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  1. The purpose of this paper is to describe the feedback particle filter algorithm for problems where there are a large number (\begin{document}$ M $\end{document}) of non-interacting agents (targets) with a large number (\begin{document}$ M $\end{document}) of non-agent specific observations (measurements) that originate from these agents. In its basic form, the problem is characterized by data association uncertainty whereby the association between the observations and agents must be deduced in addition to the agent state. In this paper, the large-\begin{document}$ M $\end{document} limit is interpreted as a problem of collective inference. This viewpoint is used to derive the equation for the empirical distribution of the hidden agent states. A feedback particle filter (FPF) algorithm for this problem is presented and illustrated via numerical simulations. Results are presented for the Euclidean and the finite state-space cases, both in continuous-time settings. The classical FPF algorithm is shown to be the special case (with \begin{document}$ M = 1 $\end{document}) of these more general results. The simulations help show that the algorithm well approximates the empirical distribution of the hidden states for large \begin{document}$ M $\end{document}.

     
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  2. Duality between estimation and optimal control is a problem of rich historical significance. The first duality principle appears in the seminal paper of Kalman-Bucy, where the problem of minimum variance estimation is shown to be dual to a linear quadratic (LQ) optimal control problem. Duality offers a constructive proof technique to derive the Kalman filter equation from the optimal control solution. This paper generalizes the classical duality result of Kalman-Bucy to the nonlinear filter: The state evolves as a continuous-time Markov process and the observation is a nonlinear function of state corrupted by an additive Gaussian noise. A dual process is introduced as a backward stochastic differential equation (BSDE). The process is used to transform the problem of minimum variance estimation into an optimal control problem. Its solution is obtained from an application of the maximum principle, and subsequently used to derive the equation of the nonlinear filter. The classical duality result of Kalman-Bucy is shown to be a special case. 
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