ML | Dampster Shafer Theory

Dempster Shafer Theory is given by Arthure P.Dempster in 1967 and his student Glenn Shafer in 1976.
This theory is being 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 uncertainity in this model is given by:-



  1. Consider all possible outcomes.
  2. Belief will lead to believe in some possiblity by bringing out some evidence.
  3. Plausibility will make evidence compatiblity with possible outcomes.

For eg:-
let us consider a room where four person are presented A, B, C, D(lets say) And suddenly lights out and when the lights come back B has been died due to stabbing in his back with the help of a knife. No one came into the room and no one has leaved the room and B has not committed suicide. Then we have to find out who is the murdrer?

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 kill him i.e; {A, C, D}
  • None of the kill him {o}(let us say).

These will be the possible evidences by which we can find the murderer by measure of plausiblity.
Using the above example we can say :
Set of possible conclusion (P): {p1, p2….pn}
where P is set of possible conclusion and cannot be exhaustive means 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)

Plaausiblity 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.
  • Ignorance is reduced in this theory by adding more and more evidences.
  • Combination rule is used to combine various types of possiblities.

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.


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