where and are

The proposed Bayesian solutions to cure the troubles produced by the usual treatment of asymmetric uncertainties is to step up from approximated methods to the more general ones (see e.g. Ref. [3], in particular the top down approximation diagram of Fig. 2.2). In this paper we shall see, for example, how and minus log-likelihood fit `rules' can be derived from the Bayesian inference formulae as approximated methods and what to do when the underlying conditions do not hold. We shall encounter a similar situation regarding standard formulae to propagate uncertainty.

Some of the issues addressed here and in Refs. [2] and [3] have been recently brought to our attention by Roger Barlow [6], who proposes frequentistic ways out. Michael Schmelling had also addressed questions related to `asymmetric errors', particularly related to the issue of weighted averages [7]. The reader is encouraged to read also these references to form his/her idea about the spotted problems and the proposed solutions.

In Sec. 2 the issue of propagation of
uncertainty is briefly reviewed at an elementary level
(just focusing on the sum of
uncertain independent variables - i.e.
no correlations considered) though taking into account
asymmetry in probability density functions (p.d.f.) of the
*input* quantities. In this way we understand
what `might have been done'
(we are rarely in the positions to exactly know ``what has been done'')
by the authors who publish
asymmetric results and what is the danger of improper use of
such a published `best value' - *as is* - in subsequent analyses.
Then, Sec. 3 we shall see in
where asymmetric uncertainties stem from and what to do in order
to overcome their potential
troubles. This will be done in an exact way and, whenever is possible,
in an approximated way.
Some rules of thumb to roughly recover sensible probabilistic
quantities (expected value and standard deviation) from
results published with asymmetric uncertainties
will be given in Sec. 4.
Finally, some conclusions will be drawn.