Optimal strategies

Now consider finding the optimal strategy, denoted by $ \pi^*$, under the nondeterministic model. The sets $ Y(\theta)$ for each $ \theta \in \Theta$ must be used to determine which nature actions are possible for each observation, $ y \in Y$. Let $ \Theta(y)$ denote this, which is obtained as

$\displaystyle \Theta(y) = \{\theta \in \Theta \;\vert\; y \in Y(\theta) \} .$ (9.23)

The optimal strategy, $ \pi^*$, is defined by setting

$\displaystyle \pi^*(y) = \operatornamewithlimits{argmin}_{u \in U} \Big\{ \max_{\theta \in \Theta(y)} \Big\{ L(u,\theta) \Big\} \Big\},$ (9.24)

for each $ y \in Y$. Compare this to (9.14), in which the maximum was taken over all $ \Theta$. The advantage of having the observation, $ y$, is that the set is restricted to $ \Theta(y)
\subseteq \Theta$.

Under the probabilistic model, an operation analogous to (9.23) must be performed. This involves computing $ P(\theta\vert y)$ from $ P(y\vert\theta)$ to determine the information that $ y$ contains regarding $ \theta $. Using Bayes' rule, (9.9), with marginalization on the denominator, the result is

$\displaystyle P(\theta\vert y) = {P(y\vert\theta) P(\theta) \over \displaystyle\strut \sum_{\theta \in \Theta} P(y\vert\theta) P(\theta)} .$ (9.25)

To see the connection between the nondeterministic and probabilistic cases, define a probability distribution, $ P(y\vert\theta)$, that is nonzero only if $ y \in Y(\theta)$ and use a uniform distribution for $ P(\theta)$. In this case, (9.25) assigns nonzero probability to precisely the elements of $ \Theta(y)$ as given in (9.23). Thus, (9.25) is just the probabilistic version of (9.23). The optimal strategy, $ \pi^*$, is specified for each $ y \in Y$ as

$\displaystyle \pi^*(y) = \operatornamewithlimits{argmin}_{u \in U} \Big\{ E_\th...
...\in U} \left\{ \sum_{\theta \in \Theta} L(u,\theta) P(\theta\vert y) \right\} .$ (9.26)

This differs from (9.15) and (9.16) by replacing $ P(\theta)$ with $ P(\theta\vert y)$. For each $ u$, the expectation in (9.26) is called the conditional Bayes' risk. The optimal strategy, $ \pi^*$, always selects the strategy that minimizes this risk. Note that $ P(\theta\vert y)$ in (9.26) can be expressed using (9.25), for which the denominator (9.26) represents $ P(y)$ and does not depend on $ u$; therefore, it does not affect the optimization. Due to this, $ P(y\vert\theta) P(\theta)$ can be used in the place of $ P(\theta\vert y)$ in (9.26), and the same $ \pi^*$ 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 2012-04-20