Unbiased results

- First, without priors, the likelihoods cannot be turned into probabilities of the values of physics quantities or of probabilities of hypotheses. Even the `mathematically harmless' uniform distribution, which gets simplified in Bayes' formula, does its important job. For this reason publishing only likelihoods does not mean publishing unbiased conclusions, but rather publishing no conclusions! Hence, one is not allowed to use this `result' for uncertainty propagation, as it has no uncertainty meaning.
- This game of avoiding the priors can be done only at the final level of the analysis. Presuming priors can be avoided for each step is absurd, in the sense that the reader has no way of completely redoing the analysis by plugging in his preferred priors at all the crucial points. If the experimenter refrains to choose a prior, but, nevertheless, he goes on in the steps of the analysis, he is in fact using uniform priors in all inferences. This may be absolutely reasonable but one has to be aware of what one is doing. If instead there are reasons for using non-uniform priors in some parts of the analysis, as his past experience suggested, the experimenter should not feel guilty. There are so many subjective and even really arbitrary ingredients in a complex analysis that we must admit, if we believe somebody's results and we use his conclusions as if they were our conclusions, it is simply because we trust him. So we are confident that his knowledge of the detector and of the measurement is superior to ours and this justifies his choices. As a matter of fact, the choice of priors is insignificant compared with all the possible choices of a complicated experiment.
- The likelihoods are probabilities of observables given a particular hypothesis. Also their evaluation has subjective (and arbitrary) contributions. Sticking to the idealistic position of providing only objective data is equivalent to stagnating research.

**Classifying hypotheses.**- In the case of a discrete number of
hypotheses, the proper quantities to report are the likelihoods
of the data for each hypothesis
dataor Bayes' factor for any of the couples
**Values of quantities.**- In this case the likelihoods are summarized by the likelihood function
data. In this case one may also calculate Bayes' factors
between any pair of values
When one publishes a likelihood function this should be clearly stated. Otherwise the temptation to turn data into data is really strong. In fact, taking the example of the neutrino mass of Section , the formula