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, ...)
```

- x
(multiple options) One of:

A matrix of draws for a single variable (iterations x chains). See

`extract_variable_matrix()`

.An

`rvar`

.

- ...
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 https://avehtari.github.io/rhat_ess/ess_comparison.html

Other diagnostics:
`ess_basic()`

,
`ess_quantile()`

,
`ess_sd()`

,
`ess_tail()`

,
`mcse_mean()`

,
`mcse_quantile()`

,
`mcse_sd()`

,
`rhat_basic()`

,
`rhat()`

,
`rstar()`

```
mu <- extract_variable_matrix(example_draws(), "mu")
ess_bulk(mu)
#> [1] 558.0173
d <- as_draws_rvars(example_draws("multi_normal"))
ess_bulk(d$Sigma)
#> [,1] [,2] [,3]
#> [1,] 742.2907 454.0657 468.3890
#> [2,] 454.0657 528.7972 434.1141
#> [3,] 468.3890 434.1141 728.9440
```