 mixing proportions
<span class="math inline">\(\lambda\)</span> in the unit <span class="math inline">\(K\)</span>-simplex, then
<span class="math display">\[
p_Y(y | \lambda, \mu, \sigma)
\ = \
\sum_{k=1}^K \lambda_k \, \mathsf{normal}(y \, | \, \mu_k, \sigma_k).
\]</span></p>
<div id="log-sum-of-exponentials-linear-sums-on-the-log-scale" class="section level3 unnumbered">
<h3>Log Sum of Exponentials: Linear Sums on the Log Scale</h3>
<p>The log sum of exponentials function is used to define mixtures on t