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

The loo() methods for arrays, matrices, and functions compute PSIS-LOO CV, efficient approximate leave-one-out (LOO) cross-validation for Bayesian models using Pareto smoothed importance sampling (PSIS). This is an implementation of the methods described in Vehtari, Gelman, and Gabry (2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2022).

The loo_i() function enables testing log-likelihood functions for use with the loo.function() method.

loo(x, ...)

# S3 method for array
loo(
  x,
  ...,
  r_eff = 1,
  save_psis = FALSE,
  cores = getOption("mc.cores", 1),
  is_method = c("psis", "tis", "sis")
)

# S3 method for matrix
loo(
  x,
  ...,
  r_eff = 1,
  save_psis = FALSE,
  cores = getOption("mc.cores", 1),
  is_method = c("psis", "tis", "sis")
)

# S3 method for `function`
loo(
  x,
  ...,
  data = NULL,
  draws = NULL,
  r_eff = 1,
  save_psis = FALSE,
  cores = getOption("mc.cores", 1),
  is_method = c("psis", "tis", "sis")
)

loo_i(i, llfun, ..., data = NULL, draws = NULL, r_eff = 1, is_method = "psis")

is.loo(x)

is.psis_loo(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.

r_eff

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-normalized 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 can be over-optimistic. If the posterior draws are (near) independent then r_eff=1 can be used. r_eff has to be a scalar (same value is used for all observations) or a vector with length equal to the number of observations. The default value is 1. See the relative_eff() helper functions for help computing r_eff.

save_psis

Should the psis object created internally by loo() be saved in the returned object? The loo() function calls psis() internally but by default discards the (potentially large) psis object after using it to compute the LOO-CV summaries. Setting save_psis=TRUE will add a psis_object component to the list returned by loo. This is useful if you plan to use the E_loo() function to compute weighted expectations after running loo. Several functions in the bayesplot package also accept psis objects.

cores

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).

is_method

The importance sampling method to use. The following methods are implemented:

"psis": Pareto-Smoothed Importance Sampling (PSIS). Default method.
"tis": Truncated Importance Sampling (TIS) with truncation at sqrt(S), where S is the number of posterior draws.
"sis": Standard Importance Sampling (SIS).

data, draws, ...

For the loo.function() method and the loo_i() function, these are the data, posterior draws, and other arguments to pass to the log-likelihood function. See the Methods (by class) section below for details on how to specify these arguments.

i

For loo_i(), an integer in 1:N.

llfun

For loo_i(), the same as x for the loo.function() method. A log-likelihood function as described in the Methods (by class) section.

Value

The loo() methods return a named list with class c("psis_loo", "loo") and components:

estimates

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

pointwise

A matrix with five columns (and number of rows equal to the number of observations) containing the pointwise contributions of the measures (elpd_loo, mcse_elpd_loo, p_loo, looic, influence_pareto_k). in addition to the three measures in estimates, we also report pointwise values of the Monte Carlo standard error of elpd_loo (mcse_elpd_loo), and statistics describing the influence of each observation on the posterior distribution (influence_pareto_k). These are the estimates of the shape parameter \(k\) of the generalized Pareto fit to the importance ratios for each leave-one-out distribution (see the pareto-k-diagnostic page for details).

diagnostics

A named list containing two vectors:

pareto_k: Importance sampling reliability diagnostics. By default, these are equal to the influence_pareto_k in pointwise. Some algorithms can improve importance sampling reliability and modify these diagnostics. See the pareto-k-diagnostic page for details.
n_eff: PSIS effective sample size estimates.

psis_object

This component will be NULL unless the save_psis argument is set to TRUE when calling loo(). In that case psis_object will be the object of class "psis" that is created when the loo() function calls psis() internally to do the PSIS procedure.

The loo_i() function returns a named list with components pointwise and diagnostics. These components have the same structure as the pointwise and diagnostics components of the object returned by loo() except they contain results for only a single observation.

Details

The loo() function is an S3 generic and methods are provided for 3-D pointwise log-likelihood arrays, pointwise log-likelihood matrices, and log-likelihood functions. The array and matrix methods are the most convenient, but for models fit to very large datasets the loo.function() method is more memory efficient and may be preferable.

Methods (by class)

loo(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.
loo(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.
loo(`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.

Defining `loo()` methods in a package

Package developers can define loo() methods for fitted models objects. See the example loo.stanfit() method in the Examples section below for an example of defining a method that calls loo.array(). The loo.stanreg() method in the rstanarm package is an example of defining a method that calls loo.function().

