... positive.¹

For examples of measurements having $\Delta_+$ and $\Delta_-$ with all combinations of signs, see public online tables of Deep Inelastic Scattering results.[1] I want to make clear since the very beginning that it is not my intention to blame experimental or theoretical teams which have reported in the past asymmetric uncertainty, because we are all victims of a bad tradition in data analysis. At least, when asymmetric uncertainties have been given, there is some chance to correct the result, as described in Sec. 4. Since some asymmetric contributions to the global uncertainties almost unavoidably happen in complex experiments, I am more worried of collaborations that never arrive to final asymmetric uncertainties, because I must imagine they have symmetrised somehow the result but, I am afraid, without applying the proper shifts to the `best value' to take into account asymmetric contributions, as it will be discussed in the present paper.

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... implicitly²

Perhaps the reader would be surprised to learn that in the conventional statistical approach there is no room for probabilistic statements about the value of physics quantities (e.g. ``the top mass is between 170 and 180 GeV with such percent probability'', or ``there is 95% probability that the Higgs mass is lighter than 200 GeV''), calibration constants, and so on, as discussed extensively in Ref. [8].

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... results³

The reader might be curious to know what would happen in case of bad combinations of input quantities with skewness of mixed signs. Clearly there will be some compensation that lowers the risk of strong bias. As an academic exercise, let think of five independent variables each described by the triangular distribution of Fig. 1 and five others each described by a p.d.f. which is its mirror reflexed around $x=0.5$ [$0\le X\le 2$, $\mbox{mode}(X)=0.5$, $\mbox{E}[X] = 0.83$ and $\sigma(X)=0.42$]. The correct combination of the ten variables gives $Y=5.00\pm 1.33$, while adding the modes and combining quadratically left and right deviations we would get $5.00\pm 1.54$.

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... function⁴

But not yet a probability function! The likelihood has the probabilistic meaning of a joined p.d.f. of the data given $\theta$, and not the other way around.

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... on.⁵

$\theta$ has not a probabilistic interpretation in the frequentistic approach, and therefore we cannot speak consistently, in that framework, about its probability, or determine expectation, standard deviation and so on. Most physicists do not even know of this problem and think these are irrelevant semantic quibbles. However, it is exactly this contradiction between intuitive thinking and cultural background[8] that causes wrong scientific conclusions, like those discussed in this paper.

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... exception.⁶

It is a matter of fact that the habit in the particle physics community of applying uncritically the $\Delta \chi^2 = 1$ or $\Delta \varphi = 1/2$ is related to the use of the software package MINUIT[10]. Indeed, MINUIT can calculate the parameter variances also from the $\chi ^2$ or $\varphi$ curvature at the minimum (that relies on the same hypothesis upon which the $\Delta \chi^2 = 1$ or $\Delta \varphi = 1/2$ rules are based). But when the $\chi ^2$ or $\varphi$ are no longer parabolic, the standard deviation calculated from the curvature differs from that of the $\Delta \chi^2 = 1$ or $\Delta \varphi = 1/2$ (in particular, when the minimized function is asymmetric the latter rules give two values, the (in-)famous $\Delta_\pm$ we are dealing with). People realize that the curvature at the minimum depends from the local behavior of the minimized curve, and the $\Delta \chi^2 = 1$ or $\Delta \varphi = 1/2$ rule is typically more stable. Therefore, in particle physics the latter rule has become de facto a standard to evaluate `confidence intervals' at different `levels of confidence' (depending of the value of the $\Delta \chi^2$ or $\Delta \varphi$). But, unfortunately, when those famous curves are not parabolic, numbers obtained by these rules might loose completely a probabilistic meaning. [Sorry, a frequentist would object that, indeed, these numbers do not have probabilistic meaning about $\theta$, but they are `confidence intervals' at such and such `confidence level', because `$\theta$ is a constant of unknown value', etc...Good luck!]

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... problem.⁷

To be precise, this approximation is valid if the parameters appear only in the argument of the exponent. In practice this means that the fitted parameters must not appear in the covariance matrix on which the $\chi ^2$ depends. As a simple example in which this approximation do not hold is that of a linear fit in which also the standard deviation $\sigma$ describing the errors along the ordinate. The joint inference about line coefficients $m$ and $c$ and $\sigma$, having observed $n$ points, is achieved by $f(m,c,\sigma)\propto \sigma^{-n} e^{-\chi^2/2}$ (see Sec. 8.2 of Ref. [3]).

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... parameters.⁸

See footnote 7 concerning a possible pitfall in the use of $k e^{-\chi^2/2}$.

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... numerically⁹

Note that sometimes people do not get asymmetric uncertainty, not because the propagation is approximately linear, but because asymmetry is hidden by the standard propagation formula! Therefore also in this case the approximation might produce a bias in the result (for example, the second order formula of the expected value of the ratio of two quantities is known to experts[12]). The merit of numerical derivatives is that at least it shows the asymmetries.

