Bayesian Filtering and Belief States › Approximate Filtering for Nonlinear Systems

Real systems are nonlinear: angles wrap, contacts switch, aerodynamics saturate. Once either the dynamics or the observation model bends away from linearity, Gaussians no longer close, and we must approximate. The two dominant classical strategies push in opposite directions: the extended Kalman filter forces the belief to stay Gaussian and linearizes locally, while particle filters abandon parametric forms entirely and carry samples. Between these poles lies a spectrum of compromises, and choosing among them is fundamentally a choice about which failure mode you can afford: the bias of a wrong parametric family, or the variance and cost of Monte Carlo approximation.

The extended Kalman filter (EKF) keeps the Gaussian belief and uses first-order Taylor expansions of the nonlinear functions to build linearized covariance and correction maps. Write the general system as

$$x_t = f(x_{t-1}, u_{t-1}) + w, \qquad o_t = h(x_t) + v,$$

with additive noise covariances $Q$ and $R$. The mean prediction uses the nonlinear transition $f$, and the predicted observation uses the nonlinear function $h$. The EKF linearizes only the covariance and correction maps, using the Jacobians

$$F_t = \left.\frac{\partial f}{\partial x}\right|_{\hat{x}_{t-1}}, \qquad H_t = \left.\frac{\partial h}{\partial x}\right|_{\hat{x}_{t|t-1}},$$

so $P_{t|t-1}=F_tP_{t-1|t-1}F_t^\top+Q$ and the correction uses $H_t$ in the Kalman gain. The intuition is that the Jacobian captures the local sensitivity of the dynamics at the operating point, so the covariance update approximates how neighboring trajectories spread apart or converge near the current estimate, even though the mean follows the full curved dynamics. A separate control-noise Jacobian is needed only when uncertainty enters through the control channel rather than as the additive state noise written above.

The EKF is cheap, ubiquitous, and fragile in equal measure. Its failure modes deserve explicit attention:

• Linearization error compounds over rollouts, because each corrected estimate becomes the expansion point for the next prediction. Small biases accumulate.
• Divergence: if the true belief is multimodal or strongly skewed, the Gaussian approximation can place its mean somewhere no plausible state exists, after which innovations may point the wrong way and the filter can walk away from the truth.
• Jacobian availability: deriving analytic derivatives is laborious; automatic differentiation helps, but discontinuous models (contact, mode switches) resist differentiation altogether.

Each failure mode suggests a partial remedy that practitioners have catalogued over decades: iterating the correction step to refine the expansion point, inflating $Q$ to hedge against unmodeled curvature, or switching to derivative-free variants when Jacobians are unavailable. None of these removes the fundamental constraint, which is the Gaussian bottleneck.

The unscented Kalman filter (UKF) attacks linearization error differently. Rather than differentiating, it propagates a small deterministic set of sigma points through the nonlinear functions and fits a Gaussian to the transformed cloud. A common construction uses $2n+1$ points for an $n$-dimensional state. It captures curvature that a single local Jacobian misses without requiring derivatives, which makes it useful for black-box simulation models. Whether it improves on an EKF depends on the system, tuning, and operating regime; both methods still force the belief into a Gaussian family. Neither can represent a multimodal belief. To see why that limitation bites, imagine localizing a robot in a symmetric corridor from ambiguous landmarks: the true posterior has two peaks, one for each consistent interpretation, and a single Gaussian summary places its mean between them.