... length.1.1
But after observation of the first sequence one would strongly suspect that the coin had two heads, if one had no means of directly checking the coin. The concept of probability will be used, in fact, to quantify the degree of such suspicion.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... measurand.1.2
It is then clear that the definition of true value implying an indefinite series of measurements with ideal instrumentation gives the illusion that the true value is unique. The ISO definition, instead, takes into account the fact that measurements are performed under real conditions and can be accompanied by all the sources of uncertainty in the above list.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... systematic errors1.3
To be more precise one should specify of unknown size', since an accurately assessed systematic error does not yield uncertainty, but only a correction to the raw result.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... all''.1.4
By the way, it is a good and recommended practice to provide the complete list of contributions to the overall uncertainty[3]; but it is also clear that, at some stage, the producer or the user of the result has to combine the uncertainty to form his idea about the interval in which the quantity of interest is believed to lie.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... require.1.5
And in fact, one can see that when there are only two or three contributions to the systematic error', there are still people who prefer to add them linearly.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... justification.1.6
Some others, including some old lecture notes of mine, try to convince the reader that the propagation is applied to the observables, in a very complicated and artificial way. Then, later, as in the game of the three cards' proposed by professional cheaters in the street, one uses the same formulae for physics quantities, hoping that the students do not notice the logical gap.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... were1.7
There are also those who express the result, making the trivial mistake of saying this means that, if I repeat the experiment a great number of times, then I will find that in roughly 68% of the cases the observed average will be in the interval . (Besides the interpretation problem, there is a missing factor of in the width of the interval ...)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... that1.8
The capital letter to indicate the average appearing in () is used because here this symbol stands for a random variable, while in () it indicated a realization of it. For the Greek symbols this distinction is not made, but the different role should be evident from the context.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... value''1.9
It is worth noting the paradoxical inversion of role between , about which we are in a state of uncertainty, considered to be a constant, and the observation , which has a certain value and which is instead considered a random quantity. This distorted way of thinking produces the statements to which we are used, such as speaking of uncertainty (or error) on the observed number'': If one observes 10 on a scaler, there is no uncertainty on this number, but on the quantity which we try to infer from the observation (e.g. of a Poisson distribution, or a rate).
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...fig:htest). 1.10
At present,-values' (or significance probabilities') are also used in place of hypothesis tests as a means of giving more information about the relationship between the data and the hypothesis than does a simple reject/do not reject decision''[9]. They consist in giving the probability of the tail(s)', as also usually done in HEP, although the name -values' has not yet entered our lexicon. Anyhow, they produce the same interpretation problems of the hypothesis test paradigm (see also example 8 of next section).
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... prevision1.11
By prevision I mean, following [11], a probabilistic prediction', which corresponds to what is usually known as expectation value (see Section ).
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... question,1.12
Personally, I find it is somehow impolite to give an answer to a question which is different from that asked. At least one should apologize for being unable to answer the original question. However, textbooks usually do not do this, and people get confused.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... journal.1.13
Example taken from Ref. [12].
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... true.1.14
This should not be confused with the probability of the actual data, which is clearly 1, since they have been observed.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... healthy!1.15
The result will be a simple application of Bayes' theorem, which will be introduced later. A crude way to check this result is to imagine performing the test on the entire population. Then the number of persons declared Positive will be all the HIV infected plus of the remaining population. In total 100 000 infected and 120 000 healthy persons. The general, Bayesian solution is given in Section
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... was:1.16
One might think that the misleading meaning of that sentence was due to unfortunate wording, but this possibility is ruled out by other statements which show clearly a quite odd point of view of probabilistic matter. In fact the DESY 1998 activity report [14] insists that the likelihood that the data produced are the result of a statistical fluctuation ...is equivalent to that of tossing a coin and throwing seven heads or tails in a row'' (replacing probability' by likelihood' does not change the sense of the message). Then, trying to explain the meaning of a statistical fluctuation, the following example is given: This process can be simulated with a die. If the number of times a die is thrown is sufficiently large, the die falls equally often on all faces, i.e. all six numbers occur equally often. The probability for each face is exactly a sixth or 16.66%, assuming the die is not loaded. If the die is thrown less often, then the probability curve for the distribution of the six die values is no longer a straight line but has peaks and troughs. The probability distribution obtained by throwing the die varies about the theoretical value of 16.66% depending on how many times it is thrown.''
