/OperatorSpace/node73.html which seems to be talking about something different.

IEEE Trans. AES, 1991, J.A. Roecker, (ee.ucd.ie/~amoni/DARIUSPaperWarehouse/roecker91combining.pdf) and “Combining Error Ellipses”, 2007, J.E. Davis (cxc.harvard.edu/csc/memos/files/Davis_ellipse.pdf) might be relevant. $\endgroup$