Compute a bulk effective sample size estimate (bulk-ESS) for a single variable. Bulk-ESS is useful as a diagnostic for the sampling efficiency in the bulk of the posterior. It is defined as the effective sample size for rank normalized values using split chains. For the tail effective sample size see ess_tail(). See Vehtari (2021) for an in-depth comparison of different effective sample size estimators.

ess_bulk(x, ...)

# S3 method for default
ess_bulk(x, ...)

# S3 method for rvar
ess_bulk(x, ...)



(multiple options) One of:


Arguments passed to individual methods (if applicable).


If the input is an array, returns a single numeric value. If any of the draws is non-finite, that is, NA, NaN, Inf, or -Inf, the returned output will be (numeric) NA. Also, if all draws within any of the chains of a variable are the same (constant), the returned output will be (numeric) NA

as well. The reason for the latter is that, for constant draws, we cannot distinguish between variables that are supposed to be constant (e.g., a diagonal element of a correlation matrix is always 1) or variables that just happened to be constant because of a failure of convergence or other problems in the sampling process.

If the input is an rvar, returns an array of the same dimensions as the rvar, where each element is equal to the value that would be returned by passing the draws array for that element of the rvar to this function.


Aki Vehtari, Andrew Gelman, Daniel Simpson, Bob Carpenter, and Paul-Christian Bürkner (2021). Rank-normalization, folding, and localization: An improved R-hat for assessing convergence of MCMC (with discussion). Bayesian Data Analysis. 16(2), 667-–718. doi:10.1214/20-BA1221

Aki Vehtari (2021). Comparison of MCMC effective sample size estimators. Retrieved from

See also


mu <- extract_variable_matrix(example_draws(), "mu")
#> [1] 558.0173

d <- as_draws_rvars(example_draws("multi_normal"))
#>          [,1]     [,2]     [,3]
#> [1,] 742.2907 454.0657 468.3890
#> [2,] 454.0657 528.7972 434.1141
#> [3,] 468.3890 434.1141 728.9440