Short summary of the problem:

As far as I know, the random effects of a multilevel multinomial logit regression follow a multivariate normal distribution with mean zero and a covariance matrix that looks like the following (image below). My main question concerns how the “categorical” family in ‘brms’ deals with the covariances.

Consider an intercept-only two-level multinomial logit regression for a categorical variable with five categories (k=1,2,3,4,5; corresponding to “none”, “low”, “some”, “high”, and “very high”), with the reference category being 3 (“some”).

Student *i* is nested in school *j*.

Here is a code that I ran

```
multinom_S <- brm(STRESS ~ 1+(1|SCHOOL), data=bigdat, family=categorical(link="logit", refcat = "some"),
prior=c(
set_prior("cauchy(0, 1)", class="sd", dpar="munone"),
set_prior("cauchy(0, 1)", class="sd", dpar="mulow"),
set_prior("cauchy(0, 1)", class="sd", dpar="muhigh"),
set_prior("cauchy(0, 1)", class="sd", dpar="muveryhigh")
),
cores=4, chains=4)
```

From get_prior(), I get the following output

I have two quetions here: First, where are the priors for the covariances? I can see that student_t(3, 0, 2.5) is set by default for all the intercepts and standard deviations, but there seems to be no prior for the covariances. Is zero covariance assumed, or am I understanding something wrong?

Also, I am not so sure why the halfCauchy priors I set do not appear in the get_priors command. Was there an error in how I specified the priors for the sd?

Thanks much in advance.

- Operating System: Windows 10
- brms Version: R3.6.3