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 applicable to only the
vector-type random variables, while certain applications rely on set-type
random variables with an unknown number of vector elements. In this paper, we
first develop BP rules for set-type random variables and demonstrate that
vector-type BP is a special case of set-type BP. We further propose factor
graphs with set-factor and set-variable nodes by devising the set-factor nodes
that can address the set-variables with random elements and cardinality, while
the number of vector elements in vector-type is known. To demonstrate the
validity of developed set-type BP, we apply it to the Poisson multi-Bernoulli
(PMB) filter for simultaneous localization and mapping (SLAM), which naturally
leads to a new set-type BP-SLAM filter. Finally, we reveal connections between
the vector-type BP-SLAM filter and the proposed set-type BP-SLAM filter and
show a performance gain of the proposed set-type BP-SLAM filter in comparison
with the vector-type BP-SLAM filter.

本文开发了用于集合类型随机变量的信念传播规则，并提出了存在随机元素和基数的集变量节点的因子图，将该方法应用于同时定位和地图制图中，并证明了提出的方法相较于标准的信念传播方法在性能上有更好的表现。