Multi-object estimation in state-space models (SSMs) wherein the system state is represented as a finite set has attracted significant interest in recent years. In Bayesian inference, the posterior density captures all information on the system trajectory since it considers the history of the states. In most multi-object SSM applications, closed-form multi-object posteriors are not available for non-standard multi-object models. Thus, functional approximation is necessary because these posteriors are very high-dimensional. This work provides a tractable multi-scan Generalized Labeled Multi-Bernoulli (GLMB) approximation that matches the trajectory cardinality distribution of the labeled multi-object posterior density. The proposed approximation is also proven to minimize the Kullback-Leibler divergence over a special class of multi-scan GLMB model. Additionally, we develop a tractable algorithm for computing the approximate multi-object posteriors over finite windows. Numerical multi-object tracking experiments, using a simulated social force model with uninformative observations, and a real-world social force pedestrian dataset are presented to validate the proposed approximation method.

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