The standard rank-normalized versions from Vehtari et al. (2021), computed
on the split chains. split_rhat() is max(bulk-Rhat, tail-Rhat): the
classical Gelman-Rubin statistic run once on the rank-normalized draws
(bulk) and once on the rank-normalized draws folded around the median
(tail, which catches chains that agree in location but disagree in
spread – something bulk-Rhat alone can miss). bulk_ess() rank-normalizes
the same way. Implemented here rather than taken from posterior to
keep the dependency surface where it is; the test suite cross-checks
against posterior when that package happens to be installed.
References
Vehtari, A., Gelman, A., Simpson, D., Carpenter, B. and Burkner, P.-C. (2021). Rank-normalization, folding, and localization. Bayesian Analysis 16(2), 667-718. doi:10.1214/20-BA1221