loo_compare: Model comparison

In loo: Efficient Leave-One-Out Cross-Validation and WAIC for Bayesian Models

Model comparison

Description

Compare fitted models based on ELPD.

Usage

loo_compare(x, ...)

## Default S3 method:
loo_compare(x, ...)

## S3 method for class 'compare.loo'
print(x, ..., digits = 1, p_worse = TRUE, simplify = TRUE)

## S3 method for class 'compare.loo_ss'
print(x, ..., digits = 1)

Arguments

x	An object of class "loo" or a list of such objects. If a list is used then the list names will be used as the model names in the output. See Examples.
...	Additional objects of class "loo", if not passed in as a single list.
digits	For the print method only, the number of digits to use when printing.
p_worse	For the print method only, should we include the normal approximation based probability of each model having worse performance than the best model? The default is TRUE.
simplify	For the print method only, should the output be simplified to only include the model names, ELPD differences, and (when p_worse = TRUE) diagnostic columns? The default is TRUE. Set to FALSE to also print the available estimate columns (pointwise ELPD, LOOIC/WAIC, and their standard errors).

Details

When comparing two fitted models, we can estimate the difference in their expected predictive accuracy by the difference in elpd_loo or elpd_waic (or multiplied by -2, if desired, to be on the deviance scale).

elpd_diff and se_diff

When using loo_compare(), the returned data frame will have one row per model and several columns of estimates. The values of elpd_diff and se_diff are computed by making pairwise comparisons between each model and the model with the largest ELPD (the model listed first). Therefore, the first elpd_diff value will always be 0 (i.e., the difference between the preferred model and itself) and the rest of the values will be negative.

To compute the standard error of the difference in ELPD — which should not be expected to equal the difference of the standard errors — we use a paired estimate to take advantage of the fact that the same set of N data points was used to fit both models. These calculations should be most useful when N is large, because then non-normality of the distribution is not such an issue when estimating the uncertainty in these sums. These standard errors, for all their flaws, should give a better sense of uncertainty than what is obtained using the current standard approach of comparing differences of deviances to a Chi-squared distribution, a practice derived for Gaussian linear models or asymptotically, and which only applies to nested models in any case.

p_worse, diag_diff, and diag_elpd

The values in the p_worse column show the probability of each model having worse ELPD than the best model. These probabilities are computed with a normal approximation using the values from elpd_diff and se_diff. Sivula et al. (2025) present the conditions when the normal approximation used for SE and se_diff is good, and the column diag_diff contains possible diagnostic messages:

• N < 100 (small data)

• ⁠|elpd_diff| < 4⁠ (models make similar predictions)

If either of these diagnostic messages is shown, the error distribution is skewed or thick tailed and the normal approximation based on elpd_diff and se_diff is not well calibrated. In that case, the probabilities p_worse are likely to be too large. However, elpd_diff and se_diff will still be indicative of the differences and uncertainties (for example, if ⁠|elpd_diff|⁠ is many times larger than se_diff the difference is quite certain). In addition, if the model is not well specificed and there are outliers, the error distribution can also be skewed or thick tailed and the normal approximation is not well calibrated. Possible model misspecification and outliers can be diagnosed with usual predictive checking methods.

The column diag_elpd shows the PSIS-LOO Pareto k diagnostic for the pointwise ELPD computations for each model. If ⁠K k_psis > 0.7⁠ is shown, where K is the number of high Pareto k values in the PSIS computation, then there may be significant bias in elpd_diff favoring models with a large number of high Pareto k values.

Warnings for many model comparisons

If more than 11 models are compared, we internally recompute the model differences using the median model by ELPD as the baseline model. We then estimate whether the differences in predictive performance are potentially due to chance as described by McLatchie and Vehtari (2023). This will flag a warning if it is deemed that there is a risk of over-fitting due to the selection process. In that case users are recommended to avoid model selection based on LOO-CV, and instead to favor model averaging/stacking or projection predictive inference.

Value

A data frame with class "compare.loo" that has its own print method. See the Details and Examples sections.

References

Vehtari, A., Gelman, A., and Gabry, J. (2017). 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. (2024). Pareto smoothed importance sampling. Journal of Machine Learning Research, 25(72):1-58. PDF

Sivula, T, Magnusson, M., Matamoros A. A., and Vehtari, A. (2025). Uncertainty in Bayesian leave-one-out cross-validation based model comparison. Bayesian Analysis. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/25-BA1569")}

McLatchie, Y., and Vehtari, A. (2024). Efficient estimation and correction of selection-induced bias with order statistics. Statistics and Computing. 34(132). \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11222-024-10442-4")}

See Also

• The FAQ page on the loo website for answers to frequently asked questions.

Examples

# very artificial example, just for demonstration!
LL <- example_loglik_array()
loo1 <- loo(LL)     # should be worst model when compared
loo2 <- loo(LL + 1) # should be second best model when compared
loo3 <- loo(LL + 2) # should be best model when compared

comp <- loo_compare(loo1, loo2, loo3)
print(comp, digits = 2)
print(comp, simplify = FALSE) # full table

# can use a list of objects with custom names
# the names will be used in the output
loo_compare(list("apple" = loo1, "banana" = loo2, "cherry" = loo3))

## Not run: 
# works for waic (and kfold) too
loo_compare(waic(LL), waic(LL - 10))

## End(Not run)

loo documentation built on July 24, 2026, 9:08 a.m.