Diagnostics for Pareto smoothed importance sampling (PSIS)

Print a diagnostic table summarizing the estimated Pareto shape parameters and PSIS effective sample sizes, find the indexes of observations for which the estimated Pareto shape parameter \(k\) is larger than some threshold value, or plot observation indexes vs. diagnostic estimates. The Details section below provides a brief overview of the diagnostics, but we recommend consulting Vehtari, Gelman, and Gabry (2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2022) for full details.

pareto_k_table(x)

pareto_k_ids(x, threshold = NULL)

pareto_k_values(x)

pareto_k_influence_values(x)

psis_n_eff_values(x)

mcse_loo(x, threshold = NULL)

# S3 method for psis_loo
plot(
  x,
  diagnostic = c("k", "ESS", "n_eff"),
  ...,
  label_points = FALSE,
  main = "PSIS diagnostic plot"
)

# S3 method for psis
plot(
  x,
  diagnostic = c("k", "ESS", "n_eff"),
  ...,
  label_points = FALSE,
  main = "PSIS diagnostic plot"
)

Arguments

x: An object created by loo() or psis().
threshold: For pareto_k_ids(), threshold is the minimum \(k\) value to flag (default is a sample size S dependend threshold 1 - 1 / log10(S)). For mcse_loo(), if any \(k\) estimates are greater than threshold the MCSE estimate is returned as NA See Details for the motivation behind these defaults.
diagnostic: For the plot method, which diagnostic should be plotted? The options are "k" for Pareto \(k\) estimates (the default), or "ESS" or "n_eff" for PSIS effective sample size estimates.
label_points, ...: For the plot() method, if label_points is TRUE the observation numbers corresponding to any values of \(k\) greater than the diagnostic threshold will be displayed in the plot. Any arguments specified in ... will be passed to graphics::text() and can be used to control the appearance of the labels.
main: For the plot() method, a title for the plot.

Value

pareto_k_table() returns an object of class "pareto_k_table", which is a matrix with columns "Count", "Proportion", and "Min. n_eff", and has its own print method.

pareto_k_ids() returns an integer vector indicating which observations have Pareto \(k\) estimates above threshold.

pareto_k_values() returns a vector of the estimated Pareto \(k\) parameters. These represent the reliability of sampling.

pareto_k_influence_values() returns a vector of the estimated Pareto \(k\) parameters. These represent influence of the observations on the model posterior distribution.

psis_n_eff_values() returns a vector of the estimated PSIS effective sample sizes.

mcse_loo() returns the Monte Carlo standard error (MCSE) estimate for PSIS-LOO. MCSE will be NA if any Pareto \(k\) values are above threshold.

The plot() method is called for its side effect and does not return anything. If x is the result of a call to loo()

or psis() then plot(x, diagnostic) produces a plot of the estimates of the Pareto shape parameters (diagnostic = "k") or estimates of the PSIS effective sample sizes (diagnostic = "ESS").

Details

The reliability and approximate convergence rate of the PSIS-based estimates can be assessed using the estimates for the shape parameter \(k\) of the generalized Pareto distribution. The diagnostic threshold for Pareto \(k\) depends on sample size \(S\) (sample size dependent threshold was introduced by Vehtari et al. (2022), and before that fixed thresholds of 0.5 and 0.7 were recommended). For simplicity, loo package uses the nominal sample size \(S\) when computing the sample size specific threshold. This provides an optimistic threshold if the effective sample size is less than 2200, but if MCMC-ESS > S/2 the difference is usually negligible. Thinning of MCMC draws can be used to improve the ratio ESS/S.

If \(k < min(1 - 1 / log10(S), 0.7)\), where \(S\) is the sample size, the PSIS estimate and the corresponding Monte Carlo standard error estimate are reliable.
If \(1 - 1 / log10(S) <= k < 0.7\), the PSIS estimate and the corresponding Monte Carlo standard error estimate are not reliable, but increasing the (effective) sample size \(S\) above 2200 may help (this will increase the sample size specific threshold \((1-1/log10(2200)>0.7\) and then the bias specific threshold 0.7 dominates).
If \(0.7 <= k < 1\), the PSIS estimate and the corresponding Monte Carlo standard error have large bias and are not reliable. Increasing the sample size may reduce the uncertainty in the \(k\) estimate.
If \(0.7 <= k < 1\), the PSIS estimate and the corresponding Monte Carlo standard error have large bias and are not reliable. Increasing the sample size may reduce the variability in \(k\) estimate, which may result in lower \(k\) estimate, too.
If \(k \geq 1\), the target distribution is estimated to have a non-finite mean. The PSIS estimate and the corresponding Monte Carlo standard error are not well defined. Increasing the sample size may reduce the variability in the \(k\) estimate, which may also result in a lower \(k\) estimate.

What if the estimated tail shape parameter \(k\) exceeds the diagnostic threshold?

Importance sampling is likely to work less well if the marginal posterior \(p(\theta^s | y)\) and LOO posterior \(p(\theta^s | y_{-i})\) are very different, which is more likely to happen with a non-robust model and highly influential observations. If the estimated tail shape parameter \(k\) exceeds the diagnostic threshold, the user should be warned. (Note: If \(k\) is greater than the diagnostic threshold then WAIC is also likely to fail, but WAIC lacks as accurate diagnostic.) When using PSIS in the context of approximate LOO-CV, we recommend one of the following actions:

With some additional computations, it is possible to transform the MCMC draws from the posterior distribution to obtain more reliable importance sampling estimates. This results in a smaller shape parameter \(k\). See loo_moment_match() and the vignette Avoiding model refits in leave-one-out cross-validation with moment matching for an example of this.
Sampling from a leave-one-out mixture distribution (see the vignette Mixture IS leave-one-out cross-validation for high-dimensional Bayesian models), directly from \(p(\theta^s | y_{-i})\) for the problematic observations \(i\), or using \(K\)-fold cross-validation (see the vignette Holdout validation and K-fold cross-validation of Stan programs with the loo package) will generally be more stable.
Using a model that is more robust to anomalous observations will generally make approximate LOO-CV more stable.

Observation influence statistics

The estimated shape parameter \(k\) for each observation can be used as a measure of the observation's influence on posterior distribution of the model. These can be obtained with pareto_k_influence_values().

Effective sample size and error estimates

In the case that we obtain the samples from the proposal distribution via MCMC the loo package also computes estimates for the Monte Carlo error and the effective sample size for importance sampling, which are more accurate for PSIS than for IS and TIS (see Vehtari et al (2022) for details). However, the PSIS effective sample size estimate will be over-optimistic when the estimate of \(k\) is greater than \(min(1-1/log10(S), 0.7)\), where \(S\) is the sample size.

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. (2022). Pareto smoothed importance sampling. preprint arXiv:1507.02646