, \tilde{y})\)</span> of the same structure, the posterior predictive density is <span class="math display">\[
p(\tilde{y} \mid \tilde{x}, x, y)
=
\int
  p(\tilde{y} \mid \tilde{x}, \theta)
  \cdot p(\theta \mid x, y)
\, \textrm{d}\theta,
\]</span></p>
<p>where <span class="math inline">\(\theta\)</span> is the vector of model parameters. This predictive density is the density of the test observations, conditioned on both the test predictors <span class="math inline">\(\tilde{x}\)</span> and the training da