Conclusions

Uncritical or wishful use of p-values can be dangerous, not to speak of unscrupulous p-hacking. While years ago these criticisms were raised by a minority of thorny Bayesians, now the effect on the results in several fields of science and technology is felt as a primary issue.³² The statement of the American Statistical Association is certainly commendable in addressing the issue, but it is in my opinion unsatisfactory not admitting that the question is inherent to all statistical methods that refuse the very idea of probability of hypotheses, or of ``probability of causes'', i.e. what Poincaré used to call ``the essential problem of the experimental method.''

While I had experienced several times in the past, including this winter, claims of possible breaking discoveries in Particle Physics simply due to misinterpretations of p-values, for the first time I have realized of a case in which judgements based on p-values strongly reduce the `significance' of important results. This happens with the gravitational wave events reported this year by the LIGO-Virgo collaboration, and in particular with the October 12 events timidly reported as a LIGO-Virgo Trigger (`Cinderella'), because of its 1.7 sigmas, in spite of the huge Bayes factor of about $10^{10}$, that should instead convince any hesitating physicist about its nature of a gravitational wave radiated by a Binary Black Hole merger, especially in the light of the other, more solid two events (`the two sisters').³³I hope than that LVT151012 will be upgraded to GW151012 and that in future searches the Bayes factor will become the principal figure of merit to rank gravitational wave candidates.

I finally conclude with some questions asked at the end of talk on which this paper is based.

• Which Bayes factor would characterize the 750GeV excess?
  The result depends on the model to explain the excess³⁴ and an answer came the week after MaxEnt 2016 by Andrew Fowlie[66]. For the model considered he got a BF around 10, the exact value being irrelevant: a weak indication, but nothing striking to force sceptics to change substantially their opinion.³⁵
• Could have CDF at Fermilab claimed to have observed the Higgs boson if they had done a Bayesian analysis?
  I am quite positive they could have it, also because the prior on the possible values of the Higgs mass was not so vague and well matching the value found later, and therefore the Bayes Factor would have been rather high (and the prior probability of a possible manifestation of the boson in the final state was high too).

Acknowledgements
This work was partially supported by a grant from Simons Foundation, which allowed me a stimulating working environment during my visit at the Isaac Newton Institute of Cambridge, UK. The understanding and/or presentation of several things of this paper has benefitted of the interactions with Pia Astone, Ariel Caticha, Kyle Cranmer, Walter Del Pozzo, Norman Fenton, Enrico Franco, Gianluca Gemme, Stefano Giagu, Massimo Giovannini, Keith Inman, Gianluca Lamanna, Paola Leaci, Marco Nardecchia, Aleandro Nisati, and Cristiano Palomba. I am particularly indebded to Allen Caldwell, Alvaro de Rujula and John Skilling for many discussions on physics, probability, epistemology and sociology of scientific communities, as well for valuable comments on the manuscript, which has also benefitted of an accurate reading by Christian Durante and Dino Esposito.