This vignette focuses on graphical posterior predictive checks (PPC). Plots of parameter estimates from MCMC draws are covered in the separate vignette Plotting MCMC draws, and MCMC diagnostics are covered in the Visual MCMC diagnostics vignette.
Graphical posterior predictive checks (PPCs)
The bayesplot package provides various plotting functions for graphical posterior predictive checking, that is, creating graphical displays comparing observed data to simulated data from the posterior predictive distribution (Gabry et al, 2019).
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 (PPCs) we simulate from the posterior predictive distribution. This is the distribution of the outcome variable implied by a model after using the observed data \(y\) (a vector of \(N\) outcome values) to update our beliefs about unknown model parameters \(\theta\). The posterior predictive distribution for observation \(\widetilde{y}\) can be written as \[p(\widetilde{y} \,|\, y) = \int
p(\widetilde{y} \,|\, \theta) \, p(\theta \,|\, y) \, d\theta.\] Typically we will also condition on \(X\) (a matrix of predictor variables).
For each draw (simulation) \(s = 1, \ldots, S\) of the parameters from the posterior distribution, \(\theta^{(s)} \sim p(\theta \,|\, y)\), we draw an entire vector of \(N\) outcomes \(\widetilde{y}^{(s)}\) from the posterior predictive distribution by simulating from the data model conditional on parameters \(\theta^{(s)}\). The result is an \(S \times N\) matrix of draws \(\widetilde{y}\).
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 (\(\widetilde{y}\) using predictors \(\widetilde{X}\)). This corresponds to the notation from Gelman et al. (2013) and is the notation used throughout the package documentation.
Using the replicated datasets drawn from the posterior predictive distribution, the functions in the bayesplot package create various graphical displays comparing the observed data \(y\) to the replications. The names of the bayesplot plotting functions for posterior predictive checking all have the prefix ppc_.
Example models
To demonstrate some of the various PPCs that can be created with the bayesplot package we’ll use an example of comparing Poisson and Negative binomial regression models from one of the rstanarm package vignettes (Gabry and Goodrich, 2017).
We want to make inferences about the efficacy of a certain pest management system at reducing the number of roaches in urban apartments. […] The regression predictors for the model are the pre-treatment number of roaches roach1, the treatment indicator treatment, and a variable senior indicating whether the apartment is in a building restricted to elderly residents. Because the number of days for which the roach traps were used is not the same for all apartments in the sample, we include it as an exposure […].
First we fit a Poisson regression model with outcome variable y representing the roach count in each apartment at the end of the experiment.
y roach1 treatment senior exposure2
1 153 308.00 1 0 0.800000
2 127 331.25 1 0 0.600000
3 7 1.67 1 0 1.000000
4 7 3.00 1 0 1.000000
5 0 2.00 1 0 1.142857
6 0 0.00 1 0 1.000000
stan_glm
family: poisson [log]
formula: y ~ roach100 + treatment + senior
observations: 262
predictors: 4
------
Median MAD_SD
(Intercept) 3.1 0.0
roach100 0.7 0.0
treatment -0.5 0.0
senior -0.4 0.0
------
* For help interpreting the printed output see ?print.stanreg
* For info on the priors used see ?prior_summary.stanreg
We’ll also fit the negative binomial model that we’ll compare to the Poisson:
stan_glm
family: neg_binomial_2 [log]
formula: y ~ roach100 + treatment + senior
observations: 262
predictors: 4
------
Median MAD_SD
(Intercept) 2.8 0.2
roach100 1.3 0.3
treatment -0.8 0.3
senior -0.3 0.3
Auxiliary parameter(s):
Median MAD_SD
reciprocal_dispersion 0.3 0.0
------
* For help interpreting the printed output see ?print.stanreg
* For info on the priors used see ?prior_summary.stanreg
Defining y and yrep
In order to use the PPC functions from the bayesplot package we need a vector y of outcome values,
and a matrix yrep of draws from the posterior predictive distribution,
[1] 500 262
[1] 500 262
Each row of the matrix is a draw from the posterior predictive distribution, i.e. a vector with one element for each of the data points in y.
Since we fit the models using rstanarm we used its special posterior_predict function, but if we were using a model fit with the rstan package we could create yrep in the generated quantities block of the Stan program or by doing simulations in R after fitting the model. Draws from the posterior predictive distribution can be used with bayesplot regardless of whether or not the model was fit using an interface to Stan. bayesplot just requires a yrep matrix that has number_of_draws rows and number_of_observations columns.
