[om-list] Sense and Certainty

[Mark Butler]

Sense and Certainty

There is an ambiguity problem using a single scalar sense measure to 
combine the notions of affirmation/denial and certainty.  It is common 
for people to be emphatically neutral about a position.  Emphatic 
neutrality really should be distinguished from ignorant neutrality.  The 
rigorous way to do this is to use a probability distribution function of 
fractional set containment, but people arguably do not track simple 
statements to that level of detail.  I suggest that we can easily fix 
this problem without running into adverse computational complexity by 
splitting sense into two separate scalar measures, sense and certainty.

The sense of a proposition ranges from affirmation [positive] to denial 
[negative].   This corresponds closely to the idea of fractional set 
membership, and the more general idea of fractional set containment.  
Certainty on the other hand, expresses the degree of belief that the 
sense is accurate, ranging from no confidence to absolutely certain.

Certainty can be expressed in either linear [0,1] or logarithmic 
[-inf,+inf] form.  If the latter is multiplied by a [-1,+1] linear sense 
measure, you get a [-inf,+inf] sense measure like I described yesterday, 
which is useful in evidence calculations on boolean propositions.

However, you can also easily generate a family of probability 
distribution functions from a [sense,certainty] pair for use in more 
sophisticated calculations where the exact degree of set membership or 
containment is critical.  Any certain statement has a pdf in the form or 
an impulse or Dirac delta function - infinitely high, infinitely narrow, 
area one, centered around the sense value. Zero certainty leads to a 
flat pdf no matter what the sense is - literally the sense doesn't 
matter.  Sense values in between can be represented by any nice function 
that peaks at the sense value and has area one over [-1,+1] and that has 
a width inversely proportional to the certainty.  Finding good 
candidates is a topic for further research - my first instinct would be 
to use normalized Bezier polynomials.

However, for most applications, simpler expedients are sufficient.  
Weighted combinations of evidence from multiple sources is probably the 
most common operation that deals with both sense and certainty.

 - Mark



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