The information geometry of two-field functional integrals

Abstract

Two-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 absence of a conserved kinematic 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, and exhibit two quantities conserved along stationary trajectories. One is a Wigner function, familiar as a semiclassical density from quantum-mechanical time-dependent density-matrix methods. The second is an overlap function, between directions of variation in an underlying distribution and those in the directions of relative large-deviation probability that can be used to interrogate the distribution, and expressed as an inner product of vector fields in the Fisher information metric. 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 pair of dual affine connections particular to Doi-Peliti theory for the way they separate the roles of the nominal distribution and likelihood ratio, distinguishing them from the standard dually-flat connection of Nagaoka and Amari defined on the importance distribution, and show that dual flatness in the affine coordinates of the coherent-state basis captures the special role played by coherent states in Doi-Peliti theory.

Rules, hypergraphs, and probabilities: the three-level analysis of chemical reaction systems and other stochastic stoichiometric population processes

“…(G7) will have static solutions and will serve as an equation of constraint. This behavior reflects the anti-causal nature of the problem of inference [105] using the response field θ in the construction of Sec. IV A 4.…”

“…a. Identifying the saddle-point stationary path associated with a "nominal distribution" in the importancesampling interpretation: As developed in [105], the field n that carries the saddle-point behavior of the largedeviation rate in this Hamilton-Jacobi theory has the interpretation of a saddle point for an importance distribution in importance sampling [139,140]. The saddle point in the corresponding nominal distribution, which carries the interpretation of the underlying "state" deformed by the large deviation, is e −θ T n , 33 for which the stationarity equation is…”

“…As developed in [105], the field n that carries the saddle-point behavior of the large-deviation rate in this Hamilton-Jacobi theory has the interpretation of a saddle point for an importance distribution in importance sampling [139, 140]. The saddle point in the corresponding nominal distribution, which carries the interpretation of the underlying “state” deformed by the large deviation, is , 33 for which the stationarity equation is For many of the simple cases in which the trajectory equations are solvable in terms of mean-field dynamics, Eq.…”

“…The conserved “phase-space” volume in the Hamilton-Jacobi theory for large deviations implies a duality between forward dynamics of nominal distributions and backward dynamics of the “response field” [100, 141] e θ familiar from adjoint fluctuation theorems [142]. That duality is reflected in the symmetry of a pair of exponential-family coordinates, interchanged by a similarity-transform [105, 143] about the stationary state under the instantaneous generating parameters.…”

“…In the large-number limit where saddle-point approximation can be used, the CGF and LDF evolve under a Hamilton-Jacobi equation [64][65][66], which is the raytheoretic approximation [69,70] for the bundles of histories that dominate the probability of the least-improbable histories that arrive at improbable states. The resulting Hamiltonian dynamical system with its conserved volume element is the saddle-point representation of the conservation of probability [105] familiar from Liouville's theorem [63].…”