Thanks Ben for your interest in my question.
I had a bit more of a think and realised that I didn’t want/need to estimate the SD of alphaj - they are “fixed effects”. So I removed sigma_issue altogether.
I still received the E-BFMI warnings with a similar sigma_eta_unif vs energy__ plot, but saw comments elsewhere on these forums that half-Cauchys aren’t so well supported as SD priors. Thinking about it more, I agreed that these put too much weight on quite extreme values in my situation, so I changed it to a half-normal prior.
I still have quite a strong apparent correlation between sigma_eta and energy__ appearing in the plot, but now E-BFMI is up around 0.35-0.4 and I do not get any warnings. I’ll consider more carefully whether half-normal is really what I want, but for now it seems to have avoided the issue.
xissues is (like xfam and xfull) just a ‘design matrix’ of 0s and 1s to help me match up the observations to their parameters; its dimensions are n_obs * n_issues. The model doesn’t include any real covariates yet, but I plan to add them in the future, which is why I set it up like this.
I don’t have a centered version of the code, but this might make clearer what model I intended to fit
\mathrm{logit}(p_{ij}) = \alpha_0 + \alpha_j + \gamma_i + \eta_{ij}, \qquad i = 1, \dotsc, n, \; j = 1, \dotsc, J
with \alpha_0 \sim \mathrm{Logistic}(0,1), \alpha_j \sim N(0,3), \gamma_i \sim N(0,\sigma_\gamma), \sigma_\gamma \sim \text{Half-}N(0,1), \eta_{i} \sim MVN_J(0,\sigma_\eta \Omega^{1/2}), \sigma_\eta \sim \text{Half-}N(0,1), \Omega \sim LKJ(2)
The main idea being that the J responses within each i are correlated in some way, and (along with the \alpha_j s), that correlation matrix \Omega is of great interest.