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

The optimal strategy, , is defined by setting

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

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

This differs from (9.15) and (9.16) by replacing with . For each , the expectation in (9.26) is called the

Steven M LaValle 2020-08-14