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

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

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

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

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

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