Compute the nested Rhat convergence diagnostic for a single variable as proposed in Margossian et al. (2023).

```
rhat_nested(x, ...)
# S3 method for default
rhat_nested(x, superchain_ids, ...)
# S3 method for rvar
rhat_nested(x, superchain_ids, ...)
```

- 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).

- superchain_ids
(numeric) Vector of length nchains specifying which superchain each chain belongs to. There should be equal numbers of chains in each superchain. All chains within the same superchain are assumed to have been initialized at the same point.

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.

Nested Rhat is a convergence diagnostic useful when running many short chains. It is calculated on superchains, which are groups of chains that have been initialized at the same point.

Note that there is a slight difference in the calculation of Rhat and nested Rhat, as nested Rhat is lower bounded by 1. This means that nested Rhat with one chain per superchain will not be exactly equal to basic Rhat (see Footnote 1 in Margossian et al. (2023)).

Charles C. Margossian, Matthew D. Hoffman, Pavel Sountsov, Lionel Riou-Durand, Aki Vehtari and Andrew Gelman (2023). Nested R-hat: Assessing the convergence of Markov chain Monte Carlo when running many short chains. arxiv:arXiv:2110.13017 (version 4)

Other diagnostics:
`ess_basic()`

,
`ess_bulk()`

,
`ess_quantile()`

,
`ess_sd()`

,
`ess_tail()`

,
`mcse_mean()`

,
`mcse_quantile()`

,
`mcse_sd()`

,
`pareto_diags()`

,
`pareto_khat()`

,
`rhat()`

,
`rhat_basic()`

,
`rstar()`

```
mu <- extract_variable_matrix(example_draws(), "mu")
rhat_nested(mu, superchain_ids = c(1, 1, 2, 2))
#> [1] 1.002846
d <- as_draws_rvars(example_draws("multi_normal"))
rhat_nested(d$Sigma, superchain_ids = c(1, 1, 2, 2))
#> [,1] [,2] [,3]
#> [1,] 1.000511 1.001590 1.000154
#> [2,] 1.001590 1.001037 1.000071
#> [3,] 1.000154 1.000071 1.000469
```