The Certainty Factor Theory uses numeric values between -1 and 1 to represent the likelihood or certainty of statements or hypotheses being true based on evidence. It was developed for artificial intelligence systems to represent uncertain or incomplete information. The Certainty Factor can be calculated based on the Measure of Belief and Measure of Disbelief of hypotheses given evidence, and formulas are provided to combine Certainty Factors from multiple pieces of evidence. However, the theory has limitations, such as difficulty accurately assigning certainty values and the limited numeric range. The Dempster-Shafer Theory was introduced to address some of the limitations of probability theory. It defines a mass function over all subsets of a set of possible conclusions to represent degrees of belief, and uses belief

1. Certainty Factor Theory
The Certainty Factor (CF) is a numeric value which tells us about how likely an
event or a statement is supposed to be true.
It was introduced for systems which works on AI
CF is a measure of Uncertainty
It is based on the numbers of observations
Confidence C tells about how likely the statement is true or false
C can be measured in 2 ways
1. Measure of Belief
2. Measure of Disbelief

2. • CF is always lies between in the interval 1 and -1
• If CF is -1 then the statement never be true
• +1 then the statement is always true
• Measure of Belief
• It is always denoted as MB[H,E], where H is Hypothesis & E is Evidence
• If MB[H,E]=0 then the H is false for E
• If MB[H,E]=1 then the H is True for E
• It always lies between interval[0,1]

3. • Measure of Disbelief
• MD[H,E]
• MD[H,E]=0, then H supports Evidence
• MD[H,E]=1, then H Does not supports Evidence
• MB[H,E], MD[H,E]
• Now calculate CF
• If MB[H,E]=1(True) then MD[H,E]=0
• CF[H,E]=1
• If MB[H,E]=0(False) then MD[H,E]=1
• CF[H,E]=0

4. Formulae for CF
• 1. Multiple Evidences and single Hypothesis
• MB[H,E1 and E2 ]=MB[H,E1]+ MB[H,E2]*[1- MB[H,E1]]
• MD[H,E1 and E2 ]=MD[H,E1]+ MD[H,E2]*[1- MD[H,E1]]
• CF[H,E1 and E2]= MB[H,E1 and E2 ]- MD[H,E1 and E2 ]

5. EXAMPLE
• MB[H,E1]=0.4
• MD[H,E1]=0 (FOR E1 EVIDANCE]
• MB[H,E21]=0.3
• MD[H,E2]=0.1 (FOR E2 EVIDANCE]

6. • MB[H,E1 and E2 ]=MB[H,E1]+ MB[H,E2]*[1- MB[H,E1]
• 0.4+0.3*(1-0.4)
• 0.4+0.3*0.6
• 0.58

7. • MD[H,E1 and E2 ]=MD[H,E1]+ MD[H,E2]*[1- MD[H,E1]
• 0+0.1*(1-0)
• 0.1
• CF=0.58-0.1
• 0.48

8. APPLICATIONS OF CF
• Practical Applications of Certainty Factor
• Certainty factor has practical applications in various fields of artificial intelligence,
including:
• Medical diagnosis: In medical diagnosis systems, certainty factors are used to evaluate
the probability of a patient having a particular disease based on the presence of specific
symptoms.
• Fraud detection: In financial institutions, certainty factors can be used to evaluate the
likelihood of fraudulent activities based on transaction patterns and other relevant
factors.
• Customer service: In customer service systems, certainty factors can be used to evaluate
customer requests or complaints and provide appropriate responses.
• Risk analysis: In risk analysis applications, certainty factors can be used to assess the
likelihood of certain events occurring based on historical data and other factors.
• Natural language processing: In natural language processing applications, certainty
factors can be used to evaluate the accuracy of language models in interpreting and
generating human language.

