## Posterior predictive check

Consider a regression model with response variable \(Y\), predictor \(X\), and a set of regression parameters \(\theta\). For example, in the model above \(\theta = (\beta_0,\beta_1,\sigma)\). Further, let \(\left\lbrace \theta^{(1)}, \theta^{(2)}, \ldots, \theta^{(N)}\right\rbrace\) be an \(N\)-length Markov chain for the posterior model of \(\theta\). Then a “good” Bayesian model will produce *predictions* of \(Y\) with features similar to the *original* \(Y\) data. To evaluate whether your model satisfies this goal: