I currently study Searle's et al. (1992) book "variance components". In appendix S.d (page 474) they define Jacobian matrix of the transformation $\Theta \rightarrow\Delta$ as
$J_{\Theta \rightarrow\Delta}=\left[_m \frac {\partial \Theta_i}{\partial \Delta_j}\right]_{i,j}$
$m$ indicates that it is a matrix with the corresponding partial derivatives at $(i,j)$. Hence, $J$ is the Jacobian matrix of a function which takes as input the vector $\Theta$ and produces as output the vector $\Delta$. Or as the authors put it: parameters in $\Theta$ "are transformed in a one-to-one manner to the vector $\Delta$"
However, if I remind myself about the structure of a Jacobian matrix, amongst others here, then I find the following:
$ J=[\frac {\partial f}{\partial x_i}…\frac {\partial f}{\partial x_n}]$
Here, J is the Jacobian matrix $[m,n]$ of $f$, where $f:ℝ^n → ℝ^m$ is a function which takes as input the vector $x$ and produces as output the vector $f(x)$.
Hence, I find the two definitions of the Jacobian matrix conflicting since
$J_{\Theta \rightarrow\Delta} = J^{-1}$
I do not believe that there is a mistake in Searle's et al. but cannot reconcile both definitions of a Jacobian matrix. What do I miss?
Best Answer
That's just because of how the Jacobian is used in probability. If $\Delta = f(\Theta)$ and $p_\Delta, p_\Theta$ are the two densities then
$$\mathbf{P}(\Delta \in A) = \int_A p_{\Delta}(\delta) \,d\delta = \mathbf{P}(\Theta \in f^{-1}(A)) = \int_{f^{-1}(A)} p_\Theta(\theta) \,d\theta $$
The change of variables theorem tells us that
$$ \int_{f^{-1}(A)} p_\Theta(\theta) \,d\theta = \int_A p_\Theta(f^{-1}(\delta))\left| \det\left[ \frac{\partial \Theta_i}{\partial \Delta_j} \right] \right| \,d\delta. $$
So
$$ p_\Delta(\delta) = p_\Theta(f^{-1}(\delta))\left| \det\left[ \frac{\partial \Theta_i}{\partial \Delta_j} \right] \right|. $$
Which means we don't want the Jacobian of $f$ as defined on Wikipedia, we want the Jacobian of $f^{-1}$.