The log
Jacobian determinant of the complete inverse transform <span class="math inline">\(x = f^{-1}(y)\)</span>
is given by</p>
<p><span class="math display">\[
\log \left| \, \det J \, \right|
=
-2 \sum_{n \leq \binom{K}{2}}
\log \cosh y
\
+
\
\frac{1}{2} \
\sum_{i &gt; j}
\log \left( 1 - \sum_{j&#39; &lt; j} x_{i,j&#39;}^2 \right)
.
\]</span></p>

</div>
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<div class="footnotes">
<hr />
<ol start="17">
<li id="fn17"><p>For convenience, a summation with no terms, such as <span class=