I am wondering what is a correct/good way to specify the equations for a linear Bayesian regression model with both population- and group-level effects, fitted in brms, defining all relevant parameters that are estimated (and reported in summary).

The model is basically fitted by:

```
brm(formula = y ~ b1 + b2 + b3 + (1|subject), family = 'lognormal')
```

Here is how I would define it in equation form:

y_i \sim \mathrm{Lognormal} (\mu_i, \sigma_\mathrm{dist})\\ \mu_i = \mathbf{X}_i \mathbf{\beta} + \mathbf{Z}_i \mathbf{u}\\ \mathbf{u} \sim \mathcal{N} (\mathbf{0}, \sigma_\mathrm{subject}^2\mathbf{I})

where \beta = [\beta_0, \beta_1, \beta_2, \beta_3]^T are the population-level parameters, and \mathbf{u} the group-level parameters that are sampled from a multivariate, zero-centered, normal distribution with covariance \sigma_\mathrm{subject}^2\mathbf{I}.

Is this a correct/good definition? Specifically, does the \sigma_\mathrm{subject} correspond to the group-level effect estimate that is reported in the brms summary output as `sd(Intercept)`

?

- Operating System: Ubuntu 16.04
- brms Version: 2.6.0