Convergence and efficiency diagnostics for Markov Chains
These functions are improved versions of the traditional Rhat (for convergence) and Effective Sample Size (for efficiency).
Rhat(sims) ess_bulk(sims) ess_tail(sims)
sims |
A two-dimensional array whose rows are equal to the number of iterations of the Markov Chain(s) and whose columns are equal to the number of Markov Chains (preferably more than one). The cells are the realized draws for a particular parameter or function of parameters. |
The Rhat function produces R-hat convergence diagnostic, which
compares the between- and within-chain estimates for model parameters
and other univariate quantities of interest. If chains have not mixed
well (ie, the between- and within-chain estimates don't agree), R-hat is
larger than 1. We recommend running at least four chains by default and
only using the sample if R-hat is less than 1.05. Stan reports R-hat
which is the maximum of rank normalized split-R-hat and rank normalized
folded-split-R-hat, which works for thick tailed distributions and is
sensitive also to differences in scale.
The ess_bulk function produces an estimated Bulk Effective Sample
Size (bulk-ESS) using rank normalized draws. Bulk-ESS is useful measure
for sampling efficiency in the bulk of the distribution (related e.g. to
efficiency of mean and median estimates), and is well defined even if
the chains do not have finite mean or variance.
The ess_tail function produces an estimated Tail Effective Sample
Size (tail-ESS) by computing the minimum of effective sample sizes for
5% and 95% quantiles. Tail-ESS is useful measure for sampling
efficiency in the tails of the distribution (related e.g. to efficiency
of variance and tail quantile estimates).
Both bulk-ESS and tail-ESS should be at least 100 (approximately) per Markov Chain in order to be reliable and indicate that estimates of respective posterior quantiles are reliable.
Paul-Christian Burkner and Aki Vehtari
Aki Vehtari, Andrew Gelman, Daniel Simpson, Bob Carpenter, and
Paul-Christian Bürkner (2019). Rank-normalization, folding, and
localization: An improved R-hat for assessing convergence of MCMC.
arXiv preprint arXiv:1903.08008.
# pretend these draws came from five actual Markov Chins sims <- matrix(rnorm(500), nrow = 100, ncol = 5) Rhat(sims) ess_bulk(sims) ess_tail(sims)
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