Know what a prediction leaves uncertain.

This is the same reference used in Technical mode. Expand it whenever you need the exact numbers, assumptions or further explanation.

Three different questions

A point prediction compresses a range of possibilities into one number. A cart may have two possible next velocities because an unobserved gust can push either way. A sensor can add error to either outcome. A model fitted on limited examples can also be wrong about both. These are different mechanisms, even when all appear as variation in a spreadsheet.

Observation noise changes what is recorded. Outcome uncertainty means several physical outcomes remain possible given the supplied state and action. Model uncertainty concerns what the fitted predictor has not learned. The distinction depends on available information: measuring the gust could turn an uncertain outcome into a predictable conditional outcome.

A small experiment with explicit ground truth

All episodes start at velocity 0.4 m/s and last 0.1 s. This lesson uses a separate fictional actuator with additional response beyond force magnitude 1 N. It is not the preceding lesson’s drag model. In the following rule, numerical constants carry the units needed to express acceleration in m/s²; force is expressed in newtons for a 1 kg cart.

Wind w is either zero or equally likely −0.6/+0.6. A measurement adds independent error of zero or equally likely −0.02/+0.02 m/s. The public diagnostic display enumerates these possibilities exactly. Their probabilities are authored simulator assumptions, not frequencies learned from a robot.

Collection supplies three forces, eight one-step episodes at each, balanced across all wind and sensor signs. This deliberately balanced dataset is not a new random sample each time. Only measured velocity and force enter fitting; hidden wind and true velocity remain diagnostic fields.

Fit a point and a small distribution

The point model learns an intercept and a slope by least squares on the 24 measured examples. Subtract each fitted value from its measured target to obtain a residual. At a probe force, add each of the 24 residuals to the point prediction and give each result mass 1/24.

This empirical distribution predicts measurements, including sensor error. Moving the same residuals to another force assumes their pattern transfers. Residuals also contain model error, so this recipe does not identify noise sources from data. The minimum and maximum are empirical support, not a 95% interval or a physical safety guarantee. Fitting and checking on the same rows can understate error.

With exact sensing and wind enabled, the mean can sit between two possible outcomes without being either outcome. At +2 N the diagnostic physical velocities are 0.612 and 0.732 m/s. With sensor noise, a reading of 0.592 m/s is possible even though physical velocity is always above 0.6 m/s. Do not confuse a probability about readings with one about motion.

Fit several models without pretending they are independent evidence

Each of five models fits a bootstrap sample: 24 episode indices drawn with replacement from the same measured collection. Some rows repeat and others are omitted. The displayed standard deviation measures a typical distance from the five predictions’ average: subtract the average, square each difference, average those squares by dividing by five, then take the square root. Its unit is m/s, matching the predictions. It is neither the spread of physical outcomes nor a calibrated probability of correctness. A constant-force bootstrap would leave slope unidentified; the implementation explicitly falls back to its observed mean.

Seeds 7, 19 and 41 change resampling, not the underlying observations. Changing a seed does not produce new environmental evidence. This bounded affine demonstration is not an implementation of the neural Deep Ensembles research method linked in the references.

Agreement can be confidently wrong

Disable wind and sensor noise, collect only −0.25, 0 and +0.25 N, then probe +2 N. All training points lie on a line, and the fitted lines agree to numerical precision. They predict 0.592 m/s while the plant gives 0.672 m/s: 0.08 m/s error despite essentially zero disagreement.

Broader collection fixes this particular endpoint. It does not make the straight line the right model everywhere: at +1 N the broader line predicts 0.532 m/s while truth is 0.492 m/s. A range box is not coverage of every condition, and data cannot make a restricted model family express every mechanism.

Decide what evidence to collect next

If sensor error dominates, improve or characterize measurement. If actual outcomes differ, preserve the distribution or observe the missing cause. If coverage is poor, collect relevant actions. If a model family misses the response shape, change the model and evaluate again. These interventions address different problems.

Record the conditions, point error, distribution assumption, ensemble spread and proposed next action. The fixed public probes are repeated development practice. Temporary captures do not authenticate learning or replace the later project assessment, independent implementation or human review.