Weight of priors and weight of evidence in real life

The box example used to introduce the Bayesian reasoning was particularly simple for two reasons. First, the updating factor was calculated from elementary probability rules in an `objective way' (in the sense that everybody would agree on a Bayes factor of 13, corresponding to a $ \Delta $JL of 1.1). Second, also the prior odds $ n_1/n_2$ were univocally determined by the formulation of the problem.

In real life the situations are never so simple. Not only priors can differ a lot from a person to another. Also the probabilities that enter the Bayes factor might not be the same for everybody. Simply because they are probabilities, and probabilities, meant as degree of belief, have an intrinsic subjective nature [10]. The very reason for this trivial remark (although not accepted by everybody, because of ideological reasons) is that probability depends on the available information and - fortunately! - there are no two identical brains in the world, made exactly the same way and sharing exactly the same information. Therefore, the same event is not expected with the same security by different subjects, and the same hypothesis is not considered equally credible.18

At most degrees of belief can be inter-subjective, because in many cases there are people or entire communities that share the same initial beliefs (the same culture), reason more or less the same way (similar brains and similar education) and have access to the same data. Finally, there are stereotyped `games' in which probabilities can even be objective, in the sense that everybody will agree on its value. But these situations have to be considered the exceptions rather than the rule (and even when we state with great security that the probability of head tossing a regular coin is exactly 1/2, we forget it could remain vertically, a possibility usually excluded but that I have personally experienced a couple of times in my life.)

Therefore, although educational games with boxes and balls might be useful to learn the grammar and syntax of probabilistic reasoning, at a given point we need to move to real situations.

Giulio D'Agostini 2010-09-30