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11.3 Bayesian
Eqn. 11.4 in boxes. If these are known, then it is straightforward to convert
the probabilities from the sonar model to the form needed for the occupancy
grid.
In some cases, such as for a planetary rover, there may be some knowledge
that produces the prior probabilities. In most cases, that knowledge isn’t
mpty
available. In those cases, it is assumed that P (Occupied ) = P (E ) =
0:5. Using that assumption, the probabilities generated for the example in
Fig. 11.5 can be transformed as follows.
j
For g r[i][ i d
:
]
P (s = jO ) = 0:62 6
ccupied
P (s = jE ) m= 0:38 p t y 6
P (O ) = 0:5
ccupied
P (E ) m= 0:5 p t y

Substituting into Eqn. 11.4 yields:

(0 :62)(0 :5)
P (O js = = = :62 6 ) 0
ccupied
(0 :62)(0 :5)+ :38)(0 :5)
(0
(0 :38)(0 :5)
mpty
P (E js = = = :38 6 ) 0
(0 :38)(0 :5)+ :62)(0 :5)
(0
The use of 0.5 for the priors made P (O js) numerically equivalent
ccupied
to P (sjOccupied ), but in general P (Hjs) 6= P (sjH).
11.3.3 Updating with Bayes’ rule

Now that there is a method for computing conditional probabilities of the
correct form, the question becomes how to fuse it with other readings. The
first update is simple. Each element in the occupancy grid is initialized with
the a priori probability of being occupied or empty. Recall that this is gener-
ally implemented as a data structure consisting of two fields. If the a priori
probabilities are not known, it is assumed P (H) = P (:H) = 0:5.The first
observation affecting g r[i][ i dcan use Bayes’ rule to compute a new proba-
]
j
bility and replace the prior P (H) :5 with the new =value. 0
But what about the second observation? Or an observation made from
another sonar at the same time? It turns out that in both cases, Bayes’ rule