Uncertainty due to systematics

Finally, and this is often the case that we see in publications, asymmetric uncertainty results from systematic effects. The Bayesian approach offers a natural and clear way to treat systematics - and I smile at the many attempts¹² of `squaring the circle' using frequentistic prescriptions...- simply because probabilistic concepts are consistently applied to all influence quantities that can have an effect on the quantity of interest and whose value is not precisely known. Therefore we can treat them using probabilistic methods. This was also recognized by metrologic organizations[14].

Indeed, there is no need to treat systematic effects in a special way. They are treated as any of the many input quantities ${\mbox{\boldmath$X$}}$ discussed in Sec. 3.2, and, in fact, their asymmetric contributions come frequently from their nonlinear influence on the quantity of interest. The only word of caution, on which I would like to insist, is to use expected value and standard deviation for each systematic effect. In fact, sometimes the uncertainty about the value of the influence quantities that contribute to systematics is intrinsically asymmetric.

I also would like to comment shortly on results where either of the $\Delta_\pm$ is negative, for example $1.0^{+0.5}_{+0.3}$ (see e.g. Ref. [1] to have an idea of the variety of signs of $\Delta_\pm$). This means that that the we are in proximity of a minimum (or a maximum if $\Delta_+$ were negative) of the function $Y=Y(X_i)$. It can be shown [2,3] that Eqs. (21)-(22) hold for this case too.¹³

For further details about meaning and treatment of uncertainties due systematics and their relations to ISO Type B uncertainties[14], see Refs. [2] and [3].