Unifying role of subjective approach

I would like to give some examples to clarify what I mean by `linguistic schizophrenia' (see Section ). Let us consider the following:

1. probability of a `6' when tossing a die;
2. probability that the 100 $$001st event will be accepted in the cuts of the analysis of simulated events, if I know that 91 $$245 out of 100 $$000 events^8.1have already been accepted;
3. probability that a real event will be accepted in the analysis, given the knowledge of point 2, and assuming that exactly the same analysis program is used, and that the Monte Carlo describes best the physics and the detector;
4. probability that an observed track is $\pi^+$, if I have learned from the Monte Carlo that ...;
5. probability that the Higgs mass is greater than 400 GeV;
6. probability that the 1000th decimal digit of $\pi$ is 5;
7. probability of rain tomorrow;
8. probability that the US dollar will be exchanged at $\ge 2$DM before the end of 1999 (statement made in spring 1998).

Let us analyse in detail the statements.

• The evaluation of point 1 is based on considerations of physical symmetry, using the combinatorial evaluation rule. The first remark is that a convinced frequentist should abstain from assessing such a probability until he has collected statistical data on that die. Otherwise he is implicitly assuming that the frequency-based definition is not a definition, but one of the possible evaluation rules (and then the concept can only be that related to the degree of belief...).

  For those who, instead, believe that probability is only related to symmetry the answer appears to be absolutely objective: 1/6. But it is clear that one is in fact giving a very precise and objective answer to something that is not real. Instead, we should only talk about reality. This example should help to clarify the de Finetti sentence quoted in Section  (``The classical view ...'', in particular, ``The original sentence becomes meaningful if reversed...'').

• Point 2 leads to a consistent answer within the frequentistic approach, which is numerically equal to the subjective one [see, for example, () and ()], whilst it has no solution in a combinatorial definition.
• Points 3 and 4 are different from point 2. The frequentistic definition is not applicable. The translation from simulated events to real events is based on beliefs, which may be as firmly based as you like, but they remain beliefs. So, although this operation is routinely carried out by every experimentalist, it is meaningful only if the probability is meant as a degree of belief and not a limit of relative frequency.
• Points 3-7 are only meaningful if probability is interpreted as a degree of belief.^8.2

The unifying role of subjective probability should be clear from these examples. All those who find statements 1-7 meaningful, are implicitly using subjective probability. If not, there is nothing wrong with them, on condition that they make probabilistic statements only in those cases where their definition of probability is applicable (essentially never in real life and in research). If, however, they still insist on speaking about probability outside the condition of validity of their definition, refusing the point of view of subjective probability, they fall into the self-declared linguistic schizophrenia of which I am talking, and they generate confusion.^8.3

Another very important point is the crucial role of coherence (see Section ), which allows the exchange of the value of the probability between rational individuals: if someone tells me that he judges the probability of a given event to be $68\%$, then I imagine that he is as confident about it as he would be about extracting a white ball from a box which contains 100 balls, 68 of which are white. This event could be related, for example, to the result of a measurement:

$$\mu = \mu_\circ \pm \sigma(\mu)\,,$$

assuming a Gaussian model. If an experimentalist feels ready to place a 2:1 bet^8.4 in favour of the statement, but not a 1:2 bet against it, it means that his assessment of probability is not coherent. In other words, he is cheating, for he knows that his result will be interpreted differently from what he really believes (he has consciously overestimated the `error bar', because he is afraid of being contradicted). If you want to know whether a result is coherent, take an interval given by $70\%$ of the quoted uncertainty and ask the experimentalist if he is ready to place a 1:1 bet in either direction.