A walking robot knows where its joints are and what its accelerometers read. This proprioceptive data — no cameras, no lidar, no GPS — is enough to estimate the robot's position and orientation in principle. In practice, the standard approach learns a measurement model from data, assumes Gaussian noise, and fuses everything through a Kalman filter.
The problem is the Gaussian assumption. Real proprioceptive noise isn't Gaussian. Foot slippage, terrain compliance, and learned-model errors produce noise distributions with heavy tails and asymmetries that Gaussian filters can't represent. When the filter assumes Gaussian noise and encounters non-Gaussian reality, it either becomes overconfident (underestimates uncertainty) or drifts (accumulates bias it can't correct).
The alternative: don't assume any distribution at all. Instead, characterize the measurement noise as a set-coverage statement — a guarantee that the true state lies within a computable set with some probability, without specifying the shape of the noise distribution. This is weaker than a full probabilistic model but more honest. It says “the answer is somewhere in here” rather than “the answer has this probability density.”
Integrating set-coverage measurements with a Gaussian filter produces a hybrid: the filter propagates a Gaussian state estimate forward in time (which is computationally cheap), but corrects it using set-membership constraints (which are distribution-free). The result maintains consistency — the estimated uncertainty actually contains the true state — while avoiding the drift that comes from distributional misspecification.
The conceptual lesson: in sensing, an honest statement of ignorance outperforms a precise statement of the wrong thing. The Gaussian model provides more information, but the information is wrong. The set model provides less information, but the information is true.