  Recently, a method to dynamically define a divergence function $D$ for a
given statistical manifold $(\mathcal{M}\,,g\,,T)$ by means of the
Hamilton-Jacobi theory associated with a suitable Lagrangian function
$\mathfrak{L}$ on $T\mathcal{M}$ has been proposed. Here we will review this
construction and lay the basis for an inverse problem where we assume the
divergence function $D$ to be known and we look for a Lagrangian function
$\mathfrak{L}$ for which $D$ is a complete solution of the associated
Hamilton-Jacobi theory. To apply these ideas to quantum systems, we have to
replace probability distributions with probability amplitudes.
