Bayesian Filtering and Belief States › Summary

• Under exact Bayesian updating and the stated POMDP assumptions, the Bayes filter alternates prediction and correction, and its belief is sufficient for future inference and decision-making.
• The Kalman filter solves the recursion exactly when the transition and observation models are linear, the initial belief is Gaussian, and the Gaussian process and observation noises are independent of one another, the initial state, and across time. The recursion reduces inference to mean and covariance updates governed by a gain that balances model trust against sensor trust.
• The EKF and UKF extend filtering to nonlinear systems by local linearization and sigma-point propagation respectively, remaining Gaussian approximations with corresponding blind spots.
• Particle filters represent beliefs with weighted samples and can preserve multimodal or non-Gaussian structure, but coverage becomes rapidly more expensive in high dimensions. Effective sample size diagnoses weight concentration and can trigger resampling.
• Learned belief updates can amortize an approximation to the composed recursion into a recurrent network. Their accuracy and calibration depend on the training distribution, architecture, and objective.
• Filters can fail smoothly and misleadingly under misspecification. Monitor normalized innovations and compare reported uncertainty with realized error.

With estimation in hand, the natural complementary question is where the transition and observation models themselves come from. System identification addresses it by fitting model parameters from observed data.