Error propagation with different plus and min errors and data fitting

I am refreshing my memory on error propagation and data fitting (Levenberg-Marquadt). You have the absolute (measurement) error, the relative (measurement) error, the population/sample standard deviation and the population/sample standard error.

These are typically written down as:

$$a \pm b = a (+b, -b).$$ But suppose you have different errors in the plus and min direction?

$$a (+b, -c).$$

What are the rules here again and how are the formula for error propagation modified? Also, how is this taken into account in data fitting? Any good resource on this is welcome.

• Interesting question. Asymmetric errors seem terribly complicated compared to our familiar errors, see page 28: phas.ubc.ca/~oser/p509/Lec_10.pdf . Dec 9, 2014 at 19:15
• "Asymmetric errors: Quite honestly, the typical physicist doesn't have a clue." That's me alright. :( Dec 9, 2014 at 20:31
• @jinawee The resource you posted is interesting, I had not come across the Barlow method. However, at least for the examples given, the results are extremely similar to propagating the positive and negative uncertainties separately. Are the differences worth worrying about unless you are very certain of the form of the pdf? But if you were, why use error propagation? Need to read the Barlow paper. arxiv.org/pdf/physics/0406120v1.pdf Dec 9, 2014 at 23:22

I think what you could do is treat the quoted value as the most likely value and the two uncertainties as semi-gaussian probability distributions.

So for example in error propagation you would have to propagate the upper and lower limits separately using the usual Gaussian error propagation formulae but keeping track of whether these propagate to the upper or lower error estimate in the thing you are trying to calculate.

e.g. the uncertainties in $a^2$ would be $+2ab$, $-2ac$, but the uncertainties in $a^{-1}$ would be $+c/a^2$, $-b/a^2$.

I am not certain (no pun intended) that this procedure is correct unless you are absolutely sure that each side of the measurement uncertainty distribution is accurately represented by a semi-Gaussian. A more general technique is discussed by Barlow (2004). I have not fully digested this, but the examples appear to be based on Poisson distributions, which are of course inherently non-Gaussian for small numbers of events.

If you knew what the actual probability distribution functions were then the best way to proceed would surely be to do a Monte-Carlo simulation.

For calculating chi-squared values in data fitting you either use the positive or negative errors depending on whether the deviation from the model is negative or positive respectively. Again, only partly satisfactory because the asymmetric error bar would usually indicate that the probability distribution of the measurement is asymmetric and thus inherently non-Gaussian.

A better (though still frequentist) route to fitting a model to such data would involve maximising the likelihood function, which is calculated by integrating over the actual probability distribution of the measurements (if you know it, or at least a better representation of it than two half Gaussians).

Monte Carlo integration would again be the method of choice if your life (or career) depended on whether the resultant confidence intervals in the parameters were 68% or 75%.

After fitting the dataset $$(x_i, y_i)$$ with the function $$\hat y_i = f(x_i, \vec \beta)$$, where $$\vec \beta$$ represent the fit parameters, we are able to calculate the fit residuals $$r_i = y_i - \hat y_i$$. Analysing the fit residuals $$r_i$$ is not only the key to optimise your fit, but it also yields the error distribution. Instead of using the fit parameters and their uncertainty, I recommend using the error distribution to propagate the error to your final quantity of interest $$g(y_i)$$. My argument is the following: Given that the fit parameters possess asymmetric uncertainties the error distribution is skewed. However, a skewed distribution is not fully described by the first two moments of the distribution (=average value and standard deviation). Using two different standard deviation (below and above the average value) is a first estimate for using a skewed distribution, however, once you learn how to simulate uncertainties using the residuals you will never go back and do analytic calculations again. The simulations are not only more precise, but also simple and less error-prone.

• "once you learn how to simulate uncertainties using the residuals you will never go back and do analytic calculations again" Do you a primer on this? Feb 25, 2023 at 8:16