Dempster Shafer Theory Introduction
What is Dempster-Shafer theory ?
- Approach to combining evidence.
- Means for combining degrees of belief derived from independent items of evidence.
- Obtaining degrees of belief for one question from subjective probabilities for a related question.
- Expert Systems 1980’s saw this approach ideally suitable for such systems.
Degree of support for an event varies from 0 to 1
- 0 => No support for the event
- 1 => Complete support for the event
How similar or different is it to Bayesian Theory ?
- Both assign non-negative weights to a set of events.
Ultimately both can be used to combine new observations with prior weights to develop new weights to update beliefs about relative probabilities of events, given new information.
- Bayesian analysis focuses on assigning non negative weights (typically probabilities) to each individual proposition from a set of mutually exclusive propositions. However DS theory assigns non negative weights (mass) to each combination of events.
- The above mentioned characteristic of DS theory allows it extra flexibility and thus DS theory can be considered as a generalization of Bayesian theory. For example :
Given : {R, G, B} = {.1, .2, .7}// Bayesian approach would require that
P( R or G ) belongs to the range {.2, .3}// However DS approach could apportion weight to (R or G)
// that was in excess of the sum of the individual weights for
// (R) & (G)
This allows for the inclusion of information with unquantified uncertainty such as I saw the object in a group of object I have high confidence are either R or G.
Bayesian approach would have to alter weights assigned to R and G to incorporate this information while DS approach can simply alter the weight of (R or G) event without altering the weights of (R) or (G).
- Dempster Shafer is basically Bayesian on steroids ie. assigning Bayesian weights to each element of the power set of propositions.
- Bayesianism uses Bayes Rule to incorporate new information on primitive weights, while DS theory uses Dempster’s rule of combination to incorporate new information. This can be considered a generalization of Bayes rule.
- DS theory maintains a dual concept of weight in Belief and Plausibility(B ≤ P) which can be thought of as a lower and upper bounds of degree of confidence on an event combination, while Bayesianism only maintains a single value which can be thought of as “expected” value.
When considering unquantified uncertainty DS may more compellingly model the range of confidence reflecting a state of knowledge, however when a single estimate is preferrable a Bayesian approach may be simpler and more appropriate.
Mass function formal definitions :
Expresses 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.
Values of m(A) pertains only to set A and makes no additional claims about any subset of A each of which by definition has its own mass.
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