Dempster-Shafer Models
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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 tex2html_wrap_inline7580

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 tex2html_wrap_inline7584 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 tex2html_wrap_inline7598 is the set of possible outcomes, then a mass probability, M, is defined for each member of the set tex2html_wrap_inline7602 and takes values in the range [0,1].

The Null set, tex2html_wrap_inline7606, is also a member of tex2html_wrap_inline7602.

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 tex2html_wrap_inline7598 but for em all subsets.

So if tex2html_wrap_inline7598 is the set { Flu (F), Cold (C), Pneumonia (P) } then tex2html_wrap_inline7602 is the set { tex2html_wrap_inline7606, {F}, {C}, {P}, {F, C}, {F, P}, {C, P}, {F, C, P} }

  • The confidence interval is then defined as [B(E),PL(E)] 
    where 
    displaymath1649
    where tex2html_wrap_inline7640 i.e. all the evidence that makes us believe in the correctness of P, and 
    eqnarray1652
    where tex2html_wrap_inline7644 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 tex2html_wrap_inline7650 and tex2html_wrap_inline7652 are two belief functions.
    • Let X be the set set of subsets of tex2html_wrap_inline7598 to which tex2html_wrap_inline7650 assigns a nonzero value and letY be a similar set for tex2html_wrap_inline7652
    • Then to get a new belief function tex2html_wrap_inline7664 from the combination of beliefs in tex2html_wrap_inline7650 and tex2html_wrap_inline7652 we do:
displaymath1662

whenever tex2html_wrap_inline7670.

NOTE: We define tex2html_wrap_inline7672 to be 0 so that the orthogonal sum remains a basic probability assignment.