Bare facts and complete state of information

As it has been extensively discussed in section 5, saying that a person has taken a camera out of thirteen is a piece of information, but it is not all, and it is not enough to update correctly our beliefs.

Figure: This histogram shows in a graphical way `a' solution to the question at the end of Appendix C (details depend, obviously, on how the initial assumptions have been modelled, but the gross features do not change if different reasonable models, consistent with the assumptions, are used - here $ x$ has been taken uniform between 9 and 11; $ y$ has been modelled with an asymmetric triangular distribution ranging between 15 and 30, with maximum belief in 20). The values of $ z$ we have to believe mostly are those around $ -0.5$, but also all others in the range $ -0.7$ to $ 0.7$ cannot be really neglected. In particularly, values around 0.5 are almost as likely as those around $ -0.5$. As we can see, there is about 50% that $ z$ occurs below 0 ($ -0.02$, to be precise) and 50% above. Note that, although the center of the distribution is around 0 ($ -0.14$, to be precise), the most believable values are far from it. In other words, even if the expected value is $ -0.14$ and the standard uncertainty (quantified by the standard deviation) is $ 0.40$, if a prize is assigned to whom predicts the interval of width 0.02 in which the uncertain number $ z$ will occur, we should place that interval at $ -0.5$. Apart from the technical complications, the message of this example is that one thing is to state the basic assumptions and subjective beliefs in some of the variables of the game, a much more complicate issue is to evaluate all logical consequences of the premises. In other words, if you agree on the premises of this problem, but not on the conclusions, you run into contradiction. Now, it is a matter of fact that contradictions of this kind are rather frequent because the evaluation of the consequences is not commonly done using formal logic and probability theory. The extension to complex belief networks is straightforward, although, as we shall see in Appendix J, also a very simple network is enough to challenge our ability to provide intuitive answers.

This is true in general, even in fields of research that are considered by outsider to be the realm of objectivity, where only `facts' count. Stated with Peter Galison words [20],

``Experiments begin and end in a matrix of beliefs. ...beliefs in instrument type, in programs of experiment enquiry, in the trained, individual judgments about every local behavior of pieces of apparatus.''
$ [$Then, taking as an example the discovery of the positron:$ ]$
``Taken out of time there is no sense to the judgment that Anderson's track 75 is a positive electron; its textbook reproduction has been denuded of the prior experience that made Anderson confident in the cloud chamber, the magnet, the optics, and the photography.''
My preferred toy examples to convey this important messages are the three box problem(s)' and the two envelopes `paradox' (see section 3.13 of Ref. [3] - I remind briefly here only the box ones). The box problems are a series of recreational/educational problems, the basic one being rather famous as `Monthy Hall problem'. The great majority of people (my usual target are physics PhD students) get mad with them because they have not been educated to take into account all available information. Therefore they have quite some difficulties to understand that if a contestant has taken one box (yet unopened) and there is still another un-opened box to choose, the probability that this box contains the prize (only one of the three boxes does) depends on whether the opened (and empty) box was got by chance or was chosen with the intention to take a box without prize.44(And there is often somebody in the audience that when he/she listens the formulation of the problem in which the box was opened by chance, he/she smiles at the others, and than gives the solution...of the version in which the conductor opens on purpose and empty box.)

I found that the issue of considering into account all available information is shown in a particular convincing way in the `three prisoner paradox' (isomorph45 to Monthy Hall, but more a headache than this, perhaps because it involves humans) and in the `thousand prisoner problem' of Ref. [21]: not only bare facts enter the evaluation of probability, but also all contextual knowledge about them, including the question asked to acquire their knowledge.

Giulio D'Agostini 2010-09-30