Conditional mode of Gaussian random effects ($\hat{b}=D{Z}^T{V}^{-1}(Y-X\beta )$)

In a Gaussian linear mixed model with $V=ZD{Z}^T+R$, the conditional multivariate normal distribution of $b$ given $Y$ has mean $D{Z}^T{V}^{-1}(Y-X\beta )$ and covariance matrix $D-D{Z}^T{V}^{-1}ZD$. When nonsingular, its mean is also its mode. Estimated parameters yield empirical conditional modes and shrink group deviations toward zero. At a zero variance component the associated effect is degenerate at zero; the covariance formula still applies, whereas formulas involving $D^{-1}$ require limits.