Feedback particle filter for collective inference

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}.