ML | 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.

Dempster Shafer Theory

Dempster Shafer Theory(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. 

The uncertainty in this model is given by:- 

1. Consider all possible outcomes.
2. Belief will lead to belief in some possibility by bringing out some evidence. (What is this supposed to mean?)
3. 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. 

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} 
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 e.g.:- 
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. 

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) 

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: 

• Uncertainty Representation : The DST is designed to handle situations where there is uncertainty of information and it provides a way to represent and reason incomplete evidence.
• Conflict of Evidence : The DST allows for the combination of multiple sources of evidence. It provides a rule, Dempster’s rule of combination, to combine belief functions from different sources.
• Decision-Making Ability : By deriving measures such as belief, probability and plausibility from the combined belief function it helps in decision making.

Advantages of Dempster Shafer Theory: 

• 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 of Dempster Shafer Theory: 

• In this, computation effort is high, as we have to deal with 2ⁿ sets