Note: VGG2017 refers to Vehtari, Gelman, and Gabry (2017). See References, below.

ELPD and elpd_loo

The ELPD is the theoretical expected log pointwise predictive density for a new dataset (Eq 1 in VGG2017), which can be estimated, e.g., using cross-validation. elpd_loo is the Bayesian LOO estimate of the expected log pointwise predictive density (Eq 4 in VGG2017) and is a sum of N individual pointwise log predictive densities. Probability densities can be smaller or larger than 1, and thus log predictive densities can be negative or positive. For simplicity the ELPD acronym is used also for expected log pointwise predictive probabilities for discrete models. Probabilities are always equal or less than 1, and thus log predictive probabilities are 0 or negative.

Standard error of elpd_loo

As elpd_loo is defined as the sum of N independent components (Eq 4 in VGG2017), we can compute the standard error by using the standard deviation of the N components and multiplying by sqrt(N) (Eq 23 in VGG2017). This standard error is a coarse description of our uncertainty about the predictive performance for unknown future data. When N is small or there is severe model misspecification, the current SE estimate is overoptimistic and the actual SE can even be twice as large. Even for moderate N, when the SE estimate is an accurate estimate for the scale, it ignores the skewness. When making model comparisons, the SE of the component-wise (pairwise) differences should be used instead (see the se_diff section below and Eq 24 in VGG2017).

Monte Carlo SE of elpd_loo

The Monte Carlo standard error is the estimate for the computational accuracy of MCMC and importance sampling used to compute elpd_loo. Usually this is negligible compared to the standard describing the uncertainty due to finite number of observations (Eq 23 in VGG2017).

p_loo (effective number of parameters)

p_loo is the difference between elpd_loo and the non-cross-validated log posterior predictive density. It describes how much more difficult it is to predict future data than the observed data. Asymptotically under certain regularity conditions, p_loo can be interpreted as the effective number of parameters. In well behaving cases p_loo < N and p_loo < p, where p is the total number of parameters in the model. p_loo > N or p_loo > p indicates that the model has very weak predictive capability and may indicate a severe model misspecification. See below for more on interpreting p_loo when there are warnings about high Pareto k diagnostic values.

Pareto k estimates

The Pareto k estimate is a diagnostic for Pareto smoothed importance sampling (PSIS), which is used to compute components of elpd_loo. In importance-sampling LOO (the full posterior distribution is used as the proposal distribution). The Pareto k diagnostic estimates how far an individual leave-one-out distribution is from the full distribution. If leaving out an observation changes the posterior too much then importance sampling is not able to give reliable estimate. If k<0.5, then the corresponding component of elpd_loo is estimated with high accuracy. If 0.5<k<0.7 the accuracy is lower, but still ok. If k>0.7, then importance sampling is not able to provide useful estimate for that component/observation. Pareto k is also useful as a measure of influence of an observation. Highly influential observations have high k values. Very high k values often indicate model misspecification, outliers or mistakes in data processing. See Section 6 of Gabry et al. (2019) for an example.

Interpreting p_loo when Pareto k is large

If k > 0.7 then we can also look at the p_loo estimate for some additional information about the problem:

  • If p_loo << p (the total number of parameters in the model), then the model is likely to be misspecified. Posterior predictive checks (PPCs) are then likely to also detect the problem. Try using an overdispersed model, or add more structural information (nonlinearity, mixture model, etc.).

  • If p_loo < p and the number of parameters p is relatively large compared to the number of observations (e.g., p>N/5), it is likely that the model is so flexible or the population prior so weak that it’s difficult to predict the left out observation (even for the true model). This happens, for example, in the simulated 8 schools (in VGG2017), random effect models with a few observations per random effect, and Gaussian processes and spatial models with short correlation lengths.

  • If p_loo > p, then the model is likely to be badly misspecified. If the number of parameters p<<N, then PPCs are also likely to detect the problem. See the case study at https://avehtari.github.io/modelselection/roaches.html for an example. If p is relatively large compared to the number of observations, say p>N/5 (more accurately we should count number of observations influencing each parameter as in hierarchical models some groups may have few observations and other groups many), it is possible that PPCs won't detect the problem.

elpd_diff

elpd_diff is the difference in elpd_loo for two models. If more than two models are compared, the difference is computed relative to the model with highest elpd_loo.

se_diff

The standard error of component-wise differences of elpd_loo (Eq 24 in VGG2017) between two models. This SE is smaller than the SE for individual models due to correlation (i.e., if some observations are easier and some more difficult to predict for all models).

References

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

Gabry, J. , Simpson, D. , Vehtari, A. , Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. J. R. Stat. Soc. A, 182: 389-402. doi:10.1111/rssa.12378 (journal version, preprint arXiv:1709.01449, code on GitHub)