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... corner.1.17
One of the odd claims related to these events was on a poster of an INFN exhibition at Palazzo delle Esposizioni in Rome: These events are absolutely impossible within the current theory ... If they will be confirmed, it will imply that....'' Some friends of mine who visited the exhibition asked me what it meant that something impossible needs to be confirmed''.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... observed?''1.18
This is as if the conclusion from the AIDS test depended not only on and on the prior probability of being infected, but also on the probability that this poor guy experienced events rarer than a mistaken analysis, like sitting next to Claudia Schiffer on an international flight, or winning the lottery, or being hit by a meteorite.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... objection.1.19
I must admit I have fully understood this point only very recently, and I thank F. James for having asked, at the end of the CERN lectures, if I agreed with the sentence The probability of data not observed is irrelevant in making inferences from an experiment.''[10] I was not really ready to give a convincing reply, apart from a few intuitions, and from the trivial comment that this does not mean that we are not allowed to use MC data (strictly speaking, frequentists should not use MC data, as discussed in Section ). In fact, in the lectures I did not talk about data+tails', but only about data'. This topic will be discussed again in Section .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... notes.1.20
As an exercise, to compare the intuitive result with what we will learn later, it may be interesting to try to calculate, in the second case of the previous example ( ), the value such that we would be in a condition of indifference (i.e. probability 50% each) with respect to the two generators.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... centuries.2.1
For example, it is interesting to report Einstein's opinion [17] about Hume's criticism: Hume saw clearly that certain concepts, as for example that of causality, cannot be deduced from the material of experience by logical methods. Kant, thoroughly convinced of the indispensability of certain concepts, took them - just as they are selected - to be necessary premises of every kind of thinking and differentiated them from concepts of empirical origin. I am convinced, however, that this differentiation is erroneous.'' In the same Autobiographical Notes [17] Einstein, explaining how he came to the idea of the arbitrary character of absolute time, acknowledges that The type of critical reasoning which was required for the discovery of this central point was decisively furthered, in my case, especially by the reading of David Hume's and Ernst Mach's philosophical writings.'' This tribute to Mach and Hume is repeated in the gemeinverständlich' of special relativity [18]: Why is it necessary to drag down from the Olympian fields of Plato the fundamental ideas of thought in natural science, and to attempt to reveal their earthly lineage? Answer: In order to free these ideas from the taboo attached to them, and thus to achieve greater freedom in the formation of ideas or concepts. It is to the immortal credit of D. Hume and E. Mach that they, above all others, introduced this critical conception.'' I would like to end this parenthesis dedicated to Hume with a last citation, this time by de Finetti[11], closer to the argument of this chapter: In the philosophical arena, the problem of induction, its meaning, use and justification, has given rise to endless controversy, which, in the absence of an appropriate probabilistic framework, has inevitably been fruitless, leaving the major issues unresolved. It seems to me that the question was correctly formulated by Hume ... and the pragmatists ... However, the forces of reaction are always poised, armed with religious zeal, to defend holy obtuseness against the possibility of intelligent clarification. No sooner had Hume begun to prise apart the traditional edifice, then came poor Kant in a desperate attempt to paper over the cracks and contain the inductive argument -- like its deductive counterpart -- firmly within the narrow confines of the logic of certainty.''
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... probability.2.2
Perhaps one may try to use instead fuzzy logic or something similar. I will only try to show that this way is productive and leads to a consistent theory of uncertainty which does not need continuous injections of extraneous matter. I am not interested in demonstrating the uniqueness of this solution, and all contributions on the subject are welcome.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... probability,2.3
For an introductory and concise presentation of the subject see also Ref. [21].
