  Using the statistical inference method, a non-relativistic, spinless,
non-linear quantum dynamical equation is derived with the Fisher information
metric substituted by the Jensen-Shannon distance information. Among all
possible implications, it is shown that the non-linear Schr\"odinger equation
preserves the symplectic structure of the complex Hilbert space, hence a
Hamiltonian dynamics. The canonically projected dynamics is obtained on the
corresponding projective Hilbert space of pure state density operators.
