Leave-One-Out (LOO) predictive checks. See the **Plot Descriptions** section,
below, and Gabry et al. (2019)
for details.

```
ppc_loo_pit_overlay(
y,
yrep,
lw,
...,
pit = NULL,
samples = 100,
size = 0.25,
alpha = 0.7,
boundary_correction = TRUE,
grid_len = 512,
bw = "nrd0",
trim = FALSE,
adjust = 1,
kernel = "gaussian",
n_dens = 1024
)
ppc_loo_pit_data(
y,
yrep,
lw,
...,
pit = NULL,
samples = 100,
bw = "nrd0",
boundary_correction = TRUE,
grid_len = 512
)
ppc_loo_pit_qq(
y,
yrep,
lw,
pit,
compare = c("uniform", "normal"),
...,
size = 2,
alpha = 1
)
ppc_loo_pit(
y,
yrep,
lw,
pit,
compare = c("uniform", "normal"),
...,
size = 2,
alpha = 1
)
ppc_loo_intervals(
y,
yrep,
psis_object,
subset = NULL,
intervals = NULL,
...,
prob = 0.5,
prob_outer = 0.9,
alpha = 0.33,
size = 1,
fatten = 2.5,
linewidth = 1,
order = c("index", "median")
)
ppc_loo_ribbon(
y,
yrep,
lw,
psis_object,
subset = NULL,
intervals = NULL,
...,
prob = 0.5,
prob_outer = 0.9,
alpha = 0.33,
size = 0.25
)
```

- y
A vector of observations. See

**Details**.- yrep
An

`S`

by`N`

matrix of draws from the posterior (or prior) predictive distribution. The number of rows,`S`

, is the size of the posterior (or prior) sample used to generate`yrep`

. The number of columns,`N`

is the number of predicted observations (`length(y)`

). The columns of`yrep`

should be in the same order as the data points in`y`

for the plots to make sense. See the**Details**and**Plot Descriptions**sections for additional advice specific to particular plots.- lw
A matrix of (smoothed) log weights with the same dimensions as

`yrep`

. See`loo::psis()`

and the associated`weights()`

method as well as the**Examples**section, below.- ...
Currently unused.

- pit
For

`ppc_loo_pit_overlay()`

and`ppc_loo_pit_qq()`

, optionally a vector of precomputed PIT values that can be specified instead of`y`

,`yrep`

, and`lw`

(these are all ignored if`pit`

is specified). If not specified the PIT values are computed internally before plotting.- samples
For

`ppc_loo_pit_overlay()`

, the number of data sets (each the same size as`y`

) to simulate from the standard uniform distribution. The default is 100. The density estimate of each dataset is plotted as a thin line in the plot, with the density estimate of the LOO PITs overlaid as a thicker dark line.- alpha, size, fatten, linewidth
Arguments passed to code geoms to control plot aesthetics. For

`ppc_loo_pit_qq()`

and`ppc_loo_pit_overlay()`

,`size`

and`alpha`

are passed to`ggplot2::geom_point()`

and`ggplot2::geom_density()`

, respectively. For`ppc_loo_intervals()`

,`size`

`linewidth`

and`fatten`

are passed to`ggplot2::geom_pointrange()`

. For`ppc_loo_ribbon()`

,`alpha`

and`size`

are passed to`ggplot2::geom_ribbon()`

.- boundary_correction
For

`ppc_loo_pit_overlay()`

, when set to`TRUE`

(the default) the function will compute boundary corrected density values via convolution and a Gaussian filter, also known as the reflection method (Boneva et al., 1971). As a result, parameters controlling the standard kernel density estimation such as`adjust`

,`kernel`

and`n_dens`

are ignored. NOTE: The current implementation only works well for continuous observations.- grid_len
For

`ppc_loo_pit_overlay()`

, when`boundary_correction`

is set to`TRUE`

this parameter specifies the number of points used to generate the estimations. This is set to 512 by default.- bw, adjust, kernel, n_dens
Optional arguments passed to

`stats::density()`

to override default kernel density estimation parameters.`n_dens`

defaults to`1024`

.- trim
Passed to

`ggplot2::stat_density()`

.- compare
For

`ppc_loo_pit_qq()`

, a string that can be either`"uniform"`

or`"normal"`

. If`"uniform"`

(the default) the Q-Q plot compares computed PIT values to the standard uniform distribution. If`compare="normal"`

, the Q-Q plot compares standard normal quantiles calculated from the PIT values to the theoretical standard normal quantiles.- psis_object
If using

**loo**version`2.0.0`

or greater, an object returned by the`psis()`

function (or by the`loo()`

function with argument`save_psis`

set to`TRUE`

).- subset
For

`ppc_loo_intervals()`

and`ppc_loo_ribbon()`

, an optional integer vector indicating which observations in`y`

(and`yrep`

) to include. Dropping observations from`y`

and`yrep`

manually before passing them to the plotting function will not work because the dimensions will not match up with the dimensions of`psis_object`

