Unbiased results

In the Bayesian approach there is a natural way of giving results in an unbiased way, so that everyone may draw his own scientific conclusion depending on his prior beliefs. One can simply present likelihoods or, for convenience, ratios of likelihoods (Bayes' factors, see Sections and ). Some remarks are needed in order not to give the impression that, at the end of this long story, we have not just ended up at likelihood methods.

• 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.

Having clarified these points, let us look at two typical cases.

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

$$P($$data$$\,\vert\,H_i)\,,$$

or Bayes' factor for any of the couples

$$\frac{P(\mbox{data}\,\vert\,H_i)}{P(\mbox{data}\,\vert\,H_j)}\,.$$

On the other hand, the likelihood for a given hypothesis alone, e.g. $P($data$\,\vert\,H_\circ)$, does not help the reader to form his idea on the hypothesis, nor on alternatives (see also Section ). Therefore, if a collaboration publishes experimental evidence against the Standard Model, suggesting some kind of explanation in terms of a new effect, it should report the likelihoods for both hypotheses. (See also 5th bullet of Section 8.8 in the case of Gaussian likelihood).

Values of quantities.

In this case the likelihoods are summarized by the likelihood function $f($data$\,\vert\,\mu)$. In this case one may also calculate Bayes' factors between any pair of values

$$\frac{f(\mbox{data}\,\vert\,\mu_i)}{f(\mbox{data}\,\vert\,\mu_j)}\,.$$

This can be interesting if only a discrete number of solutions are admissible.

When one publishes a likelihood function this should be clearly stated. Otherwise the temptation to turn $f($data$\,\vert\,\mu)$ into $f(\mu\,\vert\,$data$)$ is really strong. In fact, taking the example of the neutrino mass of Section , the formula

$$\frac{1}{\sqrt{2\,\pi}\,2}\,e^{-\frac{(-4-\mu)^2} {8}}$$

(with mass in eV) can easily be considered as if it were a result for $\mu$:

$$\frac{1}{\sqrt{2\,\pi}\,2}\,e^{-\frac{[\mu-(-4)]^2} {8}}$$

and conclude that $\mu_\nu=-4\pm 2\,$eV.

After having criticized this way of publishing the data for the second time, I try in the next section to encourage this way of presenting the result, on condition that one is well aware of what one is writing.