[om-list] Degree of Belief vs. Fuzzy Set Membership

[Mark Butler]

Hello everybody,

I was talking with Thomas about epistemology, anf thought I would repeat 
an idea here.

My preferred way of tracking degree of belief is in log odds form, using 
the following formula I found in an expert systems text:

  log odds = log10( p  /  (1 - p) )  

   where p is the probability between 0 and 1 and p / (1 - p) is the 
odds "o".

The log odds comes out as a number between minus infinity and plus 
infinity.
The following approximations apply:


   p       odds   log odds
------    ------  ------
zero      zero     -infinity
0.001     1:1000   -3
0.01      1:100    -2
0.1       1:10     -1
0.5       1:1       0
0.9       10:1      1
0.99      100:1     2
0.999     1000:1    3
...
one       infinite  +infinity        



I like this formula because it is nice and symmetrical, "0" is the value 
for no evidence one way or the other, is easy to understand and explain, 
and emphasizes the level of evidence required to move from highly 
probable to absolutely certain and from highly unlikely to absolutely false.

Now of course, degree of belief is a different concept than degree of 
set membership.  For example, there is a big difference between saying 
"I am certain the weather is mild outside" and "I don't know what the 
weather is like outside".  In the first case the degree of belief is 
infinite and the degree of membership in the "weather is warm" set  is 
~1/2, whereas is in the second case the degree of belief is zero, and 
the degree of membership is unknown.

Degree of belief is only a necessary expedient where you have incomplete 
information, whereas fuzzy set membership is a valid concept even when 
you know everything.

Comments?

  - Mark Butler