Some remarks on the use of logarithmic updating of the odds

The idea of using (natural) logarithms of the odds is quite old, going back, as far as I know, to Charles Sanders Peirce [6]. He related them to what he called feeling of belief (or intensity of belief), that, according to him, ``should be as the logarithm of the chance, this latter being the expression of the state of facts which produces the belief'' [6], where by `chance' he meant exactly probability ratios, i.e. the odds.

Peirce proposed his ''thermometer for the proper intensity of belief'' [6] for several reasons.

As far as the logarithms in question, I have done a short research on their use, which, actually, lead me to discover Peirce's Probability of Intuition [6] and Good's Probability and the weighing of Evidence [7].47As far as I understand, without pretension of completeness or historical exactness:

Personally, I think that the decibel-like definition is not very essential (decibels themselves tend already to confuse normal people, also because for some applications the factor 10 is replaced by a factor 20). Instead, as far as names are concerned: I have taken the liberty to use the expression `judgment leaning' first because it evokes the famous balance of Justice, then because all other expressions I thought about have already a specific meaning, and some of them even several meanings.50It is clear, especially comparing Eq. (36) with Eq. (24), that, besides the factor ten multiplying the base ten logarithms and the notation, I am quite in tune with Good. I have also to admit I like Peirce' `intensity of belief' to name the JL's, although it is too similar to `degree of belief', already widely used to mean something else.

So, in summary, these are the symbols and names used here:

JL$ _{i,j}(\cdot)$
is the judgement leaning in favor of hypothesis $ i$ and against $ j$, with the conditions in parenthesis. If we only consider an hypothesis ($ H$) and its opposite $ \overline H$, that could be possibly related to the occurrence of the event $ E$ or its opposite $ \overline E$, also the notation JL$ _{H}(\cdot)$, or JL$ _{E}(\cdot)$, will be used (as in table 2 of Appendix I).
(Sometimes I have also tempted to call a JL `intensity of belief' if it is clear from the contest that the expression does not refer to a probability.)
$ \Delta $JL$ _{i,j}(\cdot)$
, with the same meaning of the subscript and of the argument, is the variation of judgement leaning produced by a piece of evidence and it is called here weight of evidence, although it differs by a factor from the analogous names used by Peirce and Good51.

Giulio D'Agostini 2010-09-30