What is Around assuming when propagating asymmetric uncertainties?

Question

When exploring Around and others related, there's not much documentation when it comes to propagation of asymmetric uncertainties.

I've tried calculating

f[a_,b_]= ab
f[Around[a,{al,au}],Around[b,{bl,bu}]]

where al,au comes from alower and aupper uncertainties, respectively. The result is

$$ab^{+\sqrt{au^2 b^2 + a^2 bu^2}}_{-\sqrt{al^2 b^2 + a^2 bl^2}}$$

Immediately my question is: Why?

I guess that Around propagates $a^{+au}_{-al}$ and $b^{+bu}_{-bl}$ calculating a simple error propagation, i.e., $\delta f^2 \sim \sum\limits_i \left(\frac{\partial f}{\partial x_i} \delta x_i\right)^2$ for the upper and lower uncertainties in a separated way.

Q: What are we assuming here besides a first order linear approximation? How could this be statistically correct? (as I don't know the distributions for a and b)

My question is also related to this and might also be linked to this other.

I'm confused by the way(s) or methods we use to propagate asymmetric uncertainties, specifically putting aside MC methods. As Mathematica offers a solution, Q: I wonder what are we assuming and also (related) what is its mathematical fundament and validity.

Answer 1

You are right with your assumption that Mathematica treats the upper and lower uncertainties seperately and also uses standard first order gaussian error propagation $\sigma=\sqrt{\sum_i\left(\frac{\partial f}{\partial x_i}\sigma_i\right)^2}$.

The assumption taken by Mathematica here can easily be summarized by looking at the assumed distribution:

Around[\[Mu],{\[Sigma]m,\[Sigma]p}]["Distribution"]

SplicedDistribution[{1/2, 1/ 2}, {-[Infinity], [Mu], [Infinity]}, {NormalDistribution[[Mu], \ [Sigma]m], NormalDistribution[[Mu], [Sigma]p]}]

Plotted with some example values this looks like this:

Essentially: we assume one half of the gauss distribution for each side from the mean/median with a standard deviation given by the upper or respective lower error. Thus, it makes sense to continue using the standard approach separately for the upper and lower error since it models the gaussian distribution transformations reasonably well for most inputs (and you are anyway only interested on its effect of one side of the mean).

Sure, one can actually assume an asymmetric Distributions and try to propagate errors with it (TransformedDistributionis your friend here), but usually one runs into non-solvable moments of the transformed distribution pretty fast.

So this is a reasonable approach I think.

Note btw, that that the order of the uncertainties can get reversed under certain transformations (I discovered a bug in these routines a while back and it was fixed in 12.3 afaik):

-Around[a, {lower, upper}]

Around[-a, {upper, lower}]