ath display">\[
\frac{d}{dz_{i,j}} x_{i,j}
= \sqrt{1 - \sum_{j&#39; &lt; j} x^2_{i,j&#39;}}.
\]</span></p>
<p>So the combined density for unconstrained <span class="math inline">\(y\)</span> is</p>
<p><span class="math display">\[
p_Y(y)
= p_X(f^{-1}(y))
  \ \
  \prod_{n &lt; \binom{K}{2}} \frac{1}{(\cosh y)^2}
  \ \
  \prod_{i &gt; j} \left( 1 - \sum_{j&#39; &lt; j} x_{i,j&#39;}^2
  \right)^{1/2},
\]</span></p>
<p>where <span class="math inline">\(x = f^{-1}(y)\)</span> is used for notational convenience. 