... causes.¹

One might object that if the same cause yields different effects in different trials, then other concauses must exist, responsible for the differentiation of the effects. This point of view leads e.g. to the `hidden variables' interpretation of quantum mechanics (`à la Einstein'). I have no intention to try to solve, or even to touch all philosophical questions related to causation (for a modern and fruitful approach, see Ref. [2] and references therein) and of the fundamental aspects of quantum mechanics. The approach followed here is very pragmatic and the concept of causation is, to say, a weak one, that perhaps could be better called conditionalism: ``whenever I am sure of this, then I am also somehow confident that that will occur''. The degree of confidence on the occurrence of that might rise from past experience, just from reasoning, or from both. It is not really relevant whether this is the cause of that in a classical sense, or this and that are both due to other `true causes' and we only perceive a correlation between this and that.

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... proceeds,²

Those who believe that scientists are really `falsificationist' can find enlighting the following famous Einstein's quote: ``If you want to find out anything from the theoretical physicists about the methods they use, I advise you to stick closely to one principle: don't listen to their words, fix your attention on their deeds.''[6]. We shall come to this point in the conclusions.

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... impossible'.³

In the hypothetical experiment of one million tosses of a hypothetical `regular coin' (easily realized by a little simulation) the result of 500000 heads represents an `extraordinary event' ( $8\times 10^{-4}$ probability), as `extraordinary' are all other possible outcomes!

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... Logically,⁴

The fact that in practice these methods `often work' is a different story, as discussed in Sec. 10.8 of Ref. [1].

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... observed⁵

In other words, the reasoning based on p-values [8] constantly violates the so called likelihood principle, apart from exceptions due to numerical coincidences. In fact, making the simple example of a single-tail test based on a variable that is indeed observed, the conclusion about acceptance or rejection is made on the basis of $\int_{x_{obs}}^\infty\!f(x\,\vert\,{\mbox{\boldmath$\theta$}})\,\mbox{d}x$, where ${\mbox{\boldmath$\theta$}}$ are the model parameters. But this integral is rarely simply proportional to the likelihood $f(x_{obs}\,\vert\,{\mbox{\boldmath$\theta$}})$, i.e. integral and likelihood do not differ by just a constant factor not depending on ${\mbox{\boldmath$\theta$}}$. I would like to make clear that I dislike un-needed principles, including the likelihood one, and the maximum likelihood one above all. The reason why I refer here to the likelihood principle in my argumentation is that, generally, frequentists consider this principle with some respect, but their methods usually violate it [9]. Instead, in the probabilistic approach illustrated in the sequel, this 'principle' stems automatically from the theory.

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... construction.⁶

In statistics the variables that summarize all the information sufficient for the inference are called a sufficient statistics (classical examples are the sample average and standard deviation to infer $\mu$ and $\sigma$ of a Gaussian distribution). However, I do not know of test variables that can be considered sufficient for hypothesis tests.

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... calculated.⁷

Imagine you have to decide if the extraction of $n$ white balls in $N$ trials can be considered in agreement with the hypothesis that the box contains a given percentage $p$ of white balls. You might think that you are dealing with a binomial problem, in which $n$ plays the role of random variable, calculate the p-value and draw your conclusions. But you might get the information that the person who made the extraction had decided to go on until he/she reached $n$ white balls. In this case the random variable is $N$, the problem is modeled by a Pascal distribution (or, alternatively, by a negative binomial in which the role of random variable is played by the number $N-n$ of non-white balls) and the evaluation of the p-value differs from the previous one. This problem is known as the stopping rule problem. It can be proved that the likelihood calculated from the two reasonings differ only by a constant factor, and hence the likelihood principle tells that the two reasonings should lead to identical inferential conclusions about the unknown percentage of white balls.

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... behaviors.⁸

Just in this workshop I have met yet another invention [10]: Given three model fits to data with 40 degrees of freedom and the three resulting $\chi^2$ of 37.9, 49.1 and 52.4 for models $M_1$, $M_3$ and $M_3$, the common frequentistic wisdom says the three models are about equivalent in describing the data, because the expected $\chi^2$ is $40\pm 9$, or that none of the models can be ruled out because all p-values (0.56, 0.15 and 0.091, respectively) are above the usual critical level of significance. Nevertheless, SuperKamiokande claims that models $M_2$ and $M_3$ are `disfavored' at 3.3 and 3.8 $\sigma$'s, respectively! ( $1\times 10^{-3}$ and $1.4\times 10^{-4}$ probability.) It seems the result has been achieved using inopportunely a technique of parametric inference. Imagine a minimum $\chi^2$ fit of the parameter $\theta$ for which the data give a minimum $\chi^2$ of 37.9 at $\theta=\theta_1$, while $\chi^2(\theta_2)=49.1$ and $\chi^2(\theta_3)=52.4$ (and the $\chi^2$ curve is parabolic). It follows that $\theta_2$ and $\theta_3$ are, respectively, $\sqrt{49.1-37.9}\,\sigma\mbox{'s} =3.3\,\sigma$'s and $\sqrt{52.4-37.9}\,\sigma\mbox{'s}=3.8\,\sigma$'s far from $\theta_1$. The probability that $\theta$ differs from $\theta_1$ by more than $\vert\theta_2-\theta_1\vert$ and $\vert\theta_3-\theta_1\vert$ is then $P(\vert\theta-\theta_1\vert>\vert\theta_2-\theta_1\vert) = 1\times 10^{-3}$ and $P(\vert\theta-\theta_1\vert>\vert\theta_3-\theta_1\vert) = 1.4\times 10^{-4}$, respectively. But this is quite a different problem!

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...ADD.⁹

Reference [11] has to be taken more for its methodological contents than for the physical outcome (a tiny piece of evidence in favor of the searched for signal), for in the meanwhile I have become personally very sceptical about the experimental data on which the analysis was based, after having heard a couple of public talks by authors of those data during 2004 (one in this workshop).

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