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... taught2.4
This remark -- not completely a joke -- is due to the observation that most physicists interviewed are convinced that () is legitimate, although they maintain that probability is the limit of the frequency.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... mechanics2.5
Without entering into the open problems of quantum mechanics, let us just say that it does not matter, from the cognitive point of view, whether one believes that the fundamental laws are intrinsically probabilistic, or whether this is just due to a limitation of our knowledge, as hidden variables à la Einstein would imply. If we calculate that process has a probability of 0.9, and process 0.4, we will believe much more than .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... circularity2.6
Concerning the combinatorial definition, Poincaré's criticism [6] is remarkable:
The definition, it will be said, is very simple. The probability of an event is the ratio of the number of cases favourable to the event to the total number of possible cases. A simple example will show how incomplete this definition is: ...
...We are therefore bound to complete the definition by saying ... to the total number of possible cases, provided the cases are equally probable.' So we are compelled to define the probable by the probable. How can we know that two possible cases are equally probable? Will it be by convention? If we insert at the beginning of every problem an explicit convention, well and good! We then have nothing to do but to apply the rules of arithmetic and algebra, and we complete our calculation, when our result cannot be called in question. But if we wish to make the slightest application of this result, we must prove that our convention is legitimate, and we shall find ourselves in the presence of the very difficulty we thought we had avoided.''
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... arbitrary2.7
Perhaps this is the reason why Poincaré [6], despite his many brilliant intuitions, above all about the necessity of the priors (there are certain points which seem to be well established. To undertake the calculation of any probability, and even for that calculation to have any meaning at all, we must admit, as a point of departure, an hypothesis or convention which has always something arbitrary on it ...), concludes to ... have set several problems, and have given no solution ...''. The coherence makes the distinction between arbitrariness and subjectivity' and gives a real sense to subjective probability.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... recognized2.8
One should feel obliged to follow this recommendation as a metrology rule. It is however remarkable to hear that, in spite of the diffused cultural prejudices against subjective probability, the scientists of the ISO working groups have arrived at such a conclusion.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... less.2.9
To understand the role of implicit prior knowledge, imagine someone having no scientific or technical education at all, entering a physics laboratory and reading a number on an instrument. His scientific knowledge will not improve at all, apart from the triviality that a given instrument displayed a number (not much knowledge).
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... null.2.10
But also in this case we have learned something: the thermometer does not work.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... deduction'.2.11
To be correct, the deduction we are talking about is different from the classical one. We are dealing, in fact, with probabilistic deduction, in the sense that, given a certain cause, the effect is not univocally determined.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... observed.2.12
It is important to understand that can be evaluated before one knows the observed value . In fact, to be correct, should be interpreted as beliefs of under the hypothesis that is observed, and not only as beliefs of after is observed. Similarly, can also be built after the data have been observed, although for teaching purposes the opposite has been suggested, which corresponds to the most common case.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... of)2.13
Although I don't believe it, I leave open the possibility that there really is someone who has developed some special reasoning to avoid, deep in his mind, the category of the probable when figuring out the uncertainty on a true value.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... normal,2.14
In case of doubt it is recommended to plot it.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... considered2.15
For a general and self-contained discussion concerning the inference of the intensity of Poisson processes at the limit of the detector sensitivity, see Ref. [25].
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... events.2.16
As we shall see, the use of frequencies is absolutely legitimate in subjective probability, once the distinction between probability and frequency is properly made. In this case it works because of the Bernoulli theorem, which states that for a very large Monte Carlo sample it is very improbable that the frequency distribution will differ much from the p.d.f.'' (This is the probabilistic meaning to be attributed to tend'.)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... 2.17
For example, in the absence of random error the reading () of a voltmeter depends on the probed voltage () and on the scale offset (): . Therefore, the result from the observation of gives only a constraint between and :

If we know well (within unavoidable uncertainty), then we can learn something about . If instead the prior knowledge on is better than that on we can use the measurement to calibrate the instrument.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... errors.2.18
But, in order to give a well-defined probabilistic meaning to the result, the variations must be performed according to , and not arbitrary.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... integrate'2.19
Integrate' stands for a generic term which also includes the approximate method just described.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... notes3.1
Notes based on lectures given to graduate students in Rome (May 1995) and summer students at DESY (September 1995). The original report is Ref. [27]. In this report, notes (indicated by Note added' are used, either for clarification or to refer to those parts not contained in the original primer'.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... will3.2
The use of the future tense does not imply that this definition can only be applied for future events. Will occur'' simply means that the statement will be proven to be true'', even if it refers to the past. Think for example of the probability that it was raining in Rome on the day of the battle of Waterloo''.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... reasonable3.3
This is not always true in real life. There are also other practical problems related to betting which have been treated in the literature. Other variations of the definition have also been proposed, like the one based on the penalization rule. A discussion of the problem goes beyond the purpose of these notes.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... them3.4
We will talk later about the influence of a priori beliefs on the outcome of an experimental investigation.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... here3.5
Very important: for the meaning of the formula of conditional probability'' see Section
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... occurred3.6
should not be confused with , the probability that both events occur''. For example can be very small, but nevertheless very high. Think of the limit case

 given '' is a certain event no matter how small is, even if (in the sense of Section ).