, but if all of`y`

and`yrep`

are passed along with`subset`

then**bayesplot**can do the subsetting internally for`y`

,`yrep`

*and*`psis_object`

. See the**Examples**section for a demonstration.- intervals
For

`ppc_loo_intervals()`

and`ppc_loo_ribbon()`

, optionally a matrix of precomputed LOO predictive intervals that can be specified instead of`yrep`

and`lw`

(these are both ignored if`intervals`

is specified). If not specified the intervals are computed internally before plotting. If specified,`intervals`

must be a matrix with number of rows equal to the number of data points and five columns in the following order: lower outer interval, lower inner interval, median (50%), upper inner interval and upper outer interval (column names are ignored).- prob, prob_outer
Values between

`0`

and`1`

indicating the desired probability mass to include in the inner and outer intervals. The defaults are`prob=0.5`

and`prob_outer=0.9`

.- order
For

`ppc_loo_intervals()`

, a string indicating how to arrange the plotted intervals. The default (`"index"`

) is to plot them in the order of the observations. The alternative (`"median"`

) arranges them by median value from smallest (left) to largest (right).

A ggplot object that can be further customized using the **ggplot2** package.

`ppc_loo_pit_overlay()`

,`ppc_loo_pit_qq()`

The calibration of marginal predictions can be assessed using probability integral transformation (PIT) checks. LOO improves the check by avoiding the double use of data. See the section on marginal predictive checks in Gelman et al. (2013, p. 152--153) and section 5 of Gabry et al. (2019) for an example of using

**bayesplot**for these checks.The LOO PIT values are asymptotically uniform (for continuous data) if the model is calibrated. The

`ppc_loo_pit_overlay()`

function creates a plot comparing the density of the LOO PITs (thick line) to the density estimates of many simulated data sets from the standard uniform distribution (thin lines). See Gabry et al. (2019) for an example of interpreting the shape of the miscalibration that can be observed in these plots.The

`ppc_loo_pit_qq()`

function provides an alternative visualization of the miscalibration with a quantile-quantile (Q-Q) plot comparing the LOO PITs to the standard uniform distribution. Comparing to the uniform is not good for extreme probabilities close to 0 and 1, so it can sometimes be useful to set the`compare`

argument to`"normal"`

, which will produce a Q-Q plot comparing standard normal quantiles calculated from the PIT values to the theoretical standard normal quantiles. This can help see the (mis)calibration better for the extreme values. However, in most cases we have found that the overlaid density plot (`ppc_loo_pit_overlay()`

) function will provide a clearer picture of calibration problems than the Q-Q plot.`ppc_loo_intervals()`

,`ppc_loo_ribbon()`

Similar to

`ppc_intervals()`

and`ppc_ribbon()`

but the intervals are for the LOO predictive distribution.

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari,
A., and Rubin, D. B. (2013). *Bayesian Data Analysis.* Chapman & Hall/CRC
Press, London, third edition. (p. 152--153)

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,
arXiv preprint,
code on GitHub)

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. arXiv preprint:
https://arxiv.org/abs/1507.04544

Boneva, L. I., Kendall, D., & Stefanov, I. (1971). Spline
transformations: Three new diagnostic aids for the statistical
data-analyst. *J. R. Stat. Soc. B* (Methodological), 33(1), 1-71.
https://www.jstor.org/stable/2986005.

```
# \dontrun{
suppressPackageStartupMessages(library(rstanarm))
suppressPackageStartupMessages(library(loo))
head(radon)
#> floor county log_radon log_uranium
#> 1 1 AITKIN 0.83290912 -0.6890476
#> 2 0 AITKIN 0.83290912 -0.6890476
#> 3 0 AITKIN 1.09861229 -0.6890476
#> 4 0 AITKIN 0.09531018 -0.6890476
#> 5 0 ANOKA 1.16315081 -0.8473129
#> 6 0 ANOKA 0.95551145 -0.8473129
fit <- stan_lmer(
log_radon ~ floor + log_uranium + floor:log_uranium
+ (1 + floor | county),
data = radon,
iter = 1000,
chains = 2,
cores = 2
)
y <- radon$log_radon
yrep <- posterior_predict(fit)
loo1 <- loo(fit, save_psis = TRUE, cores = 4)
#> Warning: Found 2 observation(s) with a pareto_k > 0.7. We recommend calling 'loo' again with argument 'k_threshold = 0.7' in order to calculate the ELPD without the assumption that these observations are negligible. This will refit the model 2 times to compute the ELPDs for the problematic observations directly.
psis1 <- loo1$psis_object
lw <- weights(psis1) # normalized log weights
# marginal predictive check using LOO probability integral transform
color_scheme_set("orange")
ppc_loo_pit_overlay(y, yrep, lw = lw)
#> NOTE: The kernel density estimate assumes continuous observations and is not optimal for discrete observations.
ppc_loo_pit_qq(y, yrep, lw = lw)
ppc_loo_pit_qq(y, yrep, lw = lw, compare = "normal")
#> Warning: NaNs produced
#> Warning: Removed 5 rows containing non-finite values (`stat_qq()`).
#> Warning: Removed 5 rows containing non-finite values (`stat_qq_line()`).
# loo predictive intervals vs observations
keep_obs <- 1:50
ppc_loo_intervals(y, yrep, psis_object = psis1, subset = keep_obs)
color_scheme_set("gray")
ppc_loo_intervals(y, yrep, psis_object = psis1, subset = keep_obs,
order = "median")
# }
```