Application of Gaussian Elimination

Consider the situation where S2 is a “nuisance parameter” and interest is only on inference about S1.

$$\begin{align} \begin{bmatrix}A & B\\ C & D \end{bmatrix}\begin{bmatrix}S_{1}\\ S_{2} \end{bmatrix}=\begin{bmatrix}R_{1}\\ R_{2} \end{bmatrix} \end{align}$$

Elimination

Subtract second row premulitiple by $BD^{-1}$ from first row

$$\begin{align} \begin{bmatrix}A-BD^{-1}C & B-BD^{-1}D\\\ C & D \end{bmatrix}\begin{bmatrix}S_{1}\\ S_{2} \end{bmatrix}=\begin{bmatrix}R{1}-BD^{-1}R_{2}\\ R_{2} \end{bmatrix}\\ \begin{bmatrix}A-BD^{-1}C & 0\\ C & D \end{bmatrix}\begin{bmatrix}S_{1}\\ S_{2} \end{bmatrix}=\begin{bmatrix}R{1}-BD^{-1}R_{2}\\ R_{2} \end{bmatrix} \end{align}$$

=>

$$\begin{align} \begin{bmatrix}A-BD^{-1}C\\ \end{bmatrix} \begin{bmatrix}S_{1} \end{bmatrix}=\begin{bmatrix}R{1}-BD^{-1}R_{2} \end{bmatrix} \end{align}$$

The solution to equation (2) for S1 are identical to those from equation (1). Furthermore, the inverse elements of the left-hand-side of equation (2) are identical to the corresponding inverse elements of equation (1). Thus, suppose the equation is a mixed model equation, equation (2) can be used to draw samples from the posterior of S1.¹

¹. Fernando et al.: A class of Bayesian methods to combine large numbers of genotyped and non-genotyped animals for whole-genome analyses. Genetics Selection Evolution 2014 46:50. ↩