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Basics / July 15, 2021

Dempster-Shafer Models

This can be regarded as a more general approach to representing uncertainty than the Bayesian approach.ContentsDempster-Shafer CalculusCombining beliefs 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 […]

By · July 15, 2021 · 3 min read

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

Contents

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.