Question

In a two-class, two-action problem, if the loss function is λ11 = λ22 = 0,λ12 = 6 , and λ21 = 2, write the optimal decision rule. How does the rule change if we add a third action of reject with λ= 0.8

Answer 1

The optimal decision rule in a two-class problem is to minimize expected loss using given loss values. If a third "reject" action is added with a loss value, the decision rule includes calculating the expected loss for rejection and choosing to reject if it has the lowest expected loss.

Step-by-step explanation:

In a two-class, two-action problem with the given loss function where λ11 = λ22 = 0, λ12 = 6, and λ21 = 2, the optimal decision rule is to choose the action that minimizes the expected loss. To write this decision rule, you calculate the expected losses for each action and choose the action with the lower expected loss.

If a third action is added, namely "reject" with a loss of λ = 0.8 for both classes, the decision rule now has three actions to consider. Here, you would calculate the expected loss for each of the three actions, and if the expected loss of "rejecting" is lower than choosing either of the two classes, then "reject" becomes the optimal decision.

As an example, let's consider a situation where P(class 1) and P(class 2) are the probabilities of a data point belonging to class 1 and class 2, respectively. If P(class 1) * λ21 is less than P(class 2) * λ12, you would decide for class 1. Conversely, if P(class 1) * λ21 is greater, you choose class 2. When there is a third action, "reject", if 0.8 < min(P(class 1) * λ21, P(class 2) * λ12), you would choose to "reject".

Answer 2

The optimal decision rule in a two-action problem is to predict the class with the minimal expected loss, based on the known loss function. When a third 'reject' action is added, the decision rule includes the option to reject the decision if it has a lower risk than incorrect predictions.

Step-by-step explanation:

In a two-class, two-action problem with the following loss function: λ11 = λ22 = 0, λ12 = 6, and λ21 = 2, the optimal decision rule is to predict the class that minimizes the expected loss. This involves calculating the risk for each action, and choosing the action with the lower risk. For instance, if the probability of a true class being 1 is p, then the risk of predicting class 2 is pλ12, and the risk of predicting class 1 is (1-p)λ21. If pλ12 < (1-p)λ21, we predict class 2, else we predict class 1.

When a third action, 'reject', is added with a loss value of λ = 0.8 for both classes, the decision rule changes. The risk of rejecting a decision will always be 0.8, irrespective of the true class. The decision-maker must compare the risk of prediction against the fixed risk of rejection (0.8), and choose the action that minimizes the expected loss, which could now include opting to reject if the risks associated with a wrong prediction are greater than 0.8.