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For a mixed model of the form:

$$Y = X\beta + Z u + \epsilon$$

I know it is usually assumed in the parametric approach that:

$u \sim N(0, D)$ and $\epsilon \sim N(0, \sigma^2I)$

Where $D$ is a block diagonal matrix with all blocks corresponding to the same grouping factor the same as one another.

However, in the Bayesian approach I have read that it is assumed that:

$u \mid \sigma^2_1,\sigma^2_2,...,\sigma^2_r \sim N(0,D)$

Where D is now diagonal - Not block diagonal! In this case D is often given as $\oplus_i \sigma_i^2 I$ where $I$ is the $\frac{q}{r} \times \frac{q}{r}$ identity matrix and $q$ is the number of elements in $u$.

I do not understand how these two "D" covariance matrices relate to one another/why this would be a representation of the same model - is anyone able to explain this?

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  • $\begingroup$ What does $r$ stand for? $\oplus_i \sigma_i^2 I$? $\endgroup$ – user158565 Aug 2 at 2:45

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