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# 25 Posterior Predictive Sampling

The goal of inference is often posterior prediction, that is evaluating or sampling from the posterior predictive distribution $$p(\tilde{y} \mid y),$$ where $$y$$ is observed data and $$\tilde{y}$$ is yet to be observed data. Often there are unmodeled predictors $$x$$ and $$\tilde{x}$$ for the observed data $$y$$ and unobserved data $$\tilde{y}$$. With predictors, the posterior predictive density is $$p(\tilde{y} \mid \tilde{x}, x, y).$$ All of these variables may represent multivariate quantities.

This chapter explains how to sample from the posterior predictive distribution in Stan, including applications to posterior predictive simulation and calculating event probabilities. These techniques can be coded in Stan using random number generation in the generated quantities block. Further, a technique for fitting and performing inference in two stages is presented in a section on stand-alone generated quantities in Stan