The waic() methods can be used to compute WAIC from the pointwise log-likelihood. However, we recommend LOO-CV using PSIS (as implemented by the loo() function) because PSIS provides useful diagnostics as well as effective sample size and Monte Carlo estimates.

waic(x, ...)

# S3 method for array
waic(x, ...)

# S3 method for matrix
waic(x, ...)

# S3 method for function
waic(x, ..., data = NULL, draws = NULL)

is.waic(x)

## Arguments

x

A log-likelihood array, matrix, or function. The Methods (by class) section, below, has detailed descriptions of how to specify the inputs for each method.

draws, data, ...

For the function method only. See the Methods (by class) section below for details on these arguments.

## Value

A named list (of class c("waic", "loo")) with components:

estimates

A matrix with two columns ("Estimate", "SE") and three rows ("elpd_waic", "p_waic", "waic"). This contains point estimates and standard errors of the expected log pointwise predictive density (elpd_waic), the effective number of parameters (p_waic) and the information criterion waic (which is just -2 * elpd_waic, i.e., converted to deviance scale).

pointwise

A matrix with three columns (and number of rows equal to the number of observations) containing the pointwise contributions of each of the above measures (elpd_waic, p_waic, waic).

## Methods (by class)

• waic(array): An $$I$$ by $$C$$ by $$N$$ array, where $$I$$ is the number of MCMC iterations per chain, $$C$$ is the number of chains, and $$N$$ is the number of data points.

• waic(matrix): An $$S$$ by $$N$$ matrix, where $$S$$ is the size of the posterior sample (with all chains merged) and $$N$$ is the number of data points.

• waic(function): A function f() that takes arguments data_i and draws and returns a vector containing the log-likelihood for a single observation i evaluated at each posterior draw. The function should be written such that, for each observation i in 1:N, evaluating

f(data_i = data[i,, drop=FALSE], draws = draws)

results in a vector of length S (size of posterior sample). The log-likelihood function can also have additional arguments but data_i and draws are required.

If using the function method then the arguments data and draws must also be specified in the call to loo():

• data: A data frame or matrix containing the data (e.g. observed outcome and predictors) needed to compute the pointwise log-likelihood. For each observation i, the ith row of data will be passed to the data_i argument of the log-likelihood function.

• draws: An object containing the posterior draws for any parameters needed to compute the pointwise log-likelihood. Unlike data, which is indexed by observation, for each observation the entire object draws will be passed to the draws argument of the log-likelihood function.

• The ... can be used if your log-likelihood function takes additional arguments. These arguments are used like the draws argument in that they are recycled for each observation.

## References

Watanabe, S. (2010). Asymptotic equivalence of Bayes cross validation and widely application information criterion in singular learning theory. Journal of Machine Learning Research 11, 3571-3594.

Vehtari, A., Gelman, A., and Gabry, J. (2017a). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4 (journal version, preprint arXiv:1507.04544).

Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. preprint arXiv:1507.02646

• The loo package vignettes and Vehtari, Gelman, and Gabry (2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2019) for more details on why we prefer loo() to waic().

• loo_compare() for comparing models on approximate LOO-CV or WAIC.

## Examples

### Array and matrix methods
LLarr <- example_loglik_array()
dim(LLarr)
#> [1] 500   2  32

LLmat <- example_loglik_matrix()
dim(LLmat)
#> [1] 1000   32

waic_arr <- waic(LLarr)
#> Warning:
#> 3 (9.4%) p_waic estimates greater than 0.4. We recommend trying loo instead.
waic_mat <- waic(LLmat)
#> Warning:
#> 3 (9.4%) p_waic estimates greater than 0.4. We recommend trying loo instead.
identical(waic_arr, waic_mat)
#> [1] TRUE

# \dontrun{
log_lik1 <- extract_log_lik(stanfit1)
log_lik2 <- extract_log_lik(stanfit2)