Stochasticity and Uncertainty › Stochasticity and Uncertainty

Imagine a robot waiting at an intersection, watching a car approach the light. One question is what will the car do: brake, or continue through the yellow? Another second of observation may reveal braking, steering, or signaling, but the robot's current sensors and state representation may still omit variables that influence the decision. A second question is how well do we understand the situation: have we seen enough intersections like this one to trust our dynamics model at all? Perhaps our model was trained in a different city, or only on daylight traffic rather than midnight rain. These are different kinds of not-knowing. Residual spread in a specified data-generating conditional law, after fixing the information set and treating that law as known, is aleatoric relative to that representation; posterior-predictive spread caused by uncertainty about the law is epistemic. Uncertainty about whether the learned model applies here is epistemic; a better-specified model or additional informative data can reduce it. Confusing them leads to bad engineering because the remedies differ. Aleatoric spread calls for a suitable distribution, while epistemic uncertainty calls for better evidence, better assumptions, or a decision to stop trusting the model.

A point prediction can average across separated futures and, by itself, cannot expose whether its error comes from outcome noise or model uncertainty. Auxiliary diagnostics can help, but the point value alone has the same format in familiar and unfamiliar situations. A planner that treats a low-density conditional mean as the realized next state can then optimize for an outcome the transition scarcely supports.

This chapter develops the vocabulary and machinery for representing both kinds of uncertainty in learned world models. We will see when point predictions lose important structure, how unresolved contact branches can create multimodal futures, how ensembles and Bayesian methods approximate model uncertainty, and how uncertainty can change through long rollouts. Throughout, we keep one theme in view: an uncertainty estimate earns its place when a downstream decision procedure can use it and when its claimed probabilities are empirically checked.