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...after''3.7
Note that before'' and after'' do not really necessarily imply time ordering, but only the consideration or not of the new piece of information
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... probability3.8
The symbol could be misunderstood if one forgets that the proportionality factor depends on all likelihoods and priors [see ()]. This means that, for a given hypothesis , as the state of information changes, may change if and remain constant, and if some of the other likelihoods get modified by the new information.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... establishment''3.9
A case, concerning the search for electron compositeness in e e collisions, is discussed in Ref. [38].
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... moment3.10
For a recent delightful report, see Ref. [39].
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... results3.11
A theory needs to be confirmed by experiments. But it is also true that an experimental result needs to be confirmed by a theory.'' This sentence expresses clearly -- though paradoxically -- the idea that it is difficult to accept a result which is not rationally justified. An example of results not confirmed by the theory'' are the measurements in deep-inelastic scattering shown in Fig. . Given the conflict in this situation, physicists tend to believe more in QCD and use the low-'' extrapolations (of what?) to correct the data for the unknown values of .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... objectivity3.12
It may look paradoxical, but, due to the normative role of the coherent bet, the subjective assessments are more objective about using, without direct responsibility, someone else is formulae. For example, even the knowledge that somebody else has a different evaluation of the probability is new information which must be taken into account.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... Laplace3.13
It may help in understanding Laplace's approach if we consider that he called the theory of probability good sense turned into calculation''.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... distribution:]4.1
The symbols of the following distributions have the parameters within parentheses to indicate that the variables are continuous.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... deviation4.2
Mathematicians and statisticians prefer to take , instead of , as second parameter of the normal distribution. Here the standard deviation is preferred, since it is homogeneous to and it has a more immediate physical interpretation. So, one has to pay attention to be sure about the meaning of expressions like .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... methods5.1
This is conceptually what experimentalists do when they change all the parameters of the Monte Carlo simulation in order to estimate the systematic error''.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... variance5.2
Note added: for criticisms about the standard treatment of the small-sample problem see Ref. [].
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... notes5.3
Note added: as is easy to imagine, the problem of the outliers'' should be treated with care, and surely avoiding automatic prescriptions. Some hints can be found in Refs. [43] and [44], and references therein.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... mass5.4
In reality, often rather than m is normally distributed. In this case the terms of the problem change and a new solution should be worked out, following the trace indicated in this example.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... procedure,5.5
We consider detector and analysis machinery as a black box, no matter how complicated it is, and treat the numerical outcome as a result of a direct measurement[1].
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... probability''5.6
This concept, which is very close to the physicist's mentality, is not correct from the probabilistic -- cognitive -- point of view. According to the Bayesian scheme, in fact, the probability changes with the new observations. The final inference of , however, does not depend on the particular sequence yielding successes over trials. This can be seen in the next table where is given as a function of the number of trials , for the three sequences which give two successes (indicated by 1'') in three trials [the use of () is anticipated]:
 Sequence n 011 101 110 0 1 1 1 1 2 3
This important result, related to the concept of exchangeability, allows'' a physicist who is reluctant to give up the concept unknown constant probability'' to see the problem from his point of view, ensuring that the same numerical result is obtained.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... Assuming5.7
There is a school of thought according to which the most appropriate function is . If You think that it is reasonable for your problem, it may be a good prior. Claiming that this is the Truth'' is one of the many claims of the angels' sex determinations. For didactical purposes a uniform distribution is more than enough. Some comments about the prescription will be given when discussing the particular case .
