Understanding brms nested grouping term in multivariate response models

Interfaces brms
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2

Your model specification is almost correct. It’s missing the prior on the population-level outcome intercepts:

\delta_j \sim \operatorname{Student-t}(3, 0, 2.5)

I would also write:

y_{ij} \sim \operatorname{Bernoulli}(\operatorname{logit}^{-1}(\theta_{ij}))

to make it clear the \theta parameters are on the logit scale.

And I think you are correct that the random county intercepts (for a given outcome j) are not correlated. At the group level, correlation is between outcomes in the same county, not between counties on the same outcome.

Without group-level predictors (for example: grouping by geographic region, or median household income by county), what is the correlation structure across counties within outcome that you would like to model?

A term like (1 | outcome) is not very meaningful when there are only two outcomes. @jd_c explains why in this thread with gender as two-level grouping variable.

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