15.5 Dempster-Shafer Theory 367
If X and Y are elementary D-S distributions which can be combined, then
the combination Z is also elementary and coincides with the distribution
given by the information combination theorem. In fact, if either X or Y is
elementary, then Z will also be elementary. Dempster’s rule is therefore an
extension of the information combination theorem. The combination of the
D-S distributions P and Q defined above is given as follows:
Diagnosis P Q Distribution M Combination Z
concussion 0.5 0.6 0.42 0.81
meningitis 0 0.2 0.04 0.08
tumor 0.3 0.2 0.06 0.11
concussion-meningitis 0.2 0 0 0The only complicated entry in the computation above is the value of M(con-
cussion). This probability is the sum of two products: P(concussion)Q(con-
cussion) and P(concussion-meningitis)Q(concussion). The rationale for in-
cluding both of these in the combined probability for concussion is that both
concussion and concussion-meningitis contribute to the evidence (or belief)
in concussion because they both contain concussion.
There is some question about the role played by the empty entity. It is
sometimes interpreted as representing the degree to which one is unsure
about the overall observation. However, Dempster’s rule of combination
explicitly excludes the empty entity from any combined distribution. As a
result, the only effect in D-S theory of a nonzero probability for the empty
entity is to allow distributions to be unnormalized. The information com-
bination theorems also apply to unnormalized distributions, as we noted in
the discussion after the information combination theorems.
Summary
- D-S theory introduces a probabilistic form of concept combination.
- D-S distributions are combined by using Dempster’s rule of combination.
- Dempster’s rule of combination coincides with the discrete information
combination theorem when the distributions are elementary.