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Dual Nonlinear Filters and Entropy Production

This paper makes connections between nonlinear filtering and the entropic properties of Markov processes. It starts by developing information flow models for continuous-time, discrete-state filtering problems, identifying rates of information supply and dissipation. Time reversal yields a dual filtering problem in which these flows are interchanged. The dual problem comprises a diffusion signal with nonlinear dynamics, and observations of the point process variety, but yields a finite-dimensional nonlinear filter. The paper goes on to define an entropic time derivative for a general class of Markov processes and relates the entropic derivatives of the signal and filter to the rates of information supply and dissipation. This leads to the definition of a rate of interactive entropy production, which measures the time asymmetry of the interaction between the signal and filter. This asymmetry is of the same nature as that occurring in the theory of nonequilibrium statistical mechanics based on stochastic dynamics. In this context, the interaction between the signal and filter is nondissipative—a property intimately connected with the existence of a dual problem.