Introduction

We often see published results in the form

$$\mbox{\lq best value'} ^{+\Delta_+}_{-\Delta_-} ,$$

where $\Delta_+$ and $\Delta_-$ are usually positive.¹ As firstly pointed out in Ref. [2] and discussed in a simpler but more comprehensive way in Ref. [3], this practice is far from being acceptable and, indeed, could bias the believed value of important physics quantities. The purpose of the present paper is, summarizing and somewhat completing the work done in the above references, to remind where asymmetric uncertainty stem from and to show why, as they are usually treated, they bias the value of physical quantities, either in the published result itself or in subsequent analyses. Once the problems are spotted, the remedy is straightforward, at least within the Bayesian framework (see e.g. [3], or [4] and [5] for recent reviews). In fact the Bayesian approach is conceptually based on the intuitive idea of probability, and formally grounded on the basic rules of probability (what are usually known as the probability `axioms' and the `conditional probability definition') plus logic. Within this framework many methods of `conventional' statistics are reobtained, as approximations of general solutions, under well stated conditions of validity. Instead, in the conventional, frequentistic approach ad hoc formulae, prescriptions and un-needed principles are used, often without understanding what is behind these methods - before a `principle' there is nothing!

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 $\chi ^2$ 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.