Combining uncertain priors and uncertain weights of evidence

When we have set up our problem, listed the pieces of evidence pro and con, including the 0-th one (the prior), and attributed to each of them a weight of evidence, quantified by the corresponding $ \Delta $JL's, we can finally sum up all contributions.

As it is easy to understand, if the number of pieces of evidence becomes large, the final judgment can be rather precise and far from being perfectly balanced, even if each contribution is weak and even uncertain. This is an effect of the famous `central limit theorem' that dumps the weight of the values far from the average.22 Take

Figure: Combined effect of 10 weak and `vague' pieces of evidence, each yielding a $ \Delta $JL of $ 0.5\pm 0.5$ (see text).
for example the case of 10 JL's each uniformly23ranging between 0 and 1, i.e. $ \Delta $JL$ _{1,2}(E_i,I)=0.5\pm 0.5$. Each piece of evidence is marginal, but the sum leads to a combined $ \Delta $JL$ _{1,2}({\mbox{\boldmath $E$}},I)$ of $ 5.0\pm 1.8$, where ``$ [3.2,6.8]$'' defines now an effective range of leanings24, as depicted in figure 4. Note that in this graphical representation the 5 yellow arrows (the lighter ones if you are reading the text in black/white) do not represent individual values of JL, but its interval. These arrows have all the same width to indicate that the exact value is indifferent to us. The red arrow have instead different widths to indicate that we prefer the values around 5 and the preference goes down as we move far form it. The 12 arrows only indicate an effective range, because the full range goes from 0 to 10, although $ \Delta $JL values very far from 5 must have negligible weight in our reasoning.

Giulio D'Agostini 2010-09-30