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11.4 Dempster-Shafer Theory
An interesting property of belief functions is that the total belief mass
can be assigned to dontknow ,or m(dontknow ) = 1:0. This means the ob-
server is completely uncertain, and the belief function is humorously called
VACUOUS BELIEF the vacuous belief function. The vacuous belief function is equivalent to the
=Bayesian probabilities. It is also used to
FUNCTION P (H) P (:H) :5 assignment in 0
=
initialize the occupancy grid if there is no a priori belief.

11.4.2 Belief function for sonar
Returning the sonar model in Sec. 11.2, the Shafer belief function for a sonar
reading can be expressed as:
For Region I:

( R R r ) ) + (
ccupied
m(O ) = M occupied a x
2
mpty
(11.7) m(E ) = 0:0
m(dontknow ) = 1:00 m(O )
ccupied
For Region II:

ccupied
m(O ) = 0:0
( R R r ) ) + (
(11.8) m(E ) =
mpty
2
m(dontknow ) = 1:00 m(E ) m p t y
The main conceptual difference between the probability and the Shafer
belief function is that any uncertainty in the reading counts as belief mass for
“don’t know.”
Returning to the examples in Figs. 11.4 and 11.5, the computations are
largely the same. For Example 1:

m(O ) = 0:0
ccupied
( R r )+( ) ( 10 3:5 )+( 15 0 )
mpty
m(E ) = R = 10 15 = 0:83
2 2
m(dontknow ) = 1:0 m(E ) m= 1 p 0:83 t y = 0:17
Resulting in:
0
)
mpty
B = e m(O l c c u p i e d ; (E m) = :83 ; (dontknow m = ) :17 0= 0 0
: