I am simulating a hierarchical model with MCMC Bayesian methods. The model has three groups of individual effects modelled as random effects drawn from normal priors with mean zero and variances estimated from the data. The priors for variances are inverse gamma with different hyperparameters to assess sensitivity. The posterior distributions of random effects are not showing convergence properly, but the 3 chains run in parallel from different starting values move towards close points and then stay there. This is an example where burnin is kept:
Since "fixed" parameters in the model are converging properly, my question is to what extend this behavior of random effects can be due to a poor prior for their variance. I understand that Gelman (2006) advises against the inverse gamma prior, because IG(0.001,0.001) would results in improper posteriors, while other hyperparameters would be informative and produce results sensitive to specific values. My attempt with informative inverse gamma priors is a starting point to make the model work and then to see whether we can move to non-informative priors. Hence I would have expected convergence, even if with posteriors sensitive to IG hyperparameters.
Can the lack of convergence for random effects as in the trace plot above be due to a poor informative prior on variances?