  The problem of estimating interacting systems of multiple objects is
important to a number of different fields of mathematics, physics, and
engineering. Drawing from a range of disciplines, including statistical
physics, variational calculus, point process theory, and statistical sensor
fusion, we develop a unified probabilistic framework for modelling systems of
this nature. In order to do this, we derive a new result in variational
calculus, Faa di Bruno's formula for Gateaux differentials. Using this result,
we derive the Chapman-Kolmogorov equation and Bayes' rule for stochastic
population processes with interactions and hierarchies. We illustrate the
general approach through case studies in multi-target tracking, branching
processes and renormalization.
