  The lack-of-fit statistical reduction, developed and formulated first by
Bruce Turkington, is a general method taking Liouville equation for probability
density (detailed level) and transforming it to reduced dynamics of projected
quantities (less detailed level). In this paper the method is generalized. The
Hamiltonian Liouville equation is replaced by an arbitrary Hamiltonian
evolution combined with gradient dynamics (GENERIC), the Boltzmann entropy is
replaced by an arbitrary entropy, and the kinetic energy by an arbitrary
energy. The gradient part is a generalized gradient dynamics generated by a
dissipation potential. The reduced evolution of the projected state variables
is shown to preserve the GENERIC structure of the original (detailed level)
evolution. The dissipation potential is obtained by solving a Hamilton-Jacobi
equation. In summary, the lack-of-fit reduction can start with GENERIC and
obtain GENERIC for the reduced state variables.
