 remaining
piece of the original stick,</p>
<p><span class="math display">\[
x_K = 1 - \sum_{k=1}^{K-1} x_k.
\]</span></p>
</div>
<div id="absolute-jacobian-determinant-of-the-unit-simplex-inverse-transform" class="section level3">
<h3>Absolute Jacobian Determinant of the Unit-Simplex Inverse Transform</h3>
<p>The Jacobian <span class="math inline">\(J\)</span> of the inverse transform <span class="math inline">\(f^{-1}\)</span> is
lower-triangular, with diagonal entries</p>
<p><span class="math display">\[