Now consider finding the optimal strategy, denoted by , under
the nondeterministic model. The sets for each
must be used to determine which nature actions are possible
for each observation, . Let denote this, which
is obtained as
|
(9.23) |
The optimal strategy, , is defined by setting
|
(9.24) |
for each . Compare this to (9.14), in which the
maximum was taken over all . The advantage of having the
observation, , is that the set is restricted to
.
Under the probabilistic model, an operation analogous to
(9.23) must be performed. This involves computing
from
to determine the information that
contains regarding . Using Bayes' rule,
(9.9), with marginalization on the denominator, the
result is
|
(9.25) |
To see the connection between the nondeterministic and probabilistic
cases, define a probability distribution,
, that is
nonzero only if
and use a uniform distribution for
. In this case, (9.25) assigns nonzero
probability to precisely the elements of as given in
(9.23). Thus, (9.25) is just the
probabilistic version of (9.23). The optimal strategy,
, is specified for each as
|
(9.26) |
This differs from (9.15) and (9.16) by
replacing with
. For each , the
expectation in (9.26) is called the conditional
Bayes' risk. The optimal strategy,
, always selects the strategy that minimizes this risk. Note
that
in (9.26) can be expressed using
(9.25), for which the denominator (9.26)
represents and does not depend on ; therefore, it does not
affect the optimization. Due to this,
can be
used in the place of
in (9.26), and the
same will be obtained. If the spaces are continuous, then
probability densities are used in the place of all probability
distributions, and the method otherwise remains the same.
Steven M LaValle
2020-08-14