 the <span class="math inline">\(\theta^{(m)} \sim p(\theta \mid x, y)\)</span> are draws from the posterior given only the training data <span class="math inline">\((x, y).\)</span></p>
<p>To avoid underflow in calculations, it will be more stable to compute densities on the log scale. Taking the logarithm and pushing it through results in a stable computation, <span class="math display">\[\begin{eqnarray*}
\log p(\tilde{y} \mid \tilde{x}, x, y)
&amp; \approx &amp;
\log \frac{1}{M} \sum_{m = 1}^M p(\tilde{