... occur.¹

Remember that all events of our life were indeed VERY improbable, if observed with enough detail, because they are just points in a high dimensional configuration space!

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... uncertainties²

For the meaning of error and uncertainty see [1] and [2]. Hereafter `error' in quote marks is to remind that the noun refers in reality to uncertainty, or, more precisely, standard uncertainty.

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...tab:masseK_PDG.³

Details can be found in the 2000 edition of the PDG [11]. Moreover, comparing the two editions of the PDG and taking into account that not always the details of the experiment are publicly available, it is clear that a serious work to determine at best the charged kaon mass goes beyond the aim of this paper, being mainly methodological. Nevertheless, the uncertainty reported for the 5th result of table Tab. 1 is not a good account of the experimental result, as it will be discussed later on in this paper.

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... interpretation⁴

Note that this interpretation is valid, under hypotheses which generally hold, especially if $s_i/d_i \ll 1$ (as it happens in this case), even if the results were produced with frequentistic methods that do not contemplate the possibility of attributing probabilities to the values of physics quantities. In fact, most results obtained using standard statistics ('frequentistic') are based on the analysis of the so called likelihood around its maximum. And they can then be easily turned into probabilistic results (see e.g. [12], in particular section 12.2.1 and the related figure 12.1).

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...MeV,⁵

In most cases I stick here to two digits for the standard uncertainty.

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... indovina”.⁶

“To think badly would be to sin, but very often one gets it right”$^{(*)}$. Most Italians attribute it to Giulio Andreotti, but it seems due no less then to a pope [13].
$^{(*)}$https://forum.wordreference.com/threads/a-pensare-male-si-fa-peccato-ma-spesso-ci-si-azzecca.2397506/

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... 50 keV,⁷

Value just decided by eye looking at the figure with some experienced colleagues, and not resulting from fits or optimizations of any kind.

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... above;⁸

A possible alternative would be to allow a shift of the measured quantity. However this model seems unable to yield multimodal final pdf's, which is, in my opinion, one of the desiderata of the model, as stated here. Perhaps the question requires further study but, given the limited aims of this paper, I prefer to stick for the moment to the models of Refs. [15,16].

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... interest.⁹

For the Gauss' use of what we would nowadays call a Bayesian reasoning, starting form the concept of probabilities of the true value, see Section 6.12 of Ref. [12] based on Section III of Book II of Ref. [17] $[$see Ref. [19] for details on the missing steps between Eq.(6.53) and Eq.(6.54)$]$. Here I just want to comment on the meaning of a `flat' prior, which does not imply that it has to be interpreted as strictly constant all over the real axis. With this respect it is interesting the comment that Gauss adds after he derived the `Gaussian' as the error function characterized by good mathematical behavior and such that the posterior gets its maximum in correspondence of the arithmetic average, in the case of independent measurements characterized by the same error probability distribution: “The function just found $[$ the `Gaussian'$]$ cannot, it is true, express rigorously the probabilities of the errors: for since the possible errors are in all cases confined within certain limits, the probability of errors exceeding those limits ought always be zero, while our formula always gives some value. However, this defect, which every analytical function must, from its nature, labor under, is of no importance in practice, because the value of function decreases so rapidly, when $h\,\Delta$ $[$ ` $(x_i-\mu)/\sigma$', in modern notation$]$ has acquired a considerable magnitude, that it can safely be considered as vanishing. Besides, the nature of the subject never admits of assigning with absolute rigor the limits of error.” [18]

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... outcomes,¹⁰

In this introductory section we use $x_i$ to indicate an individual observation, while in general the $d_i$ of Tab. 1 are results of `statistical analyses' based on many direct `observations'.

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... denominator.¹¹

Let us remind that, in general, $f(\underline{x}\,\vert\,\underline{\sigma},I) = \int_{-\infty}^{+\infty}f(\mu,\underline{x}\,\vert\,\underline{\sigma},I)\,d\mu$ .

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... Sampler,¹²

Talking about the Gibbs sampler algorithm applied in probabilistic inference (and forecasting) it is impossible not to to mention the BUGS project [23], the acronym staying for Bayesian inference using Gibbs Sampler, that has been a kind of revolution in Bayesian analysis, decades ago limited to simple cases because of computational problems (see also Section 1 of [24]). In the project web site [25] it is possible to find packages with excellent Graphical User Interface, tutorials and many examples [26], which, although far from the typical interests of physicists, might help to understand the underlying reasoning and the model language, practically the same used by JAGS.

