General criteria

The dependence of Bayesian inferences on initial probability is considered by opponents as the fatal flaw in the theory. But this criticism is less severe than one might think at first sight. In fact:

• It is impossible to construct a theory of uncertainty which is not affected by this ``illness''. Those methods which are advertised as being ``objective'' tend in reality to hide the hypotheses on which they are grounded. A typical example is the maximum likelihood method, of which we will talk later.
• As the amount of information increases the dependence on initial prejudices diminishes.
• When the amount of information is very limited, or completely lacking, there is nothing to be ashamed of if the inference is dominated by a priori assumptions.

It is well known to all experienced physicists that conclusions drawn from an experimental result (and sometimes even the ``result'' itself!) often depend on prejudices about the phenomenon under study. Some examples:

• When doing quick checks on a device, a single measurement is usually performed if the value is ``what it should be'', but if it is not then many measurements tend to be made.
• Results are sometimes influenced by previous results or by theoretical predictions. See for example Fig. taken from the Particle Data Book[33]. The interesting book ``How experiments end''[37] discusses, among others, the issue of when experimentalists are ``happy with the result'' and stop ``correcting for the systematics''.
• Slight deviations from the background might be interpreted as a signal (e.g. as for the first claim of discovery of the top quark in spring '94), while larger ``signals'' are viewed with suspicion if they are unwanted by the physics ``establishment''^3.9.
• Experiments are planned and financed according to the prejudices of the moment^3.10.

Figure: Results on two physical quantities as a function of the publication date.

These comments are not intended to justify unscrupulous behaviour or sloppy analysis. They are intended, instead, to remind us -- if need be -- that scientific research is ruled by subjectivity much more than outsiders imagine. The transition from subjectivity to ``objectivity'' begins when there is a large consensus among the most influential people about how to interpret the results^3.11.

Figure: $R=\sigma_L/\sigma_T$ as a function of the Deep Inelastic Scattering variable $x$ as measured by experiments and as predicted by QCD.

In this context, the subjective approach to statistical inference at least teaches us that every assumption must be stated clearly and all available information which could influence conclusions must be weighed with the maximum attempt at objectivity^3.12.

What are the rules for choosing the ``right'' initial probabilities? As one can imagine, this is an open and debated question among scientists and philosophers. My personal point of view is that one should avoid pedantic discussion of the matter, because the idea of universally true priors reminds me terribly of the famous ``angels' sex'' debates.

If I had to give recommendations, they would be the following.

• The a priori probability should be chosen in the same spirit as the rational person who places a bet, seeking to minimize the risk of losing.
• General principles -- like those that we will discuss in a while -- may help, but since it may be difficult to apply elegant theoretical ideas in all practical situations, in many circumstances the guess of the ``expert'' can be relied on for guidance.
• To avoid using as prior the results of other experiments dealing with the same open problem, otherwise correlations between the results would prevent all comparison between the experiments and thus the detection of any systematic errors. I find that this point is generally overlooked by statisticians.