9. • Limitations of Certainty Factor
• Although the certainty factor is a useful tool for representing and reasoning about uncertain or
incomplete information in artificial intelligence, there are some limitations to its use. Here are
some of the main limitations of the certainty factor:
• Difficulty in assigning accurate certainty values: Assigning accurate certainty values to
propositions or hypotheses can be challenging, especially when dealing with complex or
ambiguous situations. This can lead to faulty results and outcomes.
• Difficulty in combining certainty values: Combining certainty values from multiple sources can be
complex and difficult to achieve accurately. Different sources may have different levels of
certainty and reliability, which can lead to inconsistent or conflicting results.
• Inability to handle conflicting evidence: In some cases, conflicting evidence may be presented,
making it difficult to determine the correct certainty value for a proposition or hypothesis.
• Limited range of values: The numerical range of the certainty factor is limited to -1 to 1, which
may not be sufficient to capture the full range of uncertainty in some situations.
• Subjectivity: The Certainty factor relies on human judgment to assign certainty values, which can
introduce subjectivity and bias into the decision-making process.

10. Dempster Shafer Theory
• Dempster-Shafer Theory was given by Arthur P. Dempster in 1967
and his student Glenn Shafer in 1976. This theory was released
because of the following reason:-
• Bayesian theory is only concerned about single evidence.
• Bayesian probability cannot describe ignorance.
• DST is an evidence theory, it combines all possible outcomes of the
problem. Hence it is used to solve problems where there may be a
chance that a piece of different evidence will lead to some different
result.

11. • The uncertainty in this model is given by:-
• Consider all possible outcomes.
• Belief will lead to belief in some possibility by bringing out some
evidence. (What is this supposed to mean?)
• Plausibility will make evidence compatible with possible outcomes.
• Example: Let us consider a room where four people are present, A, B,
C, and D. Suddenly the lights go out and when the lights come back, B
has been stabbed in the back by a knife, leading to his death. No one
came into the room and no one left the room. We know that B has
not committed suicide. Now we have to find out who the murderer
is.

12. • To solve these there are the following possibilities:
• Either {A} or {C} or {D} has killed him.
• Either {A, C} or {C, D} or {A, D} have killed him.
• Or the three of them have killed him i.e; {A, C, D}
• None of them have killed him {o} (let’s say).
• There will be possible evidence by which we can find the murderer by
the measure of plausibility.
Using the above example we can say:
Set of possible conclusion (P): {p1, p2….pn}

13. • where P is a set of possible conclusions and cannot be exhaustive, i.e.
at least one
(p) I must be true.
(p)I must be mutually exclusive.
Power Set will contain 2n elements where n is the number of elements
in the possible set.
For eg:-
If P = { a, b, c}, then Power set is given as
{o, {a}, {b}, {c}, {a, d}, {d ,c}, {a, c}, {a, c ,d }}= 23 elements.

14. • Mass function m(K): It is an interpretation of m({K or B})
i.e; it means there is evidence for {K or B} which cannot be divided
among more specific beliefs for K and B.
Belief in K: The belief in element K of Power Set is the sum of masses of
the element which are subsets of K. This can be explained through an
example
Lets say K = {a, d, c}
Bel(K) = m(a) + m(d) + m(c) + m(a, d) + m(a, c) + m(d, c) + m(a, d, c)

15. • Plausibility in K: It is the sum of masses of the set that intersects with
K.
i.e; Pl(K) = m(a) + m(d) + m(c) + m(a, d) + m(d, c) + m(a, c) + m(a, d, c)
Characteristics of Dempster Shafer Theory:
• It will ignorance part such that the probability of all events
aggregate to 1. (What is this supposed to mean?)
• Ignorance is reduced in this theory by adding more and more
evidence.
• Combination rule is used to combine various types of possibilities.

16. • Advantages:
• As we add more information, the uncertainty interval reduces.
• DST has a much lower level of ignorance.
• Diagnose hierarchies can be represented using this.
• Person dealing with such problems is free to think about evidence.
• Disadvantages:
• In this, computation effort is high, as we have to deal with 2n sets