  Through integrating the evolutionary correlations across global states in the
bidirectional recursion, an explainable Bayesian recurrent neural smoother
(EBRNS) is proposed for offline data-assisted fixed-interval state smoothing.
At first, the proposed model, containing global states in the evolutionary
interval, is transformed into an equivalent model with bidirectional memory.
This transformation incorporates crucial global state information with support
for bi-directional recursive computation. For the transformed model, the joint
state-memory-trend Bayesian filtering and smoothing frameworks are derived by
introducing the bidirectional memory iteration mechanism and offline data into
Bayesian estimation theory. The derived frameworks are implemented using the
Gaussian approximation to ensure analytical properties and computational
efficiency. Finally, the neural network modules within EBRNS and its two-stage
training scheme are designed. Unlike most existing approaches that artificially
combine deep learning and model-based estimation, the bidirectional recursion
and internal gated structures of EBRNS are naturally derived from Bayesian
estimation theory, explainably integrating prior model knowledge, online
measurement, and offline data. Experiments on representative real-world
datasets demonstrate that the high smoothing accuracy of EBRNS is accompanied
by data efficiency and a lightweight parameter scale.
