# Bayesian estimation of mixed effects models covariance matrix

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?

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