Skip to contents

Expected log pointwise predictive density (elpd)

Source: R/pred_measure-builtin.R
measure_elpd.Rd

Computes the expected log pointwise predictive density (ELPD) as the sum of pointwise log predictive density contributions (\(\mathrm{lppd}_i\)), using ptw_log_pred_density(). ELPD is returned on the utility scale (higher is better), consistent with the sign convention used throughout this package.

Usage

measure_elpd(ylp, log_weights = NULL, pointwise = NULL)

Arguments

ylp

A numeric matrix or three-dimensional array of log predictive densities or probabilities:

  • Matrix (\(S \times n\)): \(S\) posterior draws (chains merged) by \(n\) observations.

  • Array (\(I \times C \times n\)): \(I\) MCMC iterations per chain, \(C\) chains, and \(n\) observations. Converted to an \(S \times n\) matrix internally.

log_weights

Optional numeric matrix of unnormalized log-importance weights with dimensions \(S \times n\). Weights are column-normalized before computing each per-observation contribution.

pointwise

Optional numeric vector of precomputed \(\mathrm{lppd}_i\) values. When supplied, ylp and log_weights are ignored.

Value

An object of class "measure": a list with:

estimates

A 1 \(\times\) 2 matrix. The row is named after the measure, the columns are Estimate and SE (standard error).

pointwise

An \(n \times 1\) matrix of per-observation values. The column is named after the measure.

extra

Optional. Auxiliary data for the standard error of a difference. Most measures do not set it.

Attributes measure (i.e., measure name) and dims (draws \(\times\) observations) are also set. Use print() for a readable summary.

Details

$$\mathrm{elpd} = \sum_{i=1}^{n} \mathrm{lppd}_i,$$ where each \(\mathrm{lppd}_i\) is computed by ptw_log_pred_density(). The standard error is \(\sqrt{n}\,\mathrm{sd}(\mathrm{lppd}_i)\).

See also

ptw_log_pred_density(), measure_mlpd(), measure_ic()

Examples

ylp <- matrix(log(c(0.2, 0.4, 0.3, 0.8)), nrow = 2)
measure_elpd(ylp)
#> 
#> Computed from 2 draws by 2 observations.
#> 
#>      Estimate   SE
#> elpd     -1.8  0.6

# With unnormalized importance weights (e.g., PSIS-LOO)
lw <- matrix(log(c(0.7, 0.3, 0.6, 0.4)), nrow = 2)
measure_elpd(ylp, log_weights = lw)
#> 
#> Computed from 2 draws by 2 observations.
#> 
#>      Estimate   SE
#> elpd     -2.0  0.7

# From a draws x chains x observations array
LLarr <- example_loglik_array()
measure_elpd(LLarr)
#> 
#> Computed from 1000 draws by 32 observations.
#> 
#>      Estimate    SE
#> elpd    -80.3   3.2