Write mixed linear model as two level hierarchical model

Question

How do I show that this mixed linear model
$$textbf{Y}=textbf{X}beta+textbf{Zb+e},$$
where
${textbf{b}choose textbf{e}} sim Nleft({textbf{0} choose textbf{0}}, { textbf{D} 0 choose 0 Sigma}right)$
can be represented by a two-level hierarchical model
$$textbf{Y|b} sim N(textbf{X}beta+textbf{Zb},Sigma),$$
where $textbf{b} sim N(textbf{0},textbf{D}).$

jld · Answer

If you start with
$$
Ymid b simmathcal N(Xbeta + Zb, Sigma) \
b sim mathcal N(0, D)
$$
then you can use the tower property and law of total variance, along with the fact that convolutions of Gaussians are Gaussian, to work out the marginal distribution of $Y$. Once you do this, you'll have a Gaussian RV with some mean and covariance. Then compare that to the marginal distribution of $Xbeta + Zb + e$ and you should have your answer.

Write mixed linear model as two level hierarchical model

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