I was wondering if there is an efficient way to compute the conditional expectation of every element in a Gaussian random vector ?
Specifically:
For a pair of Gaussian random variables $[x,y]$, the conditional expectation is $E[x\mid y] = \mu_x - \Sigma_{x,y} \Sigma_{y,y}^{-1}[y -\mu_y]$ where $\Sigma$ is (co)variance and $\mu$ is mean.
Now given a $n$-dimensional Gaussian random vector $[x_1 \ldots x_n]$, I want to compute $E[x_i \mid \bigcup x_j, \forall j\neq i]$, that is for each component $i$ I want its expected value given every other component.
One straight forward way would be to use the conditional expectation formula for each $i$ - which would involve $n$ inversions of $n-1 \times n-1$ covariance matrices.
However, it appears to me that there is a lot of redundancy in this approach. Is a more efficient way to compute this without having to do this $n-1\times n-1$ inversion per-element ?
Thanks.