Note added: criticisms concerning so called reference priors'' can be found in Ref.[].
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... Integrating5.8
It may help to know that

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... values6.1
The choice of the adjective raw'' will become clearer later on.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... completely6.2
Note added: for criticisms about the standard treatment of the small-sample problem see Ref. [].
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... quantity6.3
By input quantity'' the ISO Guide means any of the contributions or which enter into () and ().
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... deviation6.4
This example shows a type B uncertainty originated by random errors.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... methods6.5
Note added: This is exactly the presumed paradox reported by the 1998 issue of the PDG[46] as an argument against Bayesian statistics (Section 29.6.2, p. 175: If Bayesian estimates are averaged, they do not converge to the true value, since they have all been forced to be positive'').
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... variables6.6
To make the formalism lighter, let us call both the random variable associated with the quantity and the quantity itself by the same name (instead of ).
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... quantities6.7
In this section[47] the symbol will indicate the variable associated to the -th physical quantity and its -th direct measurement; the best estimate of its value, obtained by an average over many direct measurements or indirect measurements, the standard deviation, and the value corrected for the calibration constants. The weighted average of several will be denoted by .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... covariance6.8
Note added: The '' at the denominator of () is for the same reason as the '' of the sample standard deviation. Although I do not agree with the rationale behind it, this formula can be considered a kind of standard and, anyhow, replacing '' by '' has no effect in normal applications. As already said, in these notes I will not discuss the small-sample problem; anyone is interested in my worries concerning default formulae for small samples, as well as Student t distribution may have a look at Ref. [].
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... harmless6.9
This can be seen by rewriting () as

For any , the first two terms determine the value of , and the third one binds to 1.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... effects7.1
The broadening of the distribution due to the smearing suggests a choice of larger than . It is worth mentioning that there is no need to reject events where a measured quantity has a value outside the range allowed for the physical quantity. For example, in the case of deep-inelastic scattering events, cells with or give information about the true distribution too.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... inefficiency7.2
If then will be set to zero, since the experiment is not sensitive to the cause .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... events8.1
Please note that event' is also used here according to HEP jargon (this is quite a case of homonymy to which one has to pay attention, but it has nothing to do with the linguistic schizophrenia I am talking about).
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... belief.8.2
In fact, one could use the combinatorial evaluation in point 6 as well, because of the discussed cultural reasons, but not everybody is willing to speak about the probability of something which has a very precise value, although unknown.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... confusion.8.3
See for example Refs. [46] and [60], where it is admitted that the Bayesian approach is good for decision problems, although they stick to the frequentistic approach.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... bet8.4
This corresponds to a probability of .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... bag.8.5
Sometimes this expectation is justified advocating the law of large numbers, expressed by the Bernoulli theorem. This is unacceptable, as pointed out by de Finetti: For those who seek to connect the notion of probability with that of frequency, results which relate probability and frequency in some way (and especially those results like the law of large numbers') play a pivotal rôle, providing support for the approach and for the identification of the concepts. Logically speaking, however, one cannot escape from the dilemma posed by the fact that the same thing cannot both be assumed first as a definition and then proved as a theorem; nor can one avoid the contradiction that arises from a definition which would assume as certain something that the theorem only states to be very probable.''[11]
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... objectivity.8.6
My preferred motto on this matter is no one should be allowed to speak about objectivity unless he has had 10-20 years working experience in frontier science, economics, or any other applied field''.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... too.8.7
For example, the statistician D. Berry[63] has amused himself by counting how many times Hawking uses belief', to believe', or synonyms, in his A brief history of time'. The book could have been entitled A brief history of beliefs', pointed out Berry in his talk...
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... information.8.8
Recently, I met an elderly physicist at the meeting of the Italian Physical Society, who was nostalgic about the good old times when we could see decay in emulsions, and complained that at present the sophisticated electronic experiments are based on models. It took me a while to convince him that in emulsions as well he had a model and that he was not seeing these particles either.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... physicists8.9
Outstanding physicists have no reluctance in talking explicitly about beliefs. Then, paradoxically, objective science is for those who avoid the word belief' nothing but the set of beliefs of the influential scientists to which they believe...
