Dempster-shafter theory

Dempster-shafter theory produces similar results to bayesian probabilities 20220301203417, but whereas Bayes relies on evidence being represented by probability function, Dempster represents evidence as a probabilistic function, where this function represents partial evidence. For example, when detecting an object H, we may be not detecting the entire object because some occlusion, and the possibility of the object may be higher than what we have now. Belief functions are combined using Dempster’s rule, and it indicates a term when multiple observations disagree, called conflict metric.

Belief functions

Belief functions measure the belief mass of a set of propositions called the frame of discernment (FOD). For example, for the evidence of a grid cell being occupied or not, bayes states that either be occupied (H) or not (¬H). For Dempster, the FOD θ = {H, ¬H}, but the belief mass can be distributed on combinations of all the propositions, they are not mutually exclusive. We have 4 different combinations, $H, \neg H, \empty$ and θ, where $\empty$ represents the empty set and θ represents both, or in other words, that we don’t know.

For a given belief function, we have three conditions:

  • $Bel(\empty) = 0$. There cannot be any mass for the empty set, since, although there may be ambiguos, observations happen.
  • Bel(θ) = 1. All observations are contained on the in the set that has all elements.
  • $Bel(A_1,…,A_n) = \sum_{I \subset \{1,…,n\};I\neq\empty} {(-1)^{|I|+1} Bel(\bigcap_{i \epsilon I} A_i)}$. More than one belief function contributing evidence cam be summed, and the result may be higher after the summation.

Dempster rule of combination

Dempster rule of combination is one of the many ways for combining belief functions. For two given combinations or more, the combination of both, also called joint mass, is calculated:

  • m(1, 2)(θ) = 0
  • $m_(1,2)=(m_1\oplus m_2)(A)=\frac{1}{1-K}\sum_{B \cap C=A;A\neq\phi}m_1(B)\cdot m_2(C)$ ; where K is the amount of conflict measured by $K=\sum{B \cap C = \empty}m_1(B)\cdot m_2(C)$

B ∩ C = A stands for every combination that results on an intersection that is A. $B \cap C = \empty$ is every combination that results on an empty set. Since empty sets cant happen, a normalization is needed ($\frac{1}{1 - K}$) to distribute the mass on the other possible combinations. K stands for the measure of conflict.

Notes References

20220301203417 Bayes theorem

20220301203617 INDEX - Probabilities and statistics

References