I am fitting a multi-level state-space model and am running into a situation where the Gelman-Rubin diagnostic shows acceptable convergence (R-hat < 1.01), but when I look at the trace plots of the MCMC iterations, they show a considerable lack of convergence. An example for one parameter from my model is shown below, although I am getting similar results for all of my parameters. I am wondering if this is a problem that is already known, and if this should point me to something wrong with my model. Obviously, I know that I can try adding more iterations, but I am still curious as to why the G-R statistics indicates acceptable convergence in this situation.
Basically, the Gelman-Rubin diagnostic looks at how different your repeated chains are. They are extremely similar in your case, giving you a relatively small GR diagnostic. However, at the same time your chains are obviously not converged and you need a longer burn-in (warm-up) period until you get actually useful posterior densities.
When describing the Gelman-Rubin statistic (R hat) in Bayesian Data Analysis (3rd Ed, ch. 11.4, pp. 283-285), Gelman and his coauthors say that the multiple chains used in calculating R hat should be simulated with overdispersed starting points and further that each chain should be split at the middle into two parts.
For example, even if only two chains are simulated, the diagnostic would be computed across the four half chains.
If you make either of these changes (diffuse starting points and chain splitting) to your diagnostic procedure, I think you will find the Gelman-Rubin statistic would identify that your chains have failed to converge.