Dempster-Shafer Models

This can be regarded as a more general approach to representing uncertainty than the Bayesian approach.

Bayesian methods are sometimes inappropriate:

Let A represent the proposition Demi Moore is attractive.

Then the axioms of probability insist that 

Now suppose that Andrew does not even know who Demi Moore is.

Then

• We cannot say that Andrew believes the proposition if he has no idea what it means.
• Also, It is not fair to say that he disbelieves the proposition.
• It would therefore be meaningful to denote Andrew's belief of B(A) and  as both being 0.
• Certainty factors do not allow this.

Dempster-Shafer Calculus

The basic idea in representing uncertainty in this model is:

• Set up a confidence interval -- an interval of probabilities within which the true probability lies with a certain confidence -- based on the Belief B and plausibility PL provided by some evidence E for a proposition P.
• The belief brings together all the evidence that would lead us to believe in P with some certainty.
• The plausibility brings together the evidence that is compatible with P and is not inconsistent with it.

• This method allows for further additions to the set of knowledge and does not assume disjoint outcomes.

If  is the set of possible outcomes, then a mass probability, M, is defined for each member of the set  and takes values in the range [0,1].

The Null set, , is also a member of .

NOTE: This deals wit set theory terminology that will be dealt with in a tutorial shortly. Also see exercises to get experience of problem solving in this important subject matter.

M is a probability density function defined not just for  but for em all subsets.

So if  is the set { Flu (F), Cold (C), Pneumonia (P) } then  is the set { , {F}, {C}, {P}, {F, C}, {F, P}, {C, P}, {F, C, P} }

• The confidence interval is then defined as [B(E),PL(E)] 
  where 
  
  where  i.e. all the evidence that makes us believe in the correctness of P, and 
  
  where  i.e. all the evidence that contradicts P.

Combining beliefs

• We have the ability to assign M to a set of hypotheses.
• To combine multiple sources of evidence to a single (or multiple) hypothesis do the following:
• Suppose  and  are two belief functions.
• Let X be the set set of subsets of  to which  assigns a nonzero value and letY be a similar set for 
• Then to get a new belief function  from the combination of beliefs in  and  we do:

whenever .

NOTE: We define  to be 0 so that the orthogonal sum remains a basic probability assignment.