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## 24.1 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.