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.

• First because of considerations that when the odds go to zero or to infinity, then the intensity of belief on either hypothesis goes to infinity;⁴⁶ when ``an even chance is reached [the feeling of believing] should completely vanish and not incline either toward or away from the proposition.'' [6] The logarithmic function is the simplest one to achieve the desired feature. (Another interesting feature of the odds is described in footnote 16.)
• Then because (expressing the question in our terms), if we started from a state of indifference (initial odds equal to 1), each piece of evidence should produce odds equal to its Bayes factor [our $\tilde O_{i,j}(E_i)$]. The combined odds will be the product of the individual odds [Eq. 19]. But, mixing now Pierce's and our terminology, when we combine several arguments (pieces of evidence), they ``ought to produce a belief equal to the sum of the intensities of belief which either would produce separately''. [6] Then ``because we have seen that the chances of independent concurrent arguments are to be multiplied together to get the chance of their combination, and therefore the quantities which best express the intensities of belief should be such that they are to be added when the chances are multiplied...Now, the logarithm of the chance is the only quantity which fulfills this condition''. [6]
• Finally, Peirce justifies his choice by the fact that human perceptions go often as the logarithm of the stimulus (think at subjective feeling of sound and light - even `utility', meant as the `value of money' is supposed to grow logarithmically with the amount of money): ``There is a general law of sensibility, called Fechner's psychophysical law. It is that the intensity of any sensation is proportional to the logarithm of the external force which produces it.''[6] (Table 1 provides a comparisons between the different quantities involved, to show that the human sensitivity on probabilistic judgement is indeed logarithmic, with a resolution about the first decimal digit of the base 10 logarithms.)

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:

• Peirce' `chances' are introduced as if they were our odds, but are used if they were Bayes factors (``the chances of independent concurrent arguments are to be multiplied together to get the chance of their combination'' [6]). Then he takes the natural logarithm of these `chances', to which he also associates an idea of weight of evidence (``our belief ought to be proportional to the weight of evidence, in the sense, that two arguments which are entirely independent, neither weakening nor strengthening each other, ought, when they concur, to produce a belief equal to the sum of the intensities of belief which either would produce separately'' [6]).
• According to Ref. [8] the modern use of the logarithms of the odds seem to go back to I.J. Good, who used to call log-odds the natural logarithm of the odds.<sup>48</sup>
• However, reading later Ref. [8] it is clear that Good, following a suggestion of A.M. Turing, proposes a decibel-like (db) notation<sup>49</sup>, giving proper names both to the logarithm of the odds and to the logarithm of the Bayes factor:
  • `` $(10\,\log_{10}f)$db ... may be also described as the weight of evidence or amount of information for $H$ given $E$'' [7];
  • `` $(10\,\log_{10}o)$db may be called the plausibility corresponding to odds $o$'' [7].
  It follows then that
  

  $$= . \cite{Good}$$ ``Plausibility gained (36)

  
  
• Decibel-like logarithms of the odds are used since at least forty years with under the name evidence. [23].

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:

• `plausibility' is difficult to defend, because it is too similar to probability in everyday use, and, as far as I understand, has decayed;
• `weight of evidence' seems to be a good choice, for the reasons already well clear to Peirce.
• `evidence' in the sense of Ref. [23] seems, instead, quite bad for a couple of reasons:
  • First, because `evidence' has already too many meanings, including, in the Bayesian literature, the denominator of the r.h.s. of Eq. (3).
  • Second, because this name is given to the logs of the odds (including the initial ones), but not to those of the Bayes factors to which no name is given. Therefore, the name `evidence', as used in Ref. [23] in this context, is not related to the evidence.

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.<sup>50</sup>It 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.