The **bayesplot** PPC module provides various plotting functions for creating
graphical displays comparing observed data to simulated data from the
posterior (or prior) predictive distribution. See below for a brief
discussion of the ideas behind posterior predictive checking, a description
of the structure of this package, and tips on providing an interface to
**bayesplot** from another package.

The idea behind posterior predictive checking is simple: if a model is a good fit then we should be able to use it to generate data that looks a lot like the data we observed.

To generate the data used for posterior predictive checks we simulate from
the *posterior predictive distribution*. The posterior predictive
distribution is the distribution of the outcome variable implied by a model
after using the observed data \(y\) (a vector of outcome values), and
typically predictors \(X\), to update our beliefs about the unknown
parameters \(\theta\) in the model. For each draw of the parameters
\(\theta\) from the posterior distribution
\(p(\theta \,|\, y, X)\)
we generate an entire vector of outcomes. The result is
an \(S \times N\) matrix of simulations, where \(S\) is the the
size of the posterior sample (number of draws from the posterior
distribution) and \(N\) is the number of data points in \(y\). That is,
each row of the matrix is an individual "replicated" dataset of \(N\)
observations.

When simulating from the posterior predictive distribution we can use either
the same values of the predictors \(X\) that we used when fitting the model
or new observations of those predictors. When we use the same values of
\(X\) we denote the resulting simulations by \(y^{rep}\) as they
can be thought of as *replications* of the outcome \(y\) rather than
predictions for future observations. This corresponds to the notation from
Gelman et. al. (2013) and is the notation used throughout the documentation
for this package.

Using the datasets \(y^{rep}\) drawn from the posterior predictive
distribution, the functions in the **bayesplot** package produce various
graphical displays comparing the observed data \(y\) to the replications.
For a more thorough discussion of posterior predictive checking see
Chapter 6 of Gelman et. al. (2013).

To use **bayesplot** for *prior* predictive checks you can simply use draws
from the prior predictive distribution instead of the posterior predictive
distribution. See Gabry et al. (2019) for more on prior predictive checking
and when it is reasonable to compare the prior predictive distribution to the
observed data. If you want to avoid using the observed data for prior
predictive checks, then the `y`

argument to the PPC plotting functions can be
used to provide plausible or implausible `y`

values that you want to compare
to the prior predictive realizations.

The plotting functions for prior and posterior predictive checking are organized into several categories, each with its own documentation:

Distributions: Histograms, kernel density estimates, boxplots, and other plots comparing the empirical distribution of data

`y`

to the distributions of individual simulated datasets (rows) in`yrep`

.Statistics: The distribution of a statistic, or a pair of statistics, over the simulated datasets (rows) in

`yrep`

compared to value of the statistic(s) computed from`y`

.Intervals: Interval estimates of

`yrep`

with`y`

overlaid. The x-axis variable can be optionally specified by the user (e.g. to plot against a predictor variable or over time).Predictive errors: Plots of predictive errors (

`y - yrep`

) computed from`y`

and each of the simulated datasets (rows) in`yrep`

. For binomial models binned error plots are also available.Scatterplots: Scatterplots (and similar visualizations) of the data

`y`

vs. individual simulated datasets (rows) in`yrep`

, or vs. the average value of the distributions of each data point (columns) in`yrep`

.Plots for discrete outcomes: PPC functions that can only be used if

`y`

and`yrep`

are discrete. For example, rootograms for count outcomes and bar plots for ordinal, categorical, and multinomial outcomes.LOO predictive checks: PPC functions for predictive checks based on (approximate) leave-one-out (LOO) cross-validation.

Censored data: PPC functions comparing the empirical distribution of censored data

`y`

to the distributions of individual simulated datasets (rows) in`yrep`

.

In addition to the various plotting functions, the **bayesplot** package
provides the S3 generic `pp_check()`

. Authors of R packages for
Bayesian inference are encouraged to define `pp_check()`

methods for the
fitted model objects created by their packages. See the package vignettes for
more details and a simple example, and see the **rstanarm** and **brms**
packages for full examples of `pp_check()`

methods.

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)

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. (Ch. 6)