References

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. (2022). Pareto smoothed importance sampling. preprint arXiv:1507.02646

Examples

### Array and matrix methods (using example objects included with loo package)
# Array method
LLarr <- example_loglik_array()
rel_n_eff <- relative_eff(exp(LLarr))
loo(LLarr, r_eff = rel_n_eff, cores = 2)
#> 
#> Computed from 1000 by 32 log-likelihood matrix.
#> 
#>          Estimate  SE
#> elpd_loo    -83.6 4.3
#> p_loo         3.3 1.2
#> looic       167.2 8.6
#> ------
#> MCSE of elpd_loo is 0.1.
#> MCSE and ESS estimates assume MCMC draws (r_eff in [0.6, 1.0]).
#> 
#> All Pareto k estimates are good (k < 0.67).
#> See help('pareto-k-diagnostic') for details.

# Matrix method
LLmat <- example_loglik_matrix()
rel_n_eff <- relative_eff(exp(LLmat), chain_id = rep(1:2, each = 500))
loo(LLmat, r_eff = rel_n_eff, cores = 2)
#> 
#> Computed from 1000 by 32 log-likelihood matrix.
#> 
#>          Estimate  SE
#> elpd_loo    -83.6 4.3
#> p_loo         3.3 1.2
#> looic       167.2 8.6
#> ------
#> MCSE of elpd_loo is 0.1.
#> MCSE and ESS estimates assume MCMC draws (r_eff in [0.6, 1.0]).
#> 
#> All Pareto k estimates are good (k < 0.67).
#> See help('pareto-k-diagnostic') for details.


### Using log-likelihood function instead of array or matrix
set.seed(124)

# Simulate data and draw from posterior
N <- 50; K <- 10; S <- 100; a0 <- 3; b0 <- 2
p <- rbeta(1, a0, b0)
y <- rbinom(N, size = K, prob = p)
a <- a0 + sum(y); b <- b0 + N * K - sum(y)
fake_posterior <- as.matrix(rbeta(S, a, b))
dim(fake_posterior) # S x 1
#> [1] 100   1
fake_data <- data.frame(y,K)
dim(fake_data) # N x 2
#> [1] 50  2

llfun <- function(data_i, draws) {
  # each time called internally within loo the arguments will be equal to:
  # data_i: ith row of fake_data (fake_data[i,, drop=FALSE])
  # draws: entire fake_posterior matrix
  dbinom(data_i$y, size = data_i$K, prob = draws, log = TRUE)
}

# Use the loo_i function to check that llfun works on a single observation
# before running on all obs. For example, using the 3rd obs in the data:
loo_3 <- loo_i(i = 3, llfun = llfun, data = fake_data, draws = fake_posterior)
print(loo_3$pointwise[, "elpd_loo"])
#>  elpd_loo 
#> -1.267103 

# Use loo.function method (default r_eff=1 is used as this posterior not obtained via MCMC)
loo_with_fn <- loo(llfun, draws = fake_posterior, data = fake_data)

# If we look at the elpd_loo contribution from the 3rd obs it should be the
# same as what we got above with the loo_i function and i=3:
print(loo_with_fn$pointwise[3, "elpd_loo"])
#>  elpd_loo 
#> -1.267103 
print(loo_3$pointwise[, "elpd_loo"])
#>  elpd_loo 
#> -1.267103 

# Check that the loo.matrix method gives same answer as loo.function method
log_lik_matrix <- sapply(1:N, function(i) {
  llfun(data_i = fake_data[i,, drop=FALSE], draws = fake_posterior)
})
loo_with_mat <- loo(log_lik_matrix)
all.equal(loo_with_mat$estimates, loo_with_fn$estimates) # should be TRUE!
#> [1] TRUE


# \dontrun{
### For package developers: defining loo methods

# An example of a possible loo method for 'stanfit' objects (rstan package).
# A similar method is included in the rstan package.
# In order for users to be able to call loo(stanfit) instead of
# loo.stanfit(stanfit) the NAMESPACE needs to be handled appropriately
# (roxygen2 and devtools packages are good for that).
#
loo.stanfit <-
 function(x,
         pars = "log_lik",
         ...,
         save_psis = FALSE,
         cores = getOption("mc.cores", 1)) {
  stopifnot(length(pars) == 1L)
  LLarray <- loo::extract_log_lik(stanfit = x,
                                  parameter_name = pars,
                                  merge_chains = FALSE)
  r_eff <- loo::relative_eff(x = exp(LLarray), cores = cores)
  loo::loo.array(LLarray,
                 r_eff = r_eff,
                 cores = cores,
                 save_psis = save_psis)
}
# }