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

Faa di Bruno's formula for Gateaux differentials and interacting stochastic population processes