Saddle point method for JPDA and related filters

The Bayes posterior probability distribution of many multitarget tracking filters can be written in terms of the mixed derivatives of an appropriately defined generating function. Using the Cauchy Residue Theorem for several complex variables, it is shown that these derivatives can be approximated using the saddle point method, a classical technique in analytic combinatorics. The method gives approximate weights for the particles of sequential Monte Carlo filter implementations. For JPDA the approximate particle weights are seen to be accurate to within one or two percent of the exact weights, after a global scaling factor is applied. The saddle point method has polynomial computational complexity in the number of targets and measurements, assuming that a bounded number of iterations are needed to find the saddle point.