# pareto-k-diagnostic

##### 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 (2017a,
2017b) for full details.

##### Usage

`pareto_k_table(x)`pareto_k_ids(x, threshold = 0.5)

pareto_k_values(x)

psis_n_eff_values(x)

mcse_loo(x, threshold = 0.7)

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

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

##### Arguments

- x
- threshold
For

`pareto_k_ids`

,`threshold`

is the minimum \(k\) value to flag (default is 0.5). For`mcse_loo`

, if any \(k\) estimates are greater than`threshold`

the MCSE estimate is returned as`NA`

(default is 0.7).- diagnostic
For the

`plot`

method, which diagnostic should be plotted? The options are`"k"`

for Pareto \(k\) estimates (the default) 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 0.5 will be displayed in the plot. Any arguments specified in`...`

will be passed to`text`

and can be used to control the appearance of the labels.- main
For the

`plot`

method, a title for the plot.

##### 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:

If \(k < 0.5\) then the distribution of raw importance ratios has finite variance and the central limit theorem holds. However, as \(k\) approaches \(0.5\) the RMSE of plain importance sampling (IS) increases significantly while PSIS has lower RMSE.

If \(0.5 \leq k < 1\) then the variance of the raw importance ratios is infinite, but the mean exists. TIS and PSIS estimates have finite variance by accepting some bias. The convergence of the estimate is slower with increasing \(k\). If \(k\) is between 0.5 and approximately 0.7 then we observe practically useful convergence rates and Monte Carlo error estimates with PSIS (the bias of TIS increases faster than the bias of PSIS). If \(k > 0.7\) we observe impractical convergence rates and unreliable Monte Carlo error estimates.

If \(k \geq 1\) then neither the variance nor the mean of the raw importance ratios exists. The convergence rate is close to zero and bias can be large with practical sample sizes.

**If the estimated tail shape parameter \(k\) exceeds \(0.5\), the
user should be warned, although in practice we have observed good performance
for values of \(k\) up to 0.7.** (If \(k\) is greater than \(0.5\) then
WAIC is also likely to fail, but WAIC lacks its own diagnostic.)

If using PSIS in the context of approximate LOO-CV, even if the PSIS estimate has a finite variance the user should consider sampling directly from \(p(\theta^s | y_{-i})\) for any problematic observations \(i\), use \(K\)-fold cross-validation, or use a more robust model. 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 much different, which is more likely to happen with a non-robust model and highly influential observations. A robust model may reduce the sensitivity to highly influential observations.

### Effective sample size and error estimates

In the case that we obtain the samples from the proposal distribution via
MCMC we can also compute 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 (2017b) for details). However,
the PSIS effective sample size estimate will be **over-optimistic when
the estimate of \(k\) is greater than 0.7.**

We can also compute estimates for the Monte Carlo error and the effective
sample size for importance sampling. However, the PSIS effective sample size
estimate will be **over-optimistic when the estimate of \(k\) is
greater than 0.7**. In the case that we obtain the samples from the proposal
distribution via MCMC, we need to take into account also the relative
efficiency of MCMC sampling (see Vehtari et al (2017b) for details).
Following the notation in Stan, the PSIS effective sample size is denoted
here with \(n_{eff}\), instead of \(S_{eff}\) used by Vehtari et al
(2017b).

##### 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.

`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 = "n_eff"`

).

##### 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.
(published
version, arXiv preprint).

Vehtari, A., Gelman, A., and Gabry, J. (2017b). Pareto smoothed importance sampling. arXiv preprint: http://arxiv.org/abs/1507.02646/

##### See Also

`psis`

for the implementation of the PSIS algorithm.

*Documentation reproduced from package loo, version 2.0.0, License: GPL (>= 3)*