Hi all,
I used the Bayesian probit mixed effect model, and I noticed that the posterior samples for the intercepts are highly correlated, as reflected by relatively low ESS (800 from 4000 post-warmup samples) :
rating ~ 1 +(1|subject) + (1| item)
Is there something I can do in order to account for this relationship, to enhance the efficiency of the brms sampler?
Thanks !
Hi, @Shai_Shachar: This is just an intrinsic problem with the form of the model. You have three intercepts, two of which vary by subject and item. Let’s say that \alpha is the intercept, \beta_i is the effect for subject i and \gamma_j is the effect for item j, then you can see how they are perfectly collinear in the abstract case—If I add \epsilon to \alpha, I can subtract it from all the \beta_i and get the same linear predictor.
I’m not entirely sure what brms adds in the way of priors to mitigate this. Even if you identify it by either setting \beta_1 = \gamma_1 = 0, or you identify by setting \textrm{sum}(\beta) = 0 and \textrm{sum}(\gamma) = 0, you still wind up with residual correlation. You can also just identify by imposing a prior on all the values, which if you make it the same for all the parameters, will tend to push high values into the global intercept because there’s a bigger penalty for adding them to the subject or item effects.
Second, an effective sample size of 800 from 4000 post-warmup draws should be fine. That should be more than enough for inference. We generally recommend making sure the ESS is at least 50 or 100 per chain (lower than that, the ESS estimators become unstable) and R-hat values are all near 1.