Propagating uncertainty

If uncertainty is quantified by probability, as it is commonly done
explicitly or implicitly^{2} in physics, the propagation
of uncertainty is performed using rules based on probability theory.
If we indicate by
the set
(`vector') of input quantities and by the
final quantity, given by the function
of the
input quantities, the most general propagation formula
(see e.g. [3])
is given by (we stick to continuous variables):

As it is also well known, often there is no need to go through the
analytic, numerical or Monte Carlo evaluation of Eq.(1),
since linearization of
around the expected value
of
(E[
]) makes the calculation of
expected value and variance of very easy, using the well known
standard propagation formulae, that for uncorrelated input quantities are

As far as the shape of , a Gaussian one is usually assumed, as a result of the central limit theorem. Holding this assumptions, and is all what we need. gives the `best value', and probability intervals, upper/lower limits and so on can be easily calculated. In particular, within the Gaussian approximation, the most believable value (

Anyhow, Gaussian approximation is not the main issue here and, in most real applications, characterized by several contributions to the combined uncertainty about , this approximation is a reasonable one, even when some of the input quantities individually contribute asymmetrically. My concerns in this paper are more related to the evaluation of and when

- instead of
Eqs. (2)-(3),
*ad hoc*propagation prescriptions are used in presence of asymmetric uncertainties; - linearization implicit in Eqs. (2)-(3) is not a good approximation.

*asymmetric uncertainties added in quadrature*: ;*asymmetric uncertainties added linearly*: .

The situation would have been much better if
expected value and standard deviation of and
had been reported (respectively 0.17 and 0.42). Indeed, these
are the quantities that
matter in `error propagation', because the *theorems upon which
propagation formulae rely* -- exactly in the case is a linear combination
of
, or approximately in the case linearization has been performed --
*speak of expected values and variances*.
It is easy to verify from the numbers in Fig. 1
that exactly the correct values of
and
would have been obtained.
Moreover, one can see that
expected value, mode and median of do not differ much from
each other, and the shape of resembles a somewhat
skewed Gaussian. When will be combined with other quantities
in a next analysis its slightly non-Gaussian shape
will not matter any longer. Note that we have achieved this nice
result already with only two input quantities. If we had a few
more,
already would have been much Gaussian-like. Instead,
performing a bad combination of several quantities all skewed in the
same side would yield `divergent'
results^{3}:
for we get,
using a quadratic combination of left and right deviations,
versus the correct
.

As conclusion from this section I would like to make some points:

- in case of asymmetric uncertainty on a quantity,
it should be avoided to report
*only*most probable value and a probability interval (be it 68.3%, 95%, or what else); - expected value, meant as barycenter of the distribution,
as well as standard deviations should
*always*be reported, providing also the shape of the distribution (or its summary in terms of shape parameters, or even a parameterization of the log-likelihood function in a polynomial form, as done e.g. in Ref. [9]), if the distribution is asymmetric or non trivial.

Hoping that the reader is, at this point, at least worried about the effects of badly treated asymmetric uncertainties, let us now review the sources of asymmetric uncertainties.