Prediction versus Confidence Intervals

A confidence interval bounds an estimated parameter; a prediction interval bounds a future observation and is always wider.

A distinction people conflate

Confidence intervals and prediction intervals answer different questions and have different widths. Mixing them up leads to error bars that are too narrow for their stated purpose. The difference is whether you are bounding an underlying quantity or an individual future outcome.

Confidence interval

Kronos motion — parameter scan

A confidence interval bounds a fixed but unknown parameter - a mean, a regression coefficient, a model prediction of the average response. A ninety-five percent confidence interval means that if the experiment were repeated many times, ninety-five percent of the intervals so constructed would contain the true parameter. It captures uncertainty in the estimate, which shrinks as data accumulates.

Prediction interval

A prediction interval bounds a single new observation, not a parameter. It must account for both the uncertainty in the estimated model and the inherent random scatter of an individual outcome around the mean. Because it adds this irreducible noise, a prediction interval is always wider than the corresponding confidence interval - and it does not shrink to zero even with infinite data.

The width relationship

Confidence interval width shrinks toward zero as sample size grows
Prediction interval width approaches the noise standard deviation, never zero
Prediction interval variance is roughly confidence variance plus observation variance

In UQ terms

The distinction maps onto epistemic and aleatoric uncertainty. A confidence interval reflects epistemic uncertainty about the model, which more data reduces. A prediction interval adds aleatoric noise, which more data does not reduce. A Gaussian process makes this explicit: its posterior mean has a confidence band, and adding the noise variance gives the prediction band.

Which to report

Report a prediction interval when the audience cares about an individual future outcome - the value a real machine parameter will take on a given shot. Report a confidence interval when the question is about the average or the model's estimate. In Kronos results, the choice is stated explicitly so the error bar matches the claim being made.