Intuitions versus formal, possibly computer aided, reasoning

Contrary to `robotized Bayesians'⁴³I think it is quite natural that different persons might have initially different opinions, that will necessarily influence the beliefs updated by experimental evidence, although the updating rule is well defined, because based on probability theory. But we have also seen, in a formal way, that when the combined weight of evidence in favor of either hypothesis is much larger than the prior judgement leaning, i.e. $\vert\Delta$JL$_{1,2}({\mbox{\boldmath $E$}},I)\vert\gg \vert\mbox{JL}_{1,2}(I)\vert$, then priors become irrelevant and we reach highly inter-subjective conclusions.

I am not in the position to try to discuss the internal processes of the human mind that lead us to react in a certain way to different stimuli. I only acknowledge that there are experts of different fields that can make (in most case good) decisions in an fantastically short reacting time. There is no need to think to doctors or engineer in emergency situations, football players, fighter pilots, and many other examples. It is enough to observe us in the everyday actions of driving a car or recognizing people from very partial information (and the context plays and important role! How many times has happened to you not to immediately recognize/identify a neighbor, a waiter or a clerk if you meet him/her in a place you didn't expect him/her at all?). We are brought to think that much of the way in which external information is processed is not analytical, but somehow hard-wired in the brain.

A part of the automatic reasoning of the mind is innate, as we can understand observing children, animals, or even rational adults when they are possessed by pulsions and emotions. Another part comes from the experience of the individual, where by `experience' it is meant all inputs received, of which he/she might be conscious (like education and training) or unconscious, but all processed and organized (again consciously or not) by the causality principle[17], that allows us to anticipate (again consciously or not) the consequences of our and somebody else's actions. As a matter of fact, and coming to the main issue of this paper, there is no doubt that experienced policemen, lawyers and judges, thanks to their experience, have developed kinds of automatic reasonings, that we might call instinct or intuitive behavior (see footnote 2) and that certainly help them in their work.

We have seen in section 3 that priors and even individual weights of evidence can be elicited on a pure subjective way, possibly with the help of virtual bets or of comparison to reference events. The problem arrives when the situation becomes a bit more complicate than just one cause and a couple of effects, and the network of causes-effects become complex. Appendix I shows that the little complication of considering the possibility that the evidence could be somehow reported in an erroneous way, as well known to psychologists, of even fabricated by the investigators makes the problem difficult and the intuition could fail. Appendix J shows an extension of the toy model of section 2 in which several `testimonies' need to be taken into account.

In summary, the intuition of experts is fundamental to define the priors of the problem. It can be also very important, and sometimes it is the only possibility we have, to assess the degree of belief that some causes can produce some effects, needed to evaluate the Bayes factors and, when the situation becomes complex, to set up a `network of beliefs' (see Appendix J). A different story is to process the resulting network, on the base of the acquired evidences, in order to evaluate the probabilities of interest. Intuition can be at lost, or miserably fail.

To make clearer the point consider this very rude example. Imagine you are interested in the variable $z$, that you think for some reasons is related to $x$ and $y$ by the following relation:

$$z = \frac{y\times\sin(\pi^4+x^2)} {\sqrt{x^3+y^2}}\,.$$

You might have good reason to state that $x$ is about 10, most likely not less than 9 and not more than 11, and that in this interval you have no reasons to prefer a value with respect to another one. Similarly, you might thing that the value of $y$ you trust mostly is 20, but it could go down to 15 and up to 30 with decreasing beliefs. What do you expect for $z$. Which values of $z$ should you believe, consistently with your basic assumptions? If a rich prize is give to the person that predicts the interval of width 0.02 in which $z$ will occur, which interval would you choose? What is the value of $z$ (let us call it $z_m$) such that there is 50% chance that $z$ will occur below this value? What is the probability that $z$ will be above 10? [The solution is in next page (figure 7).]

Anyway, if you consider this example a bit too `technical' you might want to check the capabilities of your intuition on the much simpler one of Appendix J. (Try first to read the caption of figure 11 and to reply the questions.)