
J_{k,k}
=
\frac{\partial x_k}{\partial y_k}
=
\frac{\partial x_k}{\partial z_k} \,
\frac{\partial z_k}{\partial y_k},
\]</span></p>
<p>where</p>
<p><span class="math display">\[
\frac{\partial z_k}{\partial y_k}
= \frac{\partial}{\partial y_k}
   \mathrm{logit}^{-1} \left(
                       y_k + \log \left( \frac{1}{K-k}
                                          \right)
                    \right)
= z_k (1 - z_k),
\]</span></p>
<p>and</p>
<p><span class="math display">\[
\frac{\partial x_k}{\partial 