z_k}
=
\left(
  1 - \sum_{k&#39; = 1}^{k-1} x_{k&#39;}
   \right)
.
\]</span></p>
<p>This definition is recursive, defining <span class="math inline">\(x_k\)</span> in terms of
<span class="math inline">\(x_{1},\ldots,x_{k-1}\)</span>.</p>
<p>Because the Jacobian <span class="math inline">\(J\)</span> of <span class="math inline">\(f^{-1}\)</span> is lower triangular and positive, its
absolute determinant reduces to</p>
<p><span class="math display">\[
\left| \, \det J \, \right|
\ = \
\prod_{k=1}^{K-1} J_{