An idea well rooted among physicists, especially nuclear and particle
physicists, is that the result of a measurement must be reported
with a corresponding *uncertainty*. What makes the
measured values subject to a degree of uncertainty
is, it is commonly said,
the effect of unavoidable measurement *errors*,
usually classified as *random* (or *statistical*) and
*systematic*
^{(7)} .

Uncertainties due to statistical errors are commonly treated using the frequentist concept of confidence intervals, although the procedure is so unnatural that the interpretation of the result is unconsciously subjective (as will be shown in a while), and there are known cases (of great relevance in frontier research) in which this approach is not applicable.

As far as uncertainties due to systematics errors are concerned,
there is no conventional consistent theory to handle them,
as is also indirectly recognized by the ISO *Guide*
[10].
The ``fashion'' at the moment is to add them quadratically if they are
considered to be independent, or to build a covariance matrix if not.
This procedure is not justified theoretically (in the frequentist approach)
and I think that it is used essentially because of the reluctance of
experimentalists to add linearly the dozens
of contributions of a complicated HEP measurement, as the old fashioned
``theory'' of maximum errors
suggests doing ^{(8)} .
The pragmatic justification for the
quadratic combination of ``systematic errors'' is that one is using
a rule (the famous ``error propagation''
formula ^{(9)} )
which is considered to be valid at least for ``statistical
errors''. But, in reality, this too is not correct.
The use of this formula is again arbitrary in the case
of ``statistical errors'', if these have been
evaluated from confidence intervals
^{(10)} .
In fact, there is no logical reason
why a probabilistic procedure proved for standard deviations of
random variables (the observables) should also be valid
for 68% confidence intervals, which is considered, somehow,
an uncertainty attributed to the true value.

These examples show quite well the
contradiction between
the cultural background on probability and the practical good sense
of physicists. Thanks to this good sense, frequentist ideas are
constantly violated, with the positive effect that at least
some results are obtained
^{(11)} .
It is interesting to notice that in simple routine applications
these results are very close, both in value and in meaning, to those
achievable starting from what I consider to be the
correct point of view for handling
uncertainty (subjective probability). There are, on the other hand,
critical cases in which scientific conclusions may be seriously
mistaken. Before discussing these cases, let us look more closely
at the terms of the claimed contradiction.

^{7} This last statement may
sound like a tautology, since ``error'' and
``uncertainty'' are often used as synonyms. This
hints to the fact that in this subject there is
neither uniformity of language, nor of methods, as is recognized
by the metrological organizations, which have made
great efforts to bring some order into the
field [8,
9,
10,
11,
12].
In particular, the International Organization for Standardization
(ISO) has published *``Guide to the expression of uncertainty
in measurement''*
[10],
containing definitions, recommendations
and practical examples. For example, *error* is defined
as ``the result of a measurement minus a true value of the measurand''
*uncertainty* ``a parameter, associated with the result of a measurement,
that characterize the dispersion of the values that could reasonably be
attributed to measurand'', and, finally, *true value* ``a value
compatible with the definition of the given particular quantity''.
One can easily see that it is not just a question of practical
definitions. It seems to me that
there is a well-thought-out philosophical
choice behind these definitions,
although it is not discussed extensively in the *Guide*.
Two issues
in the *Guide* that I find of particular importance are the
discussion on the sources of uncertainty and the admission
that all contributions to the
uncertainty are of a probabilistic nature.
The latter is strictly related to the
subjective interpretation of probability, as
admitted by the *Guide* and discussed in depth
in [7].
(The reason why these comments on the ISO *Guide* have been
placed in this long footnote is that, unfortunately,
the *Guide* is not yet
known in the HEP community and, therefore, has no influence
on the behaviour of HEP physicists about which I am going
to comment here.
This is also the reason why I will often use in this paper
typical expressions currently used in HEP and
which are in disagreement with the ISO recommendations.
But I will use these expressions preferably within quote marks,
like ``systematic error'' instead of ``uncertainty due to
a recognized systematic error of unknown size''.)
Back.

^{8} In fact, one can see that when there
are only 2 or 3
contributions to the ``systematic error'', there are still people who
prefer to add them linearly.
Back.

^{9} The most well-known version is that in
which correlations are neglected:

*Y* stands for the quantity of interest, the value
of which depends
on directly measured quantities,
calibration constants and other
systematic effects (all terms generically indicated by *X*_{i}).
This formula
comes from probability theory,
but it is valid if *X*_{i} and *Y* are
random variables,
(*X*_{i}) are
standard deviation and the linearization is reasonable.
It is very interesting to look
at text books to see how this formula is derived.
The formula is usually initially proofed referring to random variables
associated to observables and then, suddenly, it is referred
to physics quantities, without any justification.
Back.

^{10} As
far as ``systematic errors'' are concerned the situation
is much more problematic because the ``errors'' are not even
operationally well defined: they may correspond to
subjectivist standard deviations (what I consider to be correct,
and what corresponds to the ISO *type B* standard uncertainty
[10]),
but they can more easily be maximum deviations, ±50% variation
on a selection cut, or the absolute difference obtained using two
assumptions for the systematic effect.
Back.

^{11} I am strongly convinced
that a rigorous application of frequentist ideas leads nowhere.
Back.