Centered eight-schools runs show small R-hat with low tail ESS

The following are empirical simulation results reported by Vehtari et al. for the centered eight-schools funnel model; they are not theorem guarantees. In a 100-times-longer centered run, the reported $\widehat{R}$ for the hierarchical scale $\tau$ was $1.00$, with bulk ESS $1880$ and tail ESS $426$, while the sampler reported $9955$ divergent transitions. The source reports that the chains still had trouble exploring small $\tau$ values and that the estimates could not be trusted. In a separate centered run thinned by $200$, the reported $\widehat{R}$ was again $1.00$ and tail ESS was $1718$, but the posterior mean was more than three reported Monte Carlo standard errors from the non-centered reference. The source classifies the thinning result as an empirical failure of these finite diagnostics to reveal the sampling problem, not as a universal statement about thinning.

The first set of values is from the author-maintained online appendix, Appendix D, “Centered parameterization with very long chains.” The second is from “Centered parameterization with very long chains and thinning,” also in Appendix D. The main paper's §5.2 reports the related centered eight-schools simulation and its difficulties exploring small $\tau$. These observations illustrate why the reported $1.01$ and $400$ recommendations are not universal cutoffs and why source simulations must remain empirical evidence.