Belief propagation (BP) is a useful probabilistic inference algorithm for efficiently computing approximate marginal probability densities of random variables. However, in its standard form, BP is only applicable to the vector-type random variables with a fixed and known number of vector elements, while certain applications rely on RFSs with an unknown number of vector elements. In this paper, we develop BP rules for factor graphs defined on sequences of RFSs where each RFS has an unknown number of elements, with the intention of deriving novel inference methods for RFSs. Furthermore, we show that vector-type BP is a special case of set-type BP, where each RFS follows the Bernoulli process. To demonstrate the validity of developed set-type BP, we apply it to the PMB filter for SLAM, which naturally leads to new set-type BP-mapping, SLAM, multi-target tracking, and simultaneous localization and tracking filters. Finally, we explore the relationships between the vector-type BP and the proposed set-type BP PMB-SLAM implementations and show a performance gain of the proposed set-type BP PMB-SLAM filter in comparison with the vector-type BP-SLAM filter.

翻译：置信传播（BP）是一种用于高效计算随机变量近似边缘概率密度的实用概率推理算法。然而，标准形式的BP仅适用于向量元素数量固定且已知的向量型随机变量，而某些应用依赖于元素数量未知的随机有限集（RFS）。本文针对定义在RFS序列上的因子图开发了BP规则（其中每个RFS具有未知数量的元素），旨在推导RFS的新的推理方法。此外，我们证明向量型BP是集合型BP的特例，其中每个RFS服从伯努利过程。为验证所提出的集合型BP的有效性，将其应用于泊松多伯努利（PMB）SLAM滤波器，自然推导出新的集合型BP映射、SLAM、多目标跟踪以及同时定位与跟踪滤波器。最后，我们探究了向量型BP与所提出的集合型BP PMB-SLAM实现之间的关联，并展示了相较于向量型BP-SLAM滤波器，所提出的集合型BP PMB-SLAM滤波器的性能提升。