Recursive Bayesian Decoding in State Observation Models: Theory and Application in Quantum-Based Inference

Accurately estimating a sequence of latent variables in state observation models remains a challenging problem, particularly when maintaining coherence among consecutive estimates. While forward filtering and smoothing methods provide coherent marginal distributions, they often fail to maintain coherence in marginal MAP estimates. Existing methods efficiently handle discrete-state or Gaussian models. However, general models remain challenging. Recently, a recursive Bayesian decoder has been discussed, which effectively infers coherent state estimates in a wide range of models, including Gaussian and Gaussian mixture models. In this work, we analyze the theoretical properties and implications of this method, drawing connections to classical inference frameworks. The versatile applicability of mixture models and the prevailing advantage of the recursive Bayesian decoding method are demonstrated using the double-slit experiment. Rather than inferring the state of a quantum particle itself, we utilize interference patterns from the slit experiments to decode the movement of a non-stationary particle detector. Our findings indicate that, by appropriate modeling and inference, the fundamental uncertainty associated with quantum objects can be leveraged to decrease the induced uncertainty of states associated with classical objects. We thoroughly discuss the interpretability of the simulation results from multiple perspectives.