... communication¹

Note based on lectures at the University of Perugia, 15-16 April 2011 and at MAPSES School in Lecce, 23-25 November 2011

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

... Columbo',²

Besides the inappropriate reference to the Columbo's episode, I consider that article substantially well done and I recommend its reading. To those interested in the subject ``probability and the law'' I also recommend, as starting points for further navigation, refs. [4].

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

... `data',³

For non experts it is important to clarify, although this is not deeply relevant here, that the histogram's `data' are non simple `empirical observations', but a result of selections and analysis (including calibrations), after suitable definitions of physical objects, like what a `jet' is.

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

... events⁴

This number, as well as 230 that follows, was estimated from the figure - precise numbers are irrelevant for the purpose of this paper.

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

... `arithmetically'.⁵

It seems rather natural to think that, if the purpose of a `subtraction' would be that of highlight extra physical components in the spectrum, this procedure should not be simply an `arithmetic subtraction' and, in particular, it should not yield unphysical negative counts.

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

... statistic.⁶

Let us remind that if a variable is described by a $\chi^2$ distribution with $\nu$ degrees of freedom, our (probabilistic) expectation (`expected value') is $\nu$, with expectation uncertainty (`standard deviation') $\sqrt{2\nu}$. Hence if $\theta_1$ and $\theta_2$ are variables of that kind, with $\nu_1 =84$ and $\nu_2 = 20$, our expectations will be ``$84\pm 13$'' and ``$20\pm 6$'', respectively. (As a side remark, we notice that, that adding a Gaussian component to explain the `excess', the difference between expected and observed value of the test statistic increases, since the $\chi^2$ goes to 56.7 for the entire region and 10.9 for the `peak region'.)

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

... right.⁷

After years of practice in particle physics and related subjects, I have developed my rule of the thumb, which until now has never failed: ``the funnier is the name of the test used to show that there is a disagreement with the `Standard Model' (or whatever is considered firmly established), the less I believe that this is the case'' (with the corollary that ``in the future I will tend to mistrust those people'').

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

... courses.⁸

Having written quite a lot on the subject, I don't want to go through yet another introduction to the subject and refer to the `Columbo paper'[5] (someone might find useful also [14]), only reminding here some of the basic ideas.

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

... GeV,''⁹

By the way it was about 36% percent by the best or our knowledge at the beginning of 1999[18], and it has changed with time, especially during 2011!

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

... something.¹⁰

For example, when I ask about the meaning of 95% CL lower bound on Higgs mass from LEP direct search, practically everybody - and I speak of particle physicists! - `explains' the result in probabilistic terms[20], although it is well known to frequentistic experts that ``The lower bounds on the Higgs mass that are quoted for the direct Higgs searches at LEP say absolutely nothing about the probability of the Higgs mass being higher or lower than some value.''[21] (By the way, it seems that the method described in [21] is essentially the one on which the LHC collaborations have agreed to report search limits: at least you know now what these results (do not) mean!)

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

... false.¹¹

If you want to try, you can play with The ultimate confidence intervals calculator[23] and no strict follower of Neyman's teaching can blame you of the results, that asymptotically will `cover' the true value of whatever quantity you have in mind in exactly the proportion of times you pre-define.

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

... mistakes.¹²

For example, in 2000 there was some excitement at CERN because some LEP experiments were observing some events above the expectation and there was a big action against the CERN directorate, that had decided to stop LEP in order to use structures and human/financial resources for LHC. This was an email I received the 10th of November 2000, addressed to a short list of physicists:

Subject: Do you want the Higgs found next year?
As you may know CERN DG, L.Maiani, has decided to shut off LEP. The decision is to be confirmed at a CERN Committee of Council meeting on Friday 17th.

As you probably know there is evidence for a Standard Model Higgs boson seen in the data in the last few months, with a probability as a background fluctuation of 4 per mille, or 2.9 sigma.
...
In other words we are seeing exactly what we should expect if Mh=115.
...

[If you wonder why 2.9$\sigma$'s is 4 per mille, instead of 2, don't ask me.]
The message ended with a request to write to Maiani in support of extending LEP run. Here follows my instant reply:

Let me understand:
do you REALLY feel 99.6% sure that the Higgs is around 115GeV (let's say below the effective kinematical threshold at the present LEP energy)? If not, how much are you confident?
...
Running or not running is a delicate decision problem which involves beliefs and risks (both financial and sociological). Therefore, I cannot disagree much with Maiani, being in his position.

On the other hand, in the position of any LEP collaborator I would push to run, certainly! (Given the same beliefs, the risk analysis is completely different).

Being myself neither the CERN Director-General, or a LEP physicist, but, with this respect, just a physics educated tax payer, I find myself more on the side of Maiani than on that of our LEP colleagues.

To make it clear, the ``$99.6\%$'' could not be how much we had to rationally believe the Higgs was at 115GeV, because it was a 0.004 p-value incorrectly turned into probability. Estimating correctly the probability, one would have got a few percent (see e.g. [18] for the method, although the numbers had changed in the meanwhile). And with a few percent, it would have been crazy to continue the LEP run, delay LHC and so on. On the other hand, if there was really a 99.6% probability, then LEP had to go on. (As it often happens with misinterpreted frequentistic methods, the errors are not little, like getting 99.6 for what should have better been 99.1, 98.5, or even perhaps 97%! - see chapter 1 of [6] for other examples. Here one considered practically certain something that was instead almost impossible.)

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

... probability.¹³

But if you really have the chance of making real bets, don't use the fair odds: you want to maximize the expected gain! This is what insurance companies and professional bookmakers do: evaluate the fair odds and then propose the most unfair ones in a given direction, unbalanced as much as someone can still accept them.

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

... principle'¹⁴

In the Essai `principles' do no stand for what we mean now as `first principles', or `axioms', but are rather the fundamental rules of probability that Laplace had derived elsewhere.

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

... likelihood¹⁵

Note that here likelihood is the same as probability, and has nothing to do with what statisticians call `likelihood' - reading directly the original French version might help, also taking into account that two hundred years ago the nouns were not as specialized as they now are.

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

... trials¹⁶

In modern terms, the problem solved by Bayes in a quite convoluted notation [29] was the inference of the binomial parameter $p$, conditioned on $x$ successes in $n$ trials, under the assumption that all values of $p$ were a priori equally likely

$$\begin{eqnarray*} f(p\,\vert\,n,x) &=& \frac{f(x\,\vert\,n,p)} {\int_0^1 f(x\,\vert\,n,p)\,dp}\,. \end{eqnarray*}$$

Laplace solved independently this problem and, indeed, the formula that gives the expected value of $p$, i.e.

$$\mbox{E}[p]=\frac{x+1}{n+2}\,,$$

is known as Laplace's rule of succession.

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

... 114 GeV;¹⁷

As mentioned in footnote 9, the 95% CL bound has nothing to do with 95% probability that its value was above the bounds. Translating the experimental information from the direct search into probabilistic assessments is not that easy, because the number also depends on the upper limits. In particular, if there would be `no' upper bound on the mass (that obviously cannot weigh grams!) there is no way to calculate the required probability. For further details see [18] and chapter 13 of [6].

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

... elsewhere.¹⁸

And if it wouldn't exist at all? OK, formulate the alternative model and try to assess your beliefs in the alternatives.

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

... Times:¹⁹

I definitely hope that when this influential newspaper reports on probability of important, uncertain scenarios that really matter for our lives, such as economy, health, international crises, future of the Planet and so on, its experts really know what they are talking about!

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