A useful quantity of Bayesian Inference is the the Posterior Predictive Distribution (PPD). This is the distribution of expected, future data \(\tilde{Y}\) according to the posterior \(p(\theta|Y)\), which is a consequence of the model (prior & likelihood) and observed data.

Rather this is the data data the model is expecting to see after seeing the dataset \(Y\).

\[ p(\tilde{Y}|Y) = \int p(\tilde{Y}|\theta)p(\theta|Y)d\theta \]

As a consequence of integrating out over the posterior distribution of parameters, predictions will incorporate the uncertainty about our estimates.