  State estimation poses substantial challenges in robotics, often involving
encounters with multimodality in real-world scenarios. To address these
challenges, it is essential to calculate Maximum a posteriori (MAP) sequences
from joint probability distributions of latent states and observations over
time. However, it generally involves a trade-off between approximation errors
and computational complexity. In this article, we propose a new method for MAP
sequence estimation called Stein-MAP, which effectively manages multimodality
with fewer approximation errors while significantly reducing computational and
memory burdens. Our key contribution lies in the introduction of a sequential
variational inference framework designed to handle temporal dependencies among
transition states within dynamical system models. The framework integrates
Stein's identity from probability theory and reproducing kernel Hilbert space
(RKHS) theory, enabling computationally efficient MAP sequence estimation. As a
MAP sequence estimator, Stein-MAP boasts a computational complexity of O(N),
where N is the number of particles, in contrast to the O(N^2) complexity of the
Viterbi algorithm. The proposed method is empirically validated through
real-world experiments focused on range-only (wireless) localization. The
results demonstrate a substantial enhancement in state estimation compared to
existing methods. A remarkable feature of Stein-MAP is that it can attain
improved state estimation with only 40 to 50 particles, as opposed to the 1000
particles that the particle filter or its variants require.
