Widely applicable information criterion (WAIC)
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 class 'array' waic(x, ...) ## S3 method for class 'matrix' waic(x, ...) ## S3 method for class ''function'' waic(x, ..., data = NULL, draws = NULL) is.waic(x)
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. |
A named list (of class c("waic", "loo")) with components:
estimatesA 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).
pointwiseA 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).
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.
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.
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.
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.
### Array and matrix methods LLarr <- example_loglik_array() dim(LLarr) LLmat <- example_loglik_matrix() dim(LLmat) waic_arr <- waic(LLarr) waic_mat <- waic(LLmat) identical(waic_arr, waic_mat) ## Not run: log_lik1 <- extract_log_lik(stanfit1) log_lik2 <- extract_log_lik(stanfit2) (waic1 <- waic(log_lik1)) (waic2 <- waic(log_lik2)) print(compare(waic1, waic2), digits = 2) ## End(Not run)
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