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 (2019).
The loo_i() function enables testing log-likelihood
functions for use with the loo.function() method.
loo(x, ...)
## S3 method for class 'array'
loo(
x,
...,
r_eff = NULL,
save_psis = FALSE,
cores = getOption("mc.cores", 1),
is_method = c("psis", "tis", "sis")
)
## S3 method for class 'matrix'
loo(
x,
...,
r_eff = NULL,
save_psis = FALSE,
cores = getOption("mc.cores", 1),
is_method = c("psis", "tis", "sis")
)
## S3 method for class ''function''
loo(
x,
...,
data = NULL,
draws = NULL,
r_eff = NULL,
save_psis = FALSE,
cores = getOption("mc.cores", 1),
is_method = c("psis", "tis", "sis")
)
loo_i(
i,
llfun,
...,
data = NULL,
draws = NULL,
r_eff = NULL,
is_method = "psis"
)
is.loo(x)
is.psis_loo(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. |
r_eff |
Vector of relative effective sample size estimates for the
likelihood ( |
save_psis |
Should the |
cores |
The number of cores to use for parallelization. This defaults to
the option
|
is_method |
The importance sampling method to use. The following methods are implemented: |
data, draws, ... |
For the |
i |
For |
llfun |
For |
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.
The loo() methods return a named list with class
c("psis_loo", "loo") and components:
estimatesA 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).
pointwiseA 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.
diagnosticsA 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_objectThis 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.
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.
loo() methods in a packagePackage 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().
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 for demonstrations.
psis() for the underlying Pareto Smoothed Importance Sampling (PSIS)
procedure used in the LOO-CV approximation.
pareto-k-diagnostic for convenience functions for looking at diagnostics.
loo_compare() for model comparison.
### 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)
# 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)
### 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
fake_data <- data.frame(y,K)
dim(fake_data) # N x 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, r_eff = NA)
print(loo_3$pointwise[, "elpd_loo"])
# Use loo.function method (setting r_eff=NA since this posterior not obtained via MCMC)
loo_with_fn <- loo(llfun, draws = fake_posterior, data = fake_data, r_eff = NA)
# If we look at the elpd_loo contribution from the 3rd obs it should be the
# same as what we got above with the loo_i function and i=3:
print(loo_with_fn$pointwise[3, "elpd_loo"])
print(loo_3$pointwise[, "elpd_loo"])
# Check that the loo.matrix method gives same answer as loo.function method
log_lik_matrix <- sapply(1:N, function(i) {
llfun(data_i = fake_data[i,, drop=FALSE], draws = fake_posterior)
})
loo_with_mat <- loo(log_lik_matrix, r_eff = NA)
all.equal(loo_with_mat$estimates, loo_with_fn$estimates) # should be TRUE!
## Not run:
### 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)
}
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