What is the posterior predictive distribution?

hard

Answer

$p(y_{\mathrm{new}}\; \mid \;D)$
$Predicts new data marginalizing over posterior uncertainty.$
$Unlike MLE + plug-in prediction, it accounts for parameter uncertainty \to wider (better-calibrated) prediction intervals, especially with small n.$
$y_{\mathrm{new}}$
$Foundation of Bayesian forecasting and probabilistic programming.$

Check yourself — multiple choice

Same as likelihood
$p(y_{\mathrm{new}}\; \mid \;D)$ = ∫ $p(y \mid {\theta})$ $p({\theta} \mid D)$ dθ → prediction that marginalizes over posterior uncertainty; wider than plug-in; correct at small n
Random
Only frequentist

Posterior predictive: integrates over posterior; correct-uncertainty predictions.

#bayesian

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