y} \mid \tilde{x}, \theta^{(m)}),
\\[4pt]
&amp; = &amp; -\log M + \log \sum_{m = 1}^M p(\tilde{y} \mid \tilde{x}, \theta^{(m)}),
\\[4pt]
&amp; = &amp; -\log M + \log \sum_{m = 1}^M \exp(\log p(\tilde{y} \mid \tilde{x}, \theta^{(m)}))
\\[4pt]
&amp; = &amp; -\log M + \textrm{log-sum-exp}_{m = 1}^M \log p(\tilde{y} \mid \tilde{x}, \theta^{(m)})
\end{eqnarray*}\]</span> where the log sum of exponentials function is defined so as to make the above equation hold, <span class="math display">\[
\textrm{log-sum-exp}