[ ARTIFICIAL INTELLIGENCE (CENG 303) ]


 
 
Chapter 8

UNCERTAINTY IN KBS (Knowledge Based Systems)


8.1 Bayesian Approach

Consider the following hypothetical situation:

A housewife finds a stain on her husband's shirt. Her husband denies that it is a lipstick stain. She knows that applying a little methylated alcohol to a stain and softening it with washing-up liquid (detergent) almost always removes lipstick stains. She also knows that this method hardly ever works on other stains. She applies this method to the shirt and finds that it removes the stain. As a result she concludes that it was almost certainly a lipstick stain.

Question: How do we represent this type of knowledge ? reasoning?

We will use notation P(X | Y) to mean probability of X occurring given that Y has occurred (subjective probability).
 
                 P (works | lipstick) = 0.9
                 P (works) = 0.3

"works" means that the method works.
"lipstick" means that it is a lipstick stain.

We can now pose the housewife's goal as first ?? finding
 
                  P (lipstick | works)

We use Bayes' rule:

                                         P(X) * P(Y | X)
                  P (X | Y)  =  ------------------------
                                               P(Y)
 
 
                                    P (lipstick) * P (works | lipstick)
 P (lipstick | works) =  ----------------------------------
                                                P (works)

P (lipstick) not available. So let us assume that the housewife can estimate the proportion of previous stains that were lipstick.

                   P (lipstick) = 0.25
                                                           0.25 * 0.9
                   P (lipstick | works) = ------------- = 0.75
                                                                0.3

This shows how we can use one piece of evidence.

Suppose that the housewife knows that the husband was late from work when he wore the shirt.

                                                                 P (lipstick) * P (works  & late | lipstick)
                   P (lipstick | works and late) = -------------------------------------------------
                                                                             P (works and late)

Since method of working & being late are independent we use :
 
                   P (works and late) = P (works) * P (late)
                                          &
             P (works and late) = P (works | lipstick) * P (late | lipstick)
 

                                                                                           P (works | lipstick)       P (late | lipstick)
            P (works and late | lipstick) = P (lipstick) *  ------------------------ *    ---------------------
                                                                                                 P (works)                      P (late)
 

This we can see that to incorporate additional evidence, we simply multiply the current probability of the stain lipstick by an additional factor.

Thus if          P (late | lipstick) = 0.6
                     P (late) = 0.5

we obtain:
                                                                                         0.6
                    P (lipstick | works and late) = 0.75 * ------ = 0.9
                                                                                         0.5

Approaches which use Bayes' rule as a basis of combining evidence are called Bayesian.

Drawback: No indication of whether the probability is a wild guess or a judgement based on experience. For example in the above example, the housewife suggested that the probability of the husband being late from work was 0.5. she did not know, or that 50% of the time?

Several alternative approaches have been advocated to overcome such deficiencies:
These are:

  1. Dempster-Shafer Theory
  2. Fuzzy Logic
  3. Internal based approaches
  4. Logic based approaches

8.2 Dempster-Shafer Theory

Can be used to express belief in a subset of hypothesis.
For example given:

             { strawberry stain, lipstick stain, blue ink stain }

the evidence that a stain looks red could be represented by associating a belief of 0.8 with the subset.

             { strawberry stain, lipstick stain }

The approach allows the narrowing and revision of such beliefs in the light of more evidence. It also maintains a measure of plausibility for the subset of the hypotheses.

This measure gives an interval which expresses the confidence in the set of hypotheses.

Drawback: The approach is inefficient when applied to situations which require reasoning about high-level concepts in KBs which have a hierarchical structure.

8.3 Fuzzy Logic
 

This approach allows us to represent and reason with vague and fuzzy knowledge. Fuzzy set theory:

Fuzzy logic argues that there is no clear boundary between a statement true and a statement being false. It allows fuzzy truth values by employing linguistic truth values like: For example, in the previous example, the housewife could conclude that "the stain is lipstick" is more or less true.

Fuzzy logic provides fuzzy operators like the ones used in predicate calculus.

8.4 Interval Based Approaches
 

Use an interval to represent the likelihood of some statement. Thus the housewife would say that the probability of the husband being late is in the range [ 0 , 1 ] to express that she does not know; or in the range [ 0.5 , 0.5 ] to say that he is late 50 % of the time.

8.5 Logic-based Approaches
 

This approach comes from the argument that assigning probabilities is not a natural way of reasoning for human experts. These approaches provide operators such as "likely" so that. Statements :

                  lipstick "likely" when late
                                &
              method works "likely" when lipstick

could be combined to

              method works "likely" when late.
 


 
 
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