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I have a data set of 2000 $[x, F(x), \delta F(x)]$ triples, where $x$ is exact and $F$ is a measured value with an uncertainty $\delta F$. I can interpolate/fit the function however needed, and this is fine because while the interpolated values can deviate from the measured values, if the fit is good it will not often or ever be outside it's own error bars. However in order to properly propagate uncertainties I need to have an interpolated uncertainty on my interpolated data.

Naively my thought was to treat $\delta F$ as a function and interpolate it using the same method as the $F$. However my issue with this is that unless I grossly overfit, many of my current 2000 uncertainties will deviate from their correct values to smaller or larger values just as usual in data fitting. Is this okay? Allowing uncertainties to change seems dishonest. Is there a better way to interpolate errors?

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My thought would be if you are far away from any points, you should pay a penalty in the certainty. In other words, error should be smallest at the anchor points, and when you are interpolating, the farther away from measured place from both ends, the less certain you are of the values.

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