I enjoy your discussion about Cox axioms.
As you should expect it, I really admire them (indeed I do)...
but I do not share the feeling that the first one (the negation) is really
compulsary.
In his book, Jaynes presents a long discussion about the second constraint
(conditioning) ... but surprisingly all he says about the first one is what
I reproduce here (chapter 2, page 206)
I quote:
The sum rule:
...The plausibility that A is false must depend in some way on the
plausibility that it is true. ... there must exist some functional relation
plausibility(notA given B) = S(plausibility(A given B))
Evidently, qualitative correspondence with common sense requires that S(u)
be a continuous monotonic decreasing function ....
End of quote.
I feel it really unbalanced and quite shallow. 'some way' can surely be
defend but that 'some way' means what Cox stated should be seriously
justified, what Jaynes does not provide.
For instance, I could defend a weakest form where 'some way' would mean:
plausibility(A given B) + plausibility(notA given B) ¾ 1
(it holds with belief functions if 'plausibility' is understood as 'belief').
By the way, the fifth axiom of Joe's paper remembers me the 'ugly' axioms
Fine presents in his book (Theories of probability') , in chapter 2, when
he tries to produce a unique probability y measure to represent ordered
qualitative probabilities.
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Now why I don't like the negation constraint.
Coughing is a very common symptom among those who do not have lung cancer,
and still more frequent among those who have a lung cancer.
A man is behind your door, you never saw him, you know nothing else about
him except that he is a male.
What is your belief that he has a lung cancer? Let it be alpha.
Then he coughs. So you have ONE symptom.
What is your belief now that he has a lung cancer, still alpha or a little
larger. Usually people agree, and so do I, that it should be a little
larger than alpha.
But then it means that your belief that he does not have lung cancer has
decreased (as probabilities add to 1).
And thus I don't like.
Coughing supports lung cancer, but it should not have the effect of
reducing your belief that the man has no lung cancer (that he is anything
but a lung cancer case).
I feel that I had some prior belief that the patient had no lung cancert,
and learning about the coughing should not reduce this belief.
This is why I abandon additivity of beliefs (and thus the negation
constraint) and develop the transferable belief model.
All the best
Philippe
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Philippe Smets
email: psmets@ulb.ac.be
IRIDIA-CP 194/6
Universite Libre de Bruxelles
50 av. Roosevelt,
1050 Bruxelles, Belgium.
tel 32 2 650 27 29 secretary,
32 2 344 82 96 private (where I am usually)
fax: 32 2 650 27 15
GSM: 32 495 50 10 72
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