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:-
- Consider all possible outcomes.
- Belief will lead to believe in some possibility by bringing out some evidence.(What is this supposed to mean?)
- 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.
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