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... estimators8.10
It is worth remembering that, in the Bayesian approach, the complete answer is given by the final distribution. The prevision (expected value') is just a way of summarizing the result, together with the standard uncertainty. Besides motivations based on penalty rules, which we cannot discuss, a practical justification is that what matters for any further approximated analysis, are expected values and standard deviation, whose properties are used in uncertainty propagation. There is nothing wrong in providing the mode(s) of the distribution or any other quantity one finds it sensible to summarize as well.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... use8.11
I refer to the steps followed in the proof of Bayes' theorem given in Section . They should convince the reader that calculated in this way is the best we can say about . Some say that in the Bayesian inference the answer is the answer (I have heard this sentence from A. Smith at the Valencia-6 conference), in the sense that one can use all his best knowledge to evaluate the probability of an event, but then, whatever happens, cannot change the assessed probability, but, at most, it can -- and must -- be taken into account for the next assessment of a different, although analogous event.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... work!8.12
This is an actual statement I have heard by Monte Carlo-oriented HEP yuppies.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... nonsense,8.13
Zech says, more optimistically: Coverage is the magic objective of classical confidence bounds. It is an attractive property from a purely esthetic point of view but it is not obvious how to make use of this concept.''[67]
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... quotation8.14
The translation of the symbols is as follows: stands for the measured quantity ( or in these notes); stands for the true value ( or here); for .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... imply.8.15
One would object that this is, more or less, the result that we could obtain making a Bayesian analysis with a uniform prior. But it was said that this prior assumes a positive attitude of the experimenters, i.e. that the experiment was planned, financed, and operated by rational people, with the hope of observing something (see Sections and ). This topic, together with the issue of reporting experimental results in a prior-free way, is discussed in detail in Ref. [25].
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... distribution8.16
The necessity of using integrated distributions is due to the fact that the probability of observing a particular configuration is always very small, and a frequentistic test would reject the null hypotheses.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... before.8.17
See Section .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... abstract8.18
I quote here the original abstract, which appears on page 18 of the conference abstract book.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... experiment?''.8.19
I also made other comments on the general illogicality of his arguments, which you may easily imagine by reading the abstract. For these comments I even received applause from the audience, which really surprised me, until I learned that David Moore is one of the most authoritative American statisticians: only a outsider like me would have said what I said...
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... verifiability.8.20
It is interesting to realize, in the light of this reflection, that the ISO definition of true value (a value compatible with the definition of a given particular quantity'', see Sections and ) can accommodate this point of view.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... normalized.8.21
I have made use several times in these notes of improper distributions, i.e. such that

d

but, as specified, they were always thought to be the limit of proper distributions (see, for example, Section ).
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... similar8.22
There is nothing profound in the fact that the two cases give very similar results. It is just due to the numbers of these examples (i.e. ).
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... reasonable9.1
I insist on the fact that they must be reasonable, and not just any prior. The fact that absurd priors give absurd results does not invalidate the inferential framework based on subjective probability.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... translated9.2
This two-step inference is not really needed, but it helps to follow the inferential flow. One could think more directly of

When the dependence between the two quantities is not linear, a two-step inference may cause trouble: see comments in Section .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... code9.3
If you are interested in Bayesian analysis with Mathematica you may take a look at Refs. [81] and [82] (I take for responsibility on the quality of the products, as I have never used them).
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... game.9.4
This section is intentionally pedagogical. An analysis using the best physical assumptions can be found in Ref. [26]. Indeed, this analysis follows the strategy outlined here, with some variations introduced to match the information available in the real situation.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... inference9.5
A two-step inference was shown in Section for the case of monopole search. There there was no problem because and are linearly related.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... LEONES''9.6
Here are the lions'' is what the ancient Romans used to write on the parts of their maps representing unexplored regions.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... becomes9.7
Numerologists may complain that this does not correspond to exactly 95%, but the same happens when a standard uncertainty is rounded to one or two digits and the probability level calculated from the rounded number may differ a lot from the nominal 68.3% calculated from the original value. But who cares?
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
`