Beliefs from past frequencies

There is no doubt that
``where different effects have been found to follow from causes, which are to appearance exactly similar, all these various effects must occur to the mind in transferring the past to the future, and enter into our consideration, when we determine the probability of the event. Though we give the preference to that which has been found most usual, and believe that this effect will exist, we must not overlook the other effects, but must assign to each of them a particular weight and authority, in proportion as we have found it to be more or less frequent.'' [17]
However, some comments about how our minds perform these operations are in order.

Before they are turned into beliefs, observed frequencies are somehow smoothed, either intuitively or by mathematical algorithms. In both cases, consciously or unconsciously, some models of regularities are somehow `assumed' (a word that in this context means exactly `believed'). Think, for example, at an experiment in which the number of counts are recorded in a defined interval of time, under conditions apparently identical. Imagine that the numbers of counts in 20 independent measurements are: 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 1, 0, 4, 2, 0, 0, 0, 0, 1. The results are reported in the histogram. The question is ``what do we expect in the 21-st observation, provided the experimental conditions remain unchanged?''. It is rather out of discussion that, if a prize is offered on the occurrence of a count, everyone will bet on 0, because it happened most frequently. But
can we state that our belief is exactly 60% (12/20)? Moreover, I am also pretty sure that, if you were asked to place your bet on 3 or 4, you would prefer 3, although this number of count has not occurred in the first 20 observations. In an analogous way you might not believe that 5 is impossible. That is because we tend to see regularities in nature.41Therefore going from past frequencies to probabilities can be quite a sophisticated process, that requires a lot of assumptions (again priors!).

Giulio D'Agostini 2010-09-30