# 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 **uncertainity 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 2^{n} 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}}= 2^{3} 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 2
^{n}of sets.

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