Bayesian Filtering and Belief States › Limitations and Impact

The seeded observations above are sanity checks for these executed trajectories, not guarantees that the same orderings or numerical comparisons hold after changing the data, seed, or training budget. The structural assertions check array alignment and implementation contracts, not benchmark superiority.

The belief-space MDP argument is exact only for an exact belief. The EKF, UKF, finite particle set, and learned recurrent state all approximate that belief, so they restore the Markov property only to the extent that their representation preserves decision-relevant information.

No filter is better than its models. The Bayes recursion is exact only under its assumptions, and a practical filter can fail when they are violated. Understating process noise $Q$ tends to make a filter overconfident; in extreme cases its covariance collapses around a wrong estimate. Overstating $Q$ tends toward overly diffuse, underconfident beliefs. Nonstationarity invalidates fixed parameters, while heavy-tailed or outlier-corrupted measurements invalidate a Gaussian likelihood and can pull an estimate far from the truth. Drifting innovations, persistent bias, and serial structure in normalized innovations are direct warnings that the prediction and observation model no longer match the data. Robust and adaptive filters address these cases in different ways. The same residual computations carry over to learned world models, but the classical Gaussian reference distribution is a guarantee only under the corresponding probabilistic assumptions. Across these systems, the most useful warning is a gap between self-reported confidence and measured accuracy.

Compounding rollout error deserves special emphasis because it anticipates model-based reinforcement learning. During prediction with no corrective measurements, disturbances and transition-model errors accumulate. A common conservative choice is to increase $Q$ to cover some unmodeled variation, accepting wider beliefs in exchange for less overconfidence. The transferable lesson is narrower than a literal equivalence: recursively applying a learned transition model also compounds error, so its uncertainty estimates must be checked against realized rollout errors.

This controlled example uses independent zero-mean disturbances, so both empirical RMS error and the correctly predicted standard deviation grow with the square root of horizon. The understated model predicts the same shape at the wrong scale and becomes increasingly overconfident in absolute terms. Biased model errors can grow differently, including approximately linearly, so the square-root curve is not a universal rollout law. The calibration test is general: compare predicted uncertainty with realized error under the deployment distribution.

Computational scaling draws the practical map. A dense Kalman covariance update is cubic in state dimension, so large systems require sparsity or low-rank structure. Particle methods can represent more general distributions, but useful coverage becomes expensive in high dimensions unless structure is exploited. Square-root filters propagate a covariance factor for better numerical stability; ensemble filters approximate moments with a finite sample set; and auxiliary-variable methods augment the representation to make sampling or conditioning easier. Along with the unscented method introduced earlier and Rao-Blackwellized methods beyond this chapter's scope, these approaches occupy different points between exact dense algebra and unconstrained sampling. The practical lesson is to exploit the conditional structure of the problem rather than choose a method by name alone.

The GRU training cell uses all 128 simulated training trajectories for exactly 1,200 random minibatch updates of size 32. It constructs no validation split and implements neither early stopping nor hyperparameter search, so this result should not be read as a tuned architecture comparison. These recursions appear in navigation, localization, tracking, attitude estimation, data assimilation, and robot state estimation. Modern learned world models often use an analogous template: maintain a latent state, predict it forward, and correct it with new evidence. A learned recurrent state is not automatically an exact Bayesian belief, but the classical filter remains a useful reference for checking prediction residuals, calibration, and collapse under repeated rollout.