Interval/Smoothing Filters for Multiple Object Tracking Via Analytic Combinatorics

The single-object Bayesian filter for an interval, or batch, of data is extended to the multiple object case using the method of analytic combinatorics. The exact expression for the probability generating functional of the Bayes posterior process is derived. It is a nested composition of functions and functionals that is evaluated via a backward recursion. Branching and immigration processes are used to model the initial multiple object process and new object arrival processes, respectively. The exact Bayes posterior distribution and various summary statistics of the interval filter are derivatives of the generating functional. These derivatives are written in equivalent Cauchy integral form and approximated using the saddle point method.