The Risk Function

The decision rule is a function that tells us which action to take for every possible observation. When we make a decision to take an action given an observation, we might not always be making the best decision, and thus some costs may be incurred due to our decision. For example, we might want to consider taking out a 100000$ insurance for a business against fire. The insurance costs 1000$ a year. If we take the decision not to insure it and a fire does occur, then the cost is 100000$. Of course if we don’t insure and no fire occurs, then we gain 1000$. The key is to find a way to calculate the risk for every possible decision. Naturally, we may want to take the action that leads to the minimum risk. Each action had an associated risk , and is the loss incurred for deciding when the true state of nature is . To minimize the overall risk, the conditional risk is computed. The definition of conditional risk is the same for the continuous and discrete case:

In plain English, the risk given our observations for every action is the sum of all losses for that action given all the states, weighted by the probability of occurrence of each state. The action with the minimum risk is then selected.