The distribution of a new or missing quantity integrates its model over the Bayesian posterior: . Both parameter and residual uncertainty should enter predictive draws used for multiple imputation.
A posterior predictive check compares a summary of the observed data with the same summary in replicated data drawn from a model's posterior predictive distribution. The replication must match the target: replicating an existing group's observations conditional on its effect differs from predicting a new group with a newly drawn effect. Such checks expose mismatches in dispersion, tails or other features; using the observations to fit and check the model means they are not automatically calibrated frequentist tests.
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The posterior predictive distribution is a concept in Bayesian statistics used to make predictions about future observations based on a model that has been updated with observed data. It combines information about the uncertainty of the model parameters (as described by the posterior distribution) with the likelihood of new data given those parameters. Here’s a breakdown of the concept: 1. **Posterior Distribution**: After observing data, we update our beliefs about the model parameters using Bayes' theorem.