ML | Dempster Shafer Theory

What Dempster Shafer Theory was given by Arthure P.Dempster in 1967 and his student Glenn Shafer in 1976. 
This theory was released because of following reason:- 
 

• Bayesian theory is only concerned about single evidences.
• 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 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 believe in some possibility by bringing out some evidence.(What is this supposed to mean?)
3. Plausibility will make evidence compatible with possible outcomes.

For eg:- 
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, C} 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 the possible evidence by which we can find the murderer by measure of plausibility. 
Using the above example we can say: 
Set of possible conclusion (P): {p1, p2….pn} 
where P is 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 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, b}, {b, c}, {a, c}, {a, b, c}}= 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 element which are subsets of K. This can be explained through an example 
Lets say K = {a, b, c} 
Bel(K) = m(a) + m(b) + m(c) + m(a, b) + m(a, c) + m(b, c) + m(a, b, c) 

Plausibility in K: It is the sum of masses of set that intersects with K. 
i.e; Pl(K) = m(a) + m(b) + m(c) + m(a, b) + m(b, c) + m(a, c) + m(a, b, c) 

Characteristics of Dempster Shafer Theory: 

• It will ignorance part such that probability of all events aggregate to 1.(What is this supposed to mean?)
• Ignorance is reduced in this theory by adding more and more evidences.
• Combination rule is used to combine various types of possibilities.

Advantages: 

• As we add more information, uncertainty interval reduces.
• DST has much lower level of ignorance.
• Diagnose hierarchies can be represented using this.
• Person dealing with such problems is free to think about evidences.

Disadvantages: 

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