loo (version 2.7.0)

E_loo: Compute weighted expectations

Description

The E_loo() function computes weighted expectations (means, variances, quantiles) using the importance weights obtained from the PSIS smoothing procedure. The expectations estimated by the E_loo() function assume that the PSIS approximation is working well. A small Pareto k estimate is necessary, but not sufficient, for E_loo() to give reliable estimates. Additional diagnostic checks for gauging the reliability of the estimates are in development and will be added in a future release.

Usage

E_loo(x, psis_object, ...)
# S3 method for default
E_loo(
  x,
  psis_object,
  ...,
  type = c("mean", "variance", "sd", "quantile"),
  probs = NULL,
  log_ratios = NULL
)
# S3 method for matrix
E_loo(
  x,
  psis_object,
  ...,
  type = c("mean", "variance", "sd", "quantile"),
  probs = NULL,
  log_ratios = NULL
)

Value

A named list with the following components:

value

The result of the computation.

For the matrix method, value is a vector with ncol(x) elements, with one exception: when type="quantile" and multiple values are specified in probs the value component of the returned object is a length(probs) by ncol(x) matrix.

For the default/vector method the value component is scalar, with one exception: when type="quantile" and multiple values are specified in probs the value component is a vector with length(probs) elements.

pareto_k

Function-specific diagnostic.

For the matrix method it will be a vector of length ncol(x) containing estimates of the shape parameter \(k\) of the generalized Pareto distribution. For the default/vector method, the estimate is a scalar. If log_ratios is not specified when calling E_loo(), the smoothed log-weights are used to estimate Pareto-k's, which may produce optimistic estimates.

For type="mean", type="var", and type="sd", the returned Pareto-k is the maximum of the Pareto-k's for the left and right tail of \(hr\) and the right tail of \(r\), where \(r\) is the importance ratio and \(h=x\) for type="mean" and \(h=x^2\) for type="var" and type="sd". For type="quantile", the returned Pareto-k is the Pareto-k for the right tail of \(r\).

Arguments

x

A numeric vector or matrix.

psis_object

An object returned by psis().

...

Arguments passed to individual methods.

type

The type of expectation to compute. The options are "mean", "variance", "sd", and "quantile".

probs

For computing quantiles, a vector of probabilities.

log_ratios

Optionally, a vector or matrix (the same dimensions as x) of raw (not smoothed) log ratios. If working with log-likelihood values, the log ratios are the negative of those values. If log_ratios is specified we are able to compute more accurate Pareto k diagnostics specific to E_loo().

Examples

# \donttest{
if (requireNamespace("rstanarm", quietly = TRUE)) {
# Use rstanarm package to quickly fit a model and get both a log-likelihood
# matrix and draws from the posterior predictive distribution
library("rstanarm")

# data from help("lm")
ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14)
trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69)
d <- data.frame(
  weight = c(ctl, trt),
  group = gl(2, 10, 20, labels = c("Ctl","Trt"))
)
fit <- stan_glm(weight ~ group, data = d, refresh = 0)
yrep <- posterior_predict(fit)
dim(yrep)

log_ratios <- -1 * log_lik(fit)
dim(log_ratios)

r_eff <- relative_eff(exp(-log_ratios), chain_id = rep(1:4, each = 1000))
psis_object <- psis(log_ratios, r_eff = r_eff, cores = 2)

E_loo(yrep, psis_object, type = "mean")
E_loo(yrep, psis_object, type = "var")
E_loo(yrep, psis_object, type = "sd")
E_loo(yrep, psis_object, type = "quantile", probs = 0.5) # median
E_loo(yrep, psis_object, type = "quantile", probs = c(0.1, 0.9))

# We can get more accurate Pareto k diagnostic if we also provide
# the log_ratios argument
E_loo(yrep, psis_object, type = "mean", log_ratios = log_ratios)
}
# }