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... deviations¹⁰

In terms of analytically calculated derivatives, $\delta$ and $\overline{\Delta}$ are given by

$$\delta = \frac{1}{2}\left.\frac{\partial^2 Y} {\partial X^2}\right\vert _{\mbox{\small E}[X]} \sigma^2(X)$$ (19)

$$\overline{\Delta} = \left.\frac{\partial Y} {\partial X}\right\vert _{\mbox{\small E}[X]} \sigma(X) .$$ (20)

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... variable.¹¹

After what we have seen in Sec. 2 we should not forget that the input quantities could have non trivial shapes. Since skewness and kurtosis are related to 3rd and 4th moment of the distribution, Eq. (22) makes use up to the 4th moment and is definitely better that the usual propagation formula, that uses only second moments. In Ref. [2] approximated formulae are given also for skewness and kurtosis of the output variable, from which it is possible to reconstruct $f(y)$ taking into account up to 4-th order moment of the distribution.

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... attempts¹²

It has been studied by psychologists how sometimes our efforts to solve a problem are the analogous with the moves along elements of a group structure (in the mathematical sense). There is no way to reach a solution until we not break out of this kind of trapping psychological or cultural cages.[13]

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... too.¹³

In this special case there should be no doubt that a shift should be applied to the best value, since moving $X_i$ by $\pm \sigma(X_i)$ around its expected value $\mbox{E}[X_i]$ the final quantity $Y$ only moves in one side of $Y(\mbox{E}[X_i])$.

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... become¹⁴

The ISO Guide [14] recommends to give the result using the standard deviation within parenthesis, instead of using the $\pm xx$ notation. In this example we would have $Y\approx 4.69 (1.5) (0.62) = 4.69 (1.62) \Rightarrow Y\approx 4.7 (1.6)$. Personally, I do not think this is a very important issue as long as we know what the quantity $xx$ means. Anyhow, I understand the ISO rational, and perhaps the proposed notation could help to make a break with the `confidence intervals'.

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...eq:exact_corr_sigma).¹⁵

The slight difference between the standard deviations comes from rounding, since $\sigma(\mu)=1.5$ of Fig. 2 is the rounded value of 1.54. Replacing 1.5 by 1.54 in Eq. (38), we get exactly the Monte Carlo value of 1.65.

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... uncertainties¹⁶

Discussing this issues with several persons I have realized, with my great surprise, that this misconception is deeply rooted and strenuously defended by many colleagues, even by data analysis experts (they constantly reply ``yes, but...''). This attitude is probably one of the consequences of being anchored to what I call un-needed principles (namely maximum likelihood, in this case), such that even the digits resulting from these principles are taken with a kind of religious respect and it seems blasphemous to touch them.

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... intervals,¹⁷

I give the central 68.3% interval with some reluctance, because I know by experience that in many minds the short circuit

$$\mbox{\lq\lq 68\% probability interval'' $\longleftrightarrow$ \lq\lq sigma''}$$

is almost unavoidable (I have known physicists convinced - and who even taught it! - that the standard deviation only `makes sense for the Gaussian' and that it was defined via the `68% rule'). For this reason, recently I have started to appreciate thinking in terms of 50% probability intervals, also because they force people to reason in terms of better perceived fifty-to-fifty bets. I find these kind of bets very enlighting to show why practically all standard ways (including Bayesian ones!) fail to report upper/lower confidence limits in frontier case situations characterized by open likelihoods (see chapter 12 in Ref.[3]). I like to ask ``please use your method and give me a 50% C.L. upper/lower limit'', and then, when I have got it, ``are you really 50% confident that the value is below that limit and 50% confident that it is above it? Would you equally bet on either side of that limit?''. And the supporters of `objective' methods are immediately at loss. (At least those who use Bayesian formulae realize that there must be some problem with the choice of priors.)

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... case¹⁸

In the example here we have been lucky because an over-correction of the first contribution was compensated by an under-correction of the second contribution. Note also that the hypothesis about the nonlinear propagation was not correct, because we had, instead, a linear propagation of asymmetric p.d.f.'s. Anyhow the overall shift calculated by the guessed hypothesis is comparable to that calculable knowing the details of the analysis (and, in any case, using in subsequent analyses the roughly corrected result is definitely better than sticking to the published `best value').

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... result.¹⁹

Note that even if we were told that $Y$ was $6.0^{+1.0}_{-2.2}$, without further information, we could still try to apply some shift to the result, obtaining $4.8\pm 1.6$ or $5.4\pm 1.6$ depending on some guesses about the source of the asymmetry. In any case, either results are better than $6.0^{+1.0}_{-2.2}$!

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