  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.
