DA “THE REASONER“, Volume 7, Number 6 – June 2013
I. J. Good was invited by Teddy Seidenfeld and Marco Zaffalon to speak at the second International Symposium on Imprecise Probabilities and Their Applications held in Lugano, Switzerland, on July 14-17, 2003. He could not travel to the meeting, but prepared a contribution titled The accumulation of imprecise weights of evidence. In it, Good outlined a proposal which doesn’t seem to have caught on in the literature, namely to use imprecise weights of evidence, rather than imprecise probabilities, to measure second-order uncertainty.
When it comes to making decisions, by and large, we know very little. So an essential aspect of effective decision-making in real-world contexts consists in gathering information. Yet not all information is born equal. Indeed one of the key questions in both pure and applied uncertain reasoning is to do with telling apart signal from noise, that is to say distinguishing information which is relevant to a particular decision from information which is not. Now, the idea that information is not uniformly relevant to a given problem played a key role in the statistical work carried out by Alan M. Turing and his chief assistant –I.J. Good– at Bletchley Park during World War II.
As I recalled in my June 2012 column, Turing led a group of statisticians whose goal was to decipher the Enigma machine used by the Nazis for their (naval) communication. In tackling this problem Turing invented the method of Bamburismus which involved centrally what was to become known in statistics as the odd form of the Bayes Factor. Let 
Then the Bayes Factor in favour of a hypothesis H provided by evidence E given background information G is defined as
which equals to
the latter equivalence being a consequence of the above equalities together with the standard properties of probability functions. The Turing-Good definition of the weight of evidence provided by E in favour of H given G is then taken to be the logarithm of the Bayes factor, namely
The mathematical properties of the function 
Whilst interval-valued probability is a natural extension of point-valued probability when it comes to capturing certain features of second-order uncertainty, reasoning with and modelling intervals is far more complicated than dealing with real-valued probabilities. Among the many argument in support of this view, Good focusses on the unavoidable arbitrariness of the endpoints of an interval representation of uncertainty:
Since the ends of the intervals are too arbitrary I prefer a model
where imprecise logodds and weights of evidence have (level-two)
normal (Gaussian) distributions. Call this the (level-two) normal
model (for weights of evidence). This device won’t do for probability
and odds because they don’t extend from minus to plus infinity
[…] I am merely claiming that this new “normal” model is
better than the “interval valued” model. The new “normal” model has
the further advantage that the sums and differences of normal random
variables (not mixtures) are again normally distributed. (Good 2003,
pp. 4-5)
The paragraph, and indeed the paper, ends with some qualifying remarks on the conditions under which the computational advantages of imprecise weights of evidence over probability intervals are obtained. Yet the very notion of “imprecise weights of evidence” is not fleshed out at all. Nor does it seem to have attracted much subsequent work. The search “The accumulation of imprecise weights of evidence” performed on Google on 14 May 2013, returned 17 results, including no research paper. Yet the idea is indeed an intriguing one. For a crucial weakness of second-order uncertainty models, and in particular of imprecise probabilities, is to be found in the difficulty of calculating with them. As a consequence, a model allowing for relatively straightforward calculations, like the one envisaged by Good (2003), would be of great interest for the decision-relevant quantification of uncertainty.
(Thanks to Jacopo Amidei for drawing my attention to Good’s paper.)
SeAllora 