In decision theory, we assume the agent has a set of possible actions to choose from. Each of these actions has costs and benefits which will depend on the underlying state of nature . We can encode this info into a loss function , that specifies the loss we incur if we take action when the state of nature is .
Once we have specified the loss function, we can compute the posterior expected loss or risk for each possible action given all the relevant evidence, which may be a single datum or an entire data set , depending on the problem:
The optimal policy also called the Bayes estimator or Bayes decision rule specifies what action to take when presented with evidence so as to minimize the risk.
An alternative, but equivalent, way of stating this result is as follows. Let us define a utility function to be the desirability of each possible action in each possible state. If we set then the optimal policy is as follows:
This is called the maximum expected utility principle.
Classification problems
Zero-one loss
Suppose the states of nature correspond to class labels, so . Furthermore suppose the actions also corresponds to class labels, so .