loo (version 2.5.1)

E_loo: Compute weighted expectations


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.


E_loo(x, psis_object, ...)

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

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



A numeric vector or matrix.


An object returned by psis().


Arguments passed to individual methods.


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


For computing quantiles, a vector of probabilities.


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 Pareto k diagnostics specific to E_loo().


A named list with the following components:


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 is "quantile" and multiple values are specified in probs the value component is a vector with length(probs) elements.


Function-specific diagnostic.

If log_ratios is not specified when calling E_loo(), pareto_k will be NULL. Otherwise, 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.


Run this code
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

# 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)

log_ratios <- -1 * log_lik(fit)

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

# To get Pareto k diagnostic with E_loo we also need to provide the negative
# log-likelihood values using the log_ratios argument.
E_loo(yrep, psis_object, type = "mean", log_ratios = log_ratios)
# }
# }

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