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27.1 Posterior predictive distribution

Given a full Bayesian model $p(y, \theta)$, the posterior predictive density for new data $\tilde{y}$ given observed data $y$ is $$p(\tilde{y} \mid y) = \int p(\tilde{y} \mid \theta) \cdot p(\theta \mid y) \, \textrm{d}\theta.$$ The product under the integral reduces to the joint posterior density $p(\tilde{y}, \theta \mid y),$ so that the integral is simply marginalizing out the parameters $\theta,$ leaving the predictive density $p(\tilde{y} \mid y)$ of future observations given past observations.