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 the sections below for a brief discussion of the ideas behind posterior predictive checking, an overview of the available PPC plots, and tips on providing an interface to bayesplot from another package.
For plots of posterior (or prior) predictive distributions that do not include observed data see PPD-overview instead.
The plotting functions for prior and
posterior predictive checking all have the prefix ppc_ and all require
the arguments y, a vector of observations, and yrep, a matrix of
replications (in-sample predictions). The plots are organized into several
categories, each with its own documentation:
PPC-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.
PPC-test-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.
PPC-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).
PPC-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.
PPC-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.
PPC-discrete: 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.
PPC-loo: PPC functions for predictive checks based on (approximate) leave-one-out (LOO) cross-validation. '
PPC-censoring: 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.
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 you can use the bayesplot PPD plots instead,
which do not take a y argument, or you can use the PPC plots but provide
plausible or implausible y values that you want to compare to the prior
predictive realizations.
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)
Other PPCs:
PPC-censoring,
PPC-discrete,
PPC-distributions,
PPC-errors,
PPC-intervals,
PPC-loo,
PPC-scatterplots,
PPC-test-statistics