Unbiased determination of forces causing observed processes

Abstract

Using measured short time correlation functions of a stochastic process as constraints in the maximum calibre principle of Jaynes, we formulate the joint probability distribution function of the process. The Lagrange multipliers which hereby occur are determined by minimizing a time-dependent form of the (Kullback) information gain. This step can alternatively be interpreted as if our system builds a neural network which “learns” the Lagrange multipliers. Next, we proceed to determine explicit formulas-expressed in terms of the Lagrange multipliers-for the drift and diffusion coefficients appearing in the corresponding Ito-Langevin equation, which describe the forces underlying the process. Computer-simulations of two processes are presented, showing good confirmation of the theory.