DS Theory

An alternative to Bayesian Networks is Dempster-Shafer Theory which is designed to deal directly with the distinction between uncertainty and ignorance. Rather than computing probabilities of propositions, it computes probabilities that evidence supports the propositions. This theory is an approach to combining evidence. Each fact has a degree of support, between 0 and 1:

– 0 No support for the fact

– 1 full support for the fact

Set of possible conclusions:

 

 

 

Bayes Theorem concerned with evidence that supported single conclusions.(eg.evidence for each outcome in)

 

D-S theory concerned with evidences which support subsets of outcome in   

 

Frame of Discernment:

The Frame of Discernment or power set of 

is the set of all possible subsets of   

 

Then the Frame of Discernment of   is: 

 

 ,                the empty set ,has a probability of 0,Since one of the outcome has to be true.Each of the other elements in the powerset has a probability between 0 and 1.

The probability of  is 1.0.Since one has to be true.

Mass function :

m(A): (where A is a member of the power set) = proportion of all evidence that supports this element of the power set.

“The mass m(A) of a given member of the power set, A, expresses the proportion of all relevant and available evidence that supports the claim that the actual state belongs to A but to no particular subset of A.”

“The value of m(A) pertains only to the set A and makes no additional claims about any subsets of A, each of which has, by definition, its own mass.

Belief function:bel(A)

The belief in an element A of the Power set is the sum of the masses of elements which are subsets of A (including A itself).

E.g., given A={q1, q2, q3}

Bel(A)=m(q1)+m(q2)+m(q3)+m({q1,q2})+m({q2,q3}) +m({q1, q3}) +m({q1, q2, q3})

Disbelief (or Doubt) in A: dis(A)
The disbelief in A is simply bel(¬A).
It is calculated by summing all masses of elements which do
not intersect with A.

Plausibility of A: pl(A)

The plausability of an element A, pl(A), is the sum of all the masses of the sets that intersect with the set A

Let us consider an example ,

There are four people named as (B,J,S,K) are locked in room when the lights go out. When the lights come on, K is dead, stabbed with a knife. Not suicide (stabbed in the back) .No-one entered the room.  Assume only one killer.

Power Set:

Detectives, after reviewing the crime-scene, assign mass
probabilities to various elements of the power set:

 ,

 

Belief Function:

Given the mass assignments as assigned by the detectives:

bel({B}) = m({B}) = 0.1

 bel({B,J}) = m({B})+m({J})+m({B,J}) =0.1+0.2+0.1=0.4

 Result:

Plausibility of A: pl(A)

As we know that ,The plausability of an element A, pl(A), is the sum of all the masses of the sets that intersect with the set A:

E.g. pl({B,J}) = m(B)+m(J)+m(B,J)+m(B,S)  

+m(J,S)+m(B,J,S) = 0.9

Disbelief (or Doubt) in A: dis(A)

Now,we need to find the disbelief fn for that we need to calculate dis(A), by summing all masses of elements which do
not intersect with A.

Then,the plausibility of A is thus 1-dis(A):
pl(A) = 1- dis(A)

 

Belief Intervals & Probability

The probability in A falls somewhere between bel(A) and
pl(A).
– bel(A) represents the evidence we have for A directly.

So prob(A) cannot be less than this value.
– pl(A) represents the maximum share of the evidence we
could possibly have, if, for all sets that intersect with A,
the part that intersects is actually valid. So pl(A) is the
maximum possible value of prob(A).

 

Belief Intervals:

Belief intervals allow Demspter-Shafer theory to reason about the degree of certainty or certainty of our beliefs.

– A small difference between belief and plausibility shows that we are certain about our belief.
– A large difference shows that we are uncertain about our belief.
• However, even with a 0 interval, this does not mean we know which conclusion is right. Just how probable it is!