n">Evaluating posterior predictive densities</h2>
<p>Given training data <span class="math inline">\((x, y)\)</span> consisting of parallel sequences of predictors and observations and test data <span class="math inline">\((\tilde{x}, \tilde{y})\)</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