Mistaking the Solution for the Problem

Sometimes we jump into an application of decision theory prematurely. Problems like deciding the probability that a person has AIDS given a positive result on a clinical test are easy to set up and solve. In essence, both the problem and nature of the solution is pre-defined. The problem is that some people have AIDS. We have a way of testing for it with known rates of false positives and false negatives. We even know the prevalence of the disease in the population. Computing the probability of infection is straightforward.

The ease of setting up such problems for solution can lead us astray even in related problems. For instance, assume you have a test to classify human cells into normal and pre-cancerous with some rate of false positive and false negative allocations. By changing the parameters of the test, you can tradeoff the two rates. That is, you can lower the rate of false negatives (make the test more sensitive) at the cost of raising the rate of false positives. What is optimum? Most people, even professionals, would set up the classification incorrectly and bias the results for greater sensitivity. That is because we have not taken care to ask the correct question. No one really cares if a particular cell is pre-cancerous. We really want to know the probability that the person has cancer, and that is a different question with a different optimum classifier. The right response is to collect a sufficient number of cells such that the probability of cancer can be derived. For that you want to probability of a false positive to be low.

In the AIDS example, the test was for AIDS directly. In the second, the test was for a parameter related to the real question.

The difficulty of setting up a problem becomes much more difficult in everyday problems such as politics and religion. Consider a common issue in society. What can we do about the problem of urban gangs and their effect on common citizens?

You will never get anywhere starting from that construction.

Consider that humans have naturally been organized in small groups for most of their history. Even when large cities and states formed, smaller neighborhoods and tribes were always important. Given a homogeneous society probably formed from transplanted populations with no continuing affiliations, and a natural inclination to organize the population into smaller groups. But a funny thing about Mother Nature — she does not seem to care about the common citizen’s desires. She simply goes about business and creates organizations. In this sense, the gangs are not a problem; they are the solution.

The real problem lies with the common citizens who oppose the gangs without providing alternative ways for the self-organizing tendencies to manifest themselves with more desirable characteristics.

You might are might not agree with that specific example, but the point I am trying to make is that contrary to what many people think, it is easy to mis-analyze a situation and mistake the problem for the solution and vice versa. Only when a problem has been correctly set up can we apply decision theory and get reliable results.