... cancel,¹

“It is of the greatest importance, that the several positions of the heavenly body on which it is proposed to base the orbit, should not be taken from single observations, but, if possible, from several so combined that the accidental errors might, as far as may be, mutually destroy each other.” [1]

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... information.²

For example Gauss discusses the implications of averaging observations over days or weeks, during which the heavenly body has certainly changed position in the elapsed time. But the mean position in the mean time can be considered as an equivalent point in space and time, and a few of them, far apart, would be enough to determine the orbit parameters.

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... features.³

The network of all quantities involved in the model is also known as Bayesian network for two reasons: degrees of belief are assigned to all variables in the game (even to the observed ones, meant as conditional probabilities depending on the value of the others); inference and forecasting (the names are related to the purposes of our analysis - from the probabilistic point of view there is no difference) are then made by the use of probability rules, in particular the so called Bayes' rule, without the need to invent prescriptions or ad hoc `principles'.

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... issues.⁴

This is not the case, at the moment, for the charged kaon mass, as commented in Ref. [2], especially footnote 19.

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... factors,⁵

This is related to the so called Likelihood Principle, which is consider a good feature by frequentists, though not all frequentistic methods respect it [5]. In practice, it says that the result of an inference should not depend on multiplicative factors of the likelihood functions. This `principle' arises automatically in a probabilistic (`Bayesian') framework.

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... `flat',⁶

I refer again to footnote 9 of Ref. [2], reminding that the Princeps Mathematicorum derived the `Gaussian' as the error function such that the maximum of the posterior for a flat `prior' (explicitly stated) had a maximum corresponding to the arithmetic average of the observations, provided they were independent and they had “the same level of accuracy”.

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... value⁷

Citing again Gauss, he wrote explicitly of most probable value of `$\mu$', under the hypothesis that before the experiment all its values were equally probable [1].

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... likelihood;⁸

Hence the famous Maximum likelihood `principle', which is then nothing but a simple case of the more general probabilistic approach. (But the value that maximizes the likelihood is not necessarily the best value to report, as it will be commented in the conclusions.)

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...; ⁹

And it is easy to recognize, in this sub-case of the general probabilistic approach, another famous `principle'.

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... early,¹⁰

It is rather well known that the human mind has problems when dealing with randomness. For example, if you ask a person to write, at random, a long list of 0's and 1's, she will tend to `regularize' the series, which will then contain only short sequences of 0's and 1's, contrary to what happens rolling a coin, or using a (pseudo-)random generator. As an interesting book that discusses, among others, this experimental fact, Ref. [12] is recommended.

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... `error'¹¹

The quote marks are to remind that they refer, more precisely, to standard uncertainty, referring the nouns error and uncertainty to different concepts [10,11].

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... Gaussian.¹²

For example, the bimodal curve shown in the ideogram of Fig. 2 is a curious linear combination of Gaussians of $m_{K^\pm}$, although from a frequentistic point of view one should not be allowed to attribute probabilities, and hence pdf's, to true values. And there are even frequentistic `gurus' who use probability in quote marks, without explaining the reason but because they are aware that could not talk about probability, when they write “When the result of a measurement of a physical quantity is published as $R=R_0\pm\sigma_0$ without further explanation, it is implied that R is a gaussian-distributed measurement with mean $R_0$ and variance $\sigma_0^2$. This allows one to calculate various confidence intervals of given “probability”, i.e., the “probability” $P$ that the true value of $R$ is withing a given interval.” [13].

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... one.¹³

Besides rounding, the numbers slightly differ from those of Ref. [7], having applied there some ad hoc selections. But this does not change the essence of the message this note desires to convey.

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... average.¹⁴

Then the PDG adds a more intriguing curve n the reported `ideograms'[7] (see e.g. Fig. 1 of Ref. [2]). But this non-Gaussian curve has no probabilistic meaning, as discussed in [2], and, anyway, it is not used to draw no quantitative results, as far as I understand (and I hope...).

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... one.¹⁵

It seems that people are worried only if the `errors' appear small. But, as also stated by the ISO Guide [10], “Uncertainties in measurements” should be “realistic rather than safe”. In particular the method recommended by the ISO Guide “stands [...] in contrast to certain older methods that have the following two ideas in common:

• The first idea is that the uncertainty reported should be `safe' or `conservative [...] In fact, because the evaluation of the uncertainty of a measurement result is problematic, it was often made deliberately large. [...]”

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... scaling¹⁶

The shift of the average almost in the middle of the two most precise results makes the contributions to the $\chi ^2$ huge. Here are the differences of the individual measurements in unit of their standard deviations: $0.67,\, -0.37,\,0.19,\, -0.45,\, 0.41,\, -4.77,\, 1.49,\, 0.61,\, 4.52$, resulting in a $\chi ^2$ of 46.7 and a consequent p-value of $1.7\times 10^{-7}$. Therefore, also to whom who are critical against p-values [8,9], in a case of this kind, an alarm bell should sound [21]. But to all persons of good sense a similar alarm, concerning the frequentist solution of the problem, should sound too (see Ref. [2] for a sceptical alternative.)

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... beliefs¹⁷

As the historian of science Peter Galison puts it, “Experiments begin and end in a matrix of beliefs. ...beliefs in instrument type, in programs of experiment inquiry, in the trained, individual judgments about every local behavior of pieces of apparatus.” [22].

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... function.¹⁸

In the case there are only some possibilities, or we are interested on how the data support an hypothesis over others, the quantity to report is (are) the likelihood ratio(s), as recently recognized also by the European Network of Forensic Science Institutes (ENFSI) [25]. Note that a likelihood function, or likelihood ratios (also known as Bayes factors), cannot be considered as `objective numbers', because they depend on the judgments of experts. This is also recognized by the ENFSI Guidelines [25], which appears then rather advanced with respect to naive ideals of objectivism that often are instead just a defense of established procedures and methods supported by inertia and authority.

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... respectively.¹⁹

See e.g. Sec. 39.2.1 of Ref [7],
http://pdg.lbl.gov/2019/reviews/rpp2019-rev-statistics.pdf

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... sample:²⁰

The sample has been obtained with the Gaussian random number generator rnorm() of the R language[30], with the following commands
set.seed(20200102)
n = 16; mu = 3; sigma = 1
x = rnorm(n, mu, sigma)
so that we know the `true $\mu$'. (The random seed, used to make the numbers reproducible, was set to the date in which the sample was generated.) Mean and standard deviation are then calculated using R functions:
m = mean(x)
s = sd(x)

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