  2-field functional integrals (2FFI) are an important class of solution
methods for generating functions of dissipative processes, including
discrete-state stochastic processes, dissipative dynamical systems, and
decohering quantum densities. The stationary trajectories of these integrals
describe a conserved current by Liouville's theorem, despite the fact that
there is no conserved phase space current in the underlying stochastic process.
We develop the information geometry of generating functions for discrete-state
classical stochastic processes in the Doi-Peliti 2FFI form, showing that the
conserved current is a Fisher information between the underlying distribution
of the process and the tilting weight of the generating function. To give an
interpretation to the time invertibility implied by current conservation, we
use generating functions to represent importance sampling protocols, and show
that the conserved Fisher information is the differential of a sample volume
under deformations of the nominal distribution and the likelihood ratio. We
derive a new pair of dual Riemannian connections respecting the symplectic
structure of transport along stationary rays that gives rise to Liouville's
theorem, and show that dual flatness in the affine coordinates of the
coherent-state basis captures the special role played by coherent states in
many 2FFI theories. The covariant convective derivative under time translation
correctly represents the geometric invariants of generating functions under
canonical transformations of the 2FFI field variables of integration.
