Using hierarchical prior with dummies

When using categorical variables, one must omit a level from each in order to avoid overparameterizing the model. However, doing so while using a hierarchical prior means that whatever arbitrary level is left out will not be pooled with the other levels. What is the recommended way to proceed?

My intuition tells me that a weak prior on the intercept and relatively tight priors on the hierarchical variance would allow me to place a coefficient for each level, but I would appreciate more informed advice on the issue.

(I think a very similar issue is brought up in the reference manual when discussing the softmax function but it only says to use “a suitable” prior without explaining just what might one look like)

Apparently the ideal approach for this situation is to use index variables instead of dummies.

This lecture by Dr. McElreath introduces the concept nicely.