  The purpose of this paper is to describe the feedback particle filter
algorithm for problems where there are a large number ($M$) of non-interacting
agents (targets) with a large number ($M$) 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-$M$ 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 $M=1$) of these more general results. The
simulations help show that the algorithm well approximates the empirical
distribution of the hidden states for large $M$.
