Afraid of `prejudices'? Inevitability of principle and frequent practical irrelevance of the priors

Doubtless, many readers could be at a loss at having to accept that scientific conclusions may depend on prejudices about the value of a physical quantity (`prejudice' currently has a negative meaning, but in reality it simply means `scientific judgement based on previous experience'). We shall have many opportunities to enter again into discussion about this problem, but it is important to give a general overview now and to make some firm statements on the role of priors.

• First, from a theoretical point of view, it is impossible to get rid of priors; that is if we want to calculate the probability of events of practical interest, and not just solve mathematical games.
• At a more intuitive level, it is absolutely reasonable to draw conclusions in the light of some reason, rather than in a purely automatic way.
• In routine measurements the interval of prior acceptance of the possible values is so large, compared to the width of the likelihood (seen as a function of $\mu$), that, in practice, it is as if all values were equally possible. The prior is then absorbed into the normalization constant:

  $$f(x\,\vert\,\mu) \cdot f_\circ(\mu) \xrightarrow [ ]{}{f(x\,\vert\,\mu)}.$$ (2.1)

  
  

• If, instead, this is not the case, it is absolutely legitimate to believe more in personal prejudices than in empirical data. This could be when one uses an instrument of which one is not very confident, or when one does for the first time measurements in a new field, or in a new kinematical domain, and so on. For example, it is easier to believe that a student has made a trivial mistake than to conceive that he has discovered a new physical effect. An interesting case is mentioned by Poincaré [6]:

  ``The impossibility of squaring the circle was shown in 1885, but before that date all geometers considered this impossibility as so `probable' that the Académie des Sciences rejected without examination the, alas! too numerous memoirs on this subject that a few unhappy madmen sent in every year. Was the Académie wrong? Evidently not, and it knew perfectly well that by acting in this manner it did not run the least risk of stifling a discovery of moment. The Académie could not have proved that it was right, but it knew quite well that its instinct did not deceive it. If you had asked the Academicians, they would have answered: `We have compared the probability that an unknown scientist should have found out what has been vainly sought for so long, with the probability that there is one madman the more on the earth, and the latter has appeared to us the greater.'''

In conclusion, contrary to those who try to find objective priors which would give the Bayesian theory a nobler status of objectivity, I prefer to state explicitly the naturalness and necessity of subjective priors[22]. If rational people (e.g. physicists), under the guidance of coherency (i.e. they are honest), but each with unavoidable personal experience, have priors which are so different that they reach divergent conclusions, it just means that the data are still not sufficiently solid to allow a high degree of intersubjectivity (i.e. the subject is still in the area of active research rather than in that of consolidated scientific culture). On the other hand, the step from abstract objective rules to dogmatism is very short[22].

Turning now to the more practical aspect of presenting a result, I will give some recommendations about unbiased ways of doing this, in cases when priors are really critical (Section ). Nevertheless, it should be clear that:

• since the natural conclusions should be probabilistic statements on physical quantities, someone has to turn the likelihoods into probabilities, and those who have done the experiment are usually the best candidates for doing this;
• taking the spirit of publishing unbiased results -- which is in principle respectable -- to extremes, one should not publish any result, but just raw data tapes.