if there are <span class="math inline">\(K\)</span> components and the mixing
simplex <span class="math inline">\(\lambda\)</span> is symmetric, i.e.,</p>
<p><span class="math display">\[
\lambda = \left( \frac{1}{K},   \ldots, \frac{1}{K} \right).
\]</span></p>
<p>The result follows from the identity</p>
<p><span class="math display">\[
\mbox{log\_sum\_exp}(c + a, c + b)
\ = \
c + \mbox{log\_sum\_exp(a, b)}
\]</span></p>
<p>and the fact that adding a constant <span class="math inline">\(c\)</span> to the l