 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 data <span class="math inline">\((x, y).\)</span></p>
<p>This integral may be calculated with Monte Carlo methods as usual, <span class="math display">\[
p(\tilde{y} \mid \tilde{x}, x, y)
\approx
\frac{1}{M} \sum_{m = 1}^M p(\tilde{y} \mid \tilde{x}, \theta^{(m)}),
\]</span> where