Here are just some simple notes about these 2 equivalent models

MME for GBLUP

You can understand GBLUP is a “improved” version of traditional PBLUP

\[\left[ \begin{array}{cc} \mathbf{X'R}^{-1}\mathbf{X} & \mathbf{X'R}^{-1}\mathbf{Z}\\ \mathbf{Z'R}^{-1}\mathbf{X} & \mathbf{Z'R}^{-1}\mathbf{Z}+(\mathbf{G}\sigma^2_a)^{-1} \end{array}\right] \left[ \begin{array}{c} \hat{\mathbf{b}}\\ \hat{\mathbf{u}} \end{array}\right] = \left[ \begin{array}{c} \mathbf{X'R}^{-1}\mathbf{y}\\ \mathbf{Z'R}^{-1}\mathbf{y} \end{array}\right]\] \[\left[ \begin{array}{cc} \mathbf{X'}\mathbf{X} & \mathbf{X'}\mathbf{Z}\\ \mathbf{Z'}\mathbf{X} & \mathbf{Z'}\mathbf{Z}+\mathbf{G}^{-1}\lambda \end{array}\right] \left[ \begin{array}{c} \hat{\mathbf{b}}\\ \hat{\mathbf{u}} \end{array}\right] = \left[ \begin{array}{c} \mathbf{X'}\mathbf{y}\\ \mathbf{Z'}\mathbf{y} \end{array}\right]\]

where $\sigma^2_a$ is the total genetic variance, \(\lambda=\sigma^2_e/\sigma^2_a\)
\(\mathbf{u}\;\sim\;N(\mathbf{0},\mathbf{G}\sigma^2_a)\)

Model for SNP-BLUP

The marker effects $\mathbf{g}_i$ were assumed to be identically and independently distributed

\[\mathbf{y=1\mu + Mg + e}\]

where
\(\mathbf{g}\;\sim\;N(\mathbf{0},\mathbf{I}\sigma^2_g)\)
\(\mathbf{e}\;\sim\;N(\mathbf{0},\mathbf{D}\sigma^2_e)\)

where
\(d_{ii} = 1/\omega_i\)
\(\omega_i = EDC_i/\lambda\)

MME

\[\left[ \begin{array}{cc} \mathbf{X'R}^{-1}\mathbf{X} & \mathbf{X'R}^{-1}\mathbf{Z}\\ \mathbf{Z'R}^{-1}\mathbf{X} & \mathbf{Z'R}^{-1}\mathbf{Z}+(\mathbf{I}\sigma^2_g)^{-1} \end{array}\right] \left[ \begin{array}{c} \hat{\mathbf{b}}\\ \hat{\mathbf{a}} \end{array}\right] = \left[ \begin{array}{c} \mathbf{X'R}^{-1}\mathbf{y}\\ \mathbf{Z'R}^{-1}\mathbf{y} \end{array}\right]\] \[\left[ \begin{array}{cc} \mathbf{X'}\mathbf{X} & \mathbf{X'}\mathbf{Z}\\ \mathbf{Z'}\mathbf{X} & \mathbf{Z'}\mathbf{Z}+\mathbf{I}\lambda \end{array}\right] \left[ \begin{array}{c} \hat{\mathbf{b}}\\ \hat{\mathbf{a}} \end{array}\right] = \left[ \begin{array}{c} \mathbf{X'}\mathbf{y}\\ \mathbf{Z'}\mathbf{y} \end{array}\right]\]

where
\(\sigma^2_g\) is the SNP variance for each SNP
\(\lambda=\sigma^2_e/\sigma^2_g\)
\(\sigma^2_a=2\sum_{i=1}^mp_iq_i\sigma^2_g\)

The DGV

\[\mathbf{\hat{a} = 1\hat{\mu} + M\hat{g}}\]