Guttman scales/Certainty Factors [long, hard] 3 of 4

["Robert J. Chassell" <bob@GNU.AI.MIT.EDU>]

This is the 3rd of 4 related messages.

Here is an example of `certainty factors' as an interval scale.

This message follows after, and is best understood, within the context
of two other messages I just posted, called `Guttman Scales' and
`Guttman scales examples'


Now, I know full well that it is hard to translate from English to
Lojban; but would one of you experts out there care to translate this?
You would get a chance to work with numbers and equations, and other
things we are telling new comers not to bother with.  :)

(I don't know whether I could translate this if I had the time--I
might not be able to.  I just ran this through the Unix `style'
program and it tells me that this is readable at the 10th through 13th
grade level --- this is considerably harder to follow than my usual
writing, which is aimed at the 9th and 10th grade level, a level that
is easy and comfortable for readers who went to college.)


Certainty Factor example
........................

In the mid 1980s, David McAllister, at MIT, developed a metric for
`certainty factors' for use in an `expert system' (a type of computer
program).  This example will give you a sense of what can be
done with an interval scale when you cannot make ratios.

Metrics of this sort will become more commonplace in the years ahead,
not only in diagnosing problems with oil refineries and illnesses in
people, but also in mortgage lending and mutual fund investment.

A certainty factor is used to express how accurate, truthful, or
reliable you judge a predicate to be.  It is your judgement of how
good your evidence is.  The issue is how to combine various
judgements.

Note that a certainty factor is neither a probability nor a truth
value.

Consider the expression `George is suffering from hypoxia'.

Based on warnings given to pilots, we would speak of there being
`strongly suggestive evidence' that George is suffering from hypoxia
when he is flying in an unpressurized airplane at 12,000 feet and his
judgement, memory, altertness, and coordination are off.

Note, we are not saying "there is an eighty percent chance that George
suffers hypoxia"; that is a probability estimate.  We are talking
about our judgement of certainty.  You may be able to generate
statements of probability, such as: "80% of US Air Force student
pilots will fail to maintain altitude within 100 feet when they fly
higher than... feet without supplementary oxygen, and this will
indicate they suffer from hypoxia."  But this is a different sort of
statement than one involving certainty factors.

What I am doing is in this example of uncertainty is taking what I was
taught as a student pilot and creating from that information a
mechanism for diagnosing hypoxia.  I don't know the probability that a
person of my health and age will suffer hypoxia at 12,000 feet, but I
do know the symptoms, which, however, may be weak, or have other
causes.

In McAllister's scheme, a certainty factor is a number from 0.0 to
1.0.  A phrase such as `suggestive evidence' is given a number such as
0.6; `strongly suggestive evidence' is given a number such as 0.8.
The person making the judgement uses the scale more or less as an
ordinal scale.  The numbers are used in a metric to permit a computer
to make calculations.

McAllister's rules for combining certainty factors are such that you
can add new evidence to existing evidence.  If the evidence is
positive, this increases your certainty, as you would expect.  But you
never become 100% certain.

Continuing our hypoxia example: George tells us that he feels
wonderful.  This is `suggestive evidence' that George suffers from
hypoxia.  (Pilots are warned of this: "if you feel euphoric, consider
hypoxia: you may be flying too high without oxygen, or suffering
carbon monoxide poisoning from a broken heater."  Of course, there are
many good reasons to become euphoric when you fly; hypoxia is
insidiously dangerous.)

McAllister defined the rule for adding two positive certainty factors
like this:

    CFcombine (CFa CFb) = CFa + CFb(1 - Cfa)

I.e., reduce the influence of the second certainty factor by the
remaining uncertainty of the first, and add the result to the
certainty of the first.

In our example, the altitude and loss of judgment are strongly
suggestive evidence, with a certainty factor of 0.8; and euphoria is
suggestive evidence, with a certainty factor of 0.8.  The combined
certainty factor is:

                .92     =  .6 + .8(1 - .6)

(Incidentally, it does not matter which factor you start with first:

       .8 + .6(1 - .8)  =  .6 + .8(1 - .6)  = .92

Both sequences produce the same result.)

McAllister also has rules for adding two negative certainties, and for
adding a positive and a negative certainty.  A negative certainty is
the degree to which you are certain the case is not so.

The rule for adding two negative certainties is simple:
Treat the two factors as positive and negate the result

    CFcombine (CFe CFf) = -(CFcombine (-CFe -CFf))

The rule for adding positive and negative certainty factors is more
complex:

    CFcombine (CFg CFh) =  (CFg + CFh) / (1 - min{|CFg|, |CFn|})

Thus if your certainty for an instance is 0.88 and your certainty
factor against it is 0.90, the result is:

               -.17     = (.88 - -.90) / (1 - min(.88, .90))

                        =   -.02 / .12

I.e. take the difference, and then multiply that value by the
reciprocal of the smallest remaining uncertainty.

These three rules provide an interval scale for certainty factors.

You will note that you cannot say that a certainty factor of 0.8 is
twice the certainty of 0.4; the rules of this metric only involve
those of addition and subtraction that I have shown, no others.

    Robert J. Chassell               bob@gnu.ai.mit.edu
    25 Rattlesnake Mountain Road     bob@rattlesnake.com
    Stockbridge, MA 01262-0693 USA   (413) 298-4725