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... case,¹³

But in frontier research it is not difficult to imagine cases in which this is not true.

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... interest.¹⁴

My preferred vademecum of Probability Distributions is the homonymous app [30].

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... positive.¹⁵

Also a mass, as many other physics quantities, is positively defined, and in principle one has to pay attention, either in the sampling steps or when the resulting chain is analyzed, that it does not get negative But this problem does not occur in practice if the the average value $\overline{x}$ is many standard $\sigma_C$ above zero. Anyway, packages like JAGS allow also sharp constrains on the priors. (This is general problem when we use Gaussians to describe positively defined quantities, already realized by Gauss and reminded in footnote 9.)

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... script¹⁶

If these lines are saved in a file, e.g. kaon_mass_naive.R, then the script can be run with the command source('kaon_mass_naive.R').

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...list¹⁷

A `list' is a very interesting object of R, which can contain other objects, also of different kinds and different lengths; the element of a `list' can be accessed either by name, as we do here, or by indices.

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...sceptical1999,¹⁸

For easier comparison with the results of Ref. [15] for the Gamma parameters we use hereafter $\delta$ and $\lambda$ instead of the standard $\alpha$ and $\beta$.

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... average.¹⁹

An example in which the sceptical combination produces a result narrower that the weighted average is shown in the Appendix.

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... figure,²⁰

The histogram with the over-imposed profile was produced by

chain.df <- as.data.frame( as.mcmc(chain) )
hist(chain.df$mu,nc=100,prob=TRUE,xlab='K mass (MeV)',ylab='f(m)', col='cyan',main=”)
lines(density(chain.df$mu, adjust=1.0), lwd=3)

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... required.²¹

Presently the value of the charged kaon mass, with relative uncertainty of around 26 ppm, is not critical for fundamental issues. For example its contribution to $\vert V_{us}\vert$ of the Standard Model is of the order of 66 ppm, to be combined in quadrature with the relative uncertainties of the other quantities from which $\vert V_{us}\vert$ depends (the branching ratios of interest depend on $M_{K^\pm}^5\cdot\vert V_{us}\vert^2$ and hence the relative uncertainty on $M_{K^\pm}$ is propagated with a factor $5/2$ into the relative uncertainty on $\vert V_{us}\vert$).

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... freedom,²²

Providing just `$\chi^2/\nu$' is, as now well understood, rather misleading, because the $\chi^2$ does not scale with $\nu$. Therefore, though a ratio of 2.31 would be a clear alarm bell for $\nu=100$, it is quite `in the norm' for $\nu=3$.

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... rounding²³

Indeed, if we just calculate weighted averages and related standard deviations, with no arbitrary scaling, the result does not change if we use the individual results or we group them in steps. This is related to the important concept of `statistical sufficiency', that will be treated in detail, for the Gaussian case, in the forthcoming Ref. [35].

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... parameters:²⁴

Someone would be surprised about the possibility of inferring a number of parameters superior to the number of the data points. This is not really a conceptual problem, as long as we understand that they are correlated, often in a complicate way and of which the correlation matrix is just a first order representation (and we have to be careful when using it in further analyses [36]).

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... matrix:²⁵

Technical remark: the correlation matrix has been obtained by the R function cor(), applied to the chain after a suitable transformation. For example, one can transform it into a data frame and then apply cor() to it:

  
> chain.df <- as.data.frame( as.mcmc(chain) )
> round(cor(chain.df),2)

that includes the rounding at two decimal digits ('$>$' is the R prompt).

Or, more simply, we can convert the chain into a matrix, each column containing the occurrences of each variable during the sample, and calculate then the correlations between them. This is how to do it in short, with nested calls to functions (remember also print(), if the command hat to be included into a script):

  
> round(cor(as.matrix(chain)),2)

And here are some useful commands to understand what is going on:

  
> chain.M <- as.matrix(chain)
> str(chain.M)
> dimnames(chain.M)
> mean(chain.M[,"mu"])
> mean(chain.M[,1])
> mean(chain.M[,"r[9]"])
> mean(chain.M[,10])
> cor(chain.M[,"mu"], chain.M[,"r[9]"])
> cor(chain.M[,1], chain.M[,10])

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... `statistical'²⁶

For example it is important to understand how the `errors' were evaluated, also because we are aware of the old custom (maintained also presently by several experimental teams) of using for `systematic errors' extreme variations for sake of safety, thus providing very conservative `error', instead than standard uncertainties [1,2].

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