Efficient approximate leave-one-out cross-validation (LOO) using subsampling

loo_subsample(x, ...)

# S3 method for function
loo_subsample(
x,
...,
data = NULL,
draws = NULL,
observations = 400,
log_p = NULL,
log_g = NULL,
r_eff = NULL,
save_psis = FALSE,
cores = getOption("mc.cores", 1),
loo_approximation = "plpd",
loo_approximation_draws = NULL,
estimator = "diff_srs",
llhess = NULL
)

## Arguments

x A function. The Methods (by class) section, below, has detailed descriptions of how to specify the inputs. For loo_subsample.function(), these are the data, posterior draws, and other arguments to pass to the log-likelihood function. The subsample observations to use. The argument can take four (4) types of arguments: NULL to use all observations. The algorithm then just uses standard loo() or loo_approximate_posterior(). A single integer to specify the number of observations to be subsampled. A vector of integers to provide the indices used to subset the data. These observations need to be subsampled with the same scheme as given by the estimator argument. A psis_loo_ss object to use the same observations that were used in a previous call to loo_subsample(). Should be supplied only if approximate posterior draws are used. The default (NULL) indicates draws are from "true" posterior (i.e. using MCMC). If not NULL then they should be specified as described in loo_approximate_posterior(). Vector of relative effective sample size estimates for the likelihood (exp(log_lik)) of each observation. This is related to the relative efficiency of estimating the normalizing term in self-normalizing importance sampling when using posterior draws obtained with MCMC. If MCMC draws are used and r_eff is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates will be over-optimistic. If the posterior draws are independent then r_eff=1 and can be omitted. See the relative_eff() helper functions for computing r_eff. Should the "psis" object created internally by loo_subsample() be saved in the returned object? See loo() for details. The number of cores to use for parallelization. This defaults to the option mc.cores which can be set for an entire R session by options(mc.cores = NUMBER). The old option loo.cores is now deprecated but will be given precedence over mc.cores until loo.cores is removed in a future release. As of version 2.0.0 the default is now 1 core if mc.cores is not set, but we recommend using as many (or close to as many) cores as possible. Note for Windows 10 users: it is strongly recommended to avoid using the .Rprofile file to set mc.cores (using the cores argument or setting mc.cores interactively or in a script is fine). What type of approximation of the loo_i's should be used? The default is "plpd" (the log predictive density using the posterior expectation). There are six different methods implemented to approximate loo_i's (see the references for more details): "plpd": uses the lpd based on point estimates (i.e., $$p(y_i|\hat{\theta})$$). "lpd": uses the lpds (i,e., $$p(y_i|y)$$). "tis": uses truncated importance sampling to approximate PSIS-LOO. "waic": uses waic (i.e., $$p(y_i|y) - p_{waic}$$). "waic_grad_marginal": uses waic approximation using first order delta method and posterior marginal variances to approximate $$p_{waic}$$ (ie. $$p(y_i|\hat{\theta})$$-p_waic_grad_marginal). Requires gradient of likelihood function. "waic_grad": uses waic approximation using first order delta method and posterior covariance to approximate $$p_{waic}$$ (ie. $$p(y_i|\hat{\theta})$$-p_waic_grad). Requires gradient of likelihood function. "waic_hess": uses waic approximation using second order delta method and posterior covariance to approximate $$p_{waic}$$ (ie. $$p(y_i|\hat{\theta})$$-p_waic_grad). Requires gradient and Hessian of likelihood function. As point estimates of $$\hat{\theta}$$, the posterior expectations of the parameters are used. The number of posterior draws used when integrating over the posterior. This is used if loo_approximation is set to "lpd", "waic", or "tis". How should elpd_loo, p_loo and looic be estimated? The default is "diff_srs". "diff_srs": uses the difference estimator with simple random sampling (srs). p_loo is estimated using standard srs. "hh": uses the Hansen-Hurwitz estimator with sampling proportional to size, where abs of loo_approximation is used as size. "srs": uses simple random sampling and ordinary estimation. The gradient of the log-likelihood. This is only used when loo_approximation is "waic_grad", "waic_grad_marginal", or "waic_hess". The default is NULL. The hessian of the log-likelihood. This is only used with loo_approximation = "waic_hess". The default is NULL.

## Value

loo_subsample() returns a named list with class c("psis_loo_ss", "psis_loo", "loo"). This has the same structure as objects returned by loo() but with the additional slot:

• loo_subsampling: A list with two vectors, log_p and log_g, of the same length containing the posterior density and the approximation density for the individual draws.

## Details

The loo_subsample() function is an S3 generic and a methods is currently provided for log-likelihood functions. The implementation works for both MCMC and for posterior approximations where it is possible to compute the log density for the approximation.

## Methods (by class)

• 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

Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2019). Leave-One-Out Cross-Validation for Large Data. In International Conference on Machine Learning

Magnusson, M., Riis Andersen, M., Jonasson, J. and Vehtari, A. (2019). Leave-One-Out Cross-Validation for Model Comparison in Large Data.

loo(), psis(), loo_compare()