Given the variance components, the Bayes estimate of the secular trend is a linear combination of the fitted mean vector and the fitted residual

\[
\tilde{\mu}=P Y+L \Sigma^{-1} Q Y \text {, }
\]

where \(P Y\) and \(Q Y\) are independent Gaussian vectors. Use this representation to approximate \(\operatorname{cov}(\tilde{\mu})\).