Decision-theoretic modeling

Decision theoretic and decision analytic methods quantify uncertainty by probability. It is quite important for a decision modeler to understand the meaning of probability. There are three fundamental interpretations of probability:

•Frequentist interpretation

Probability of an event in this view is defined as the limiting frequency of occurrence of this event in an infinite number of trials. For example, the probability of heads in a single coin toss is the proportion of heads in an infinite number of coin tosses.

•Propensity interpretation

Probability of an event in this view is determined by physical, objective properties of the object or the process generating the event. For example, the probability of heads in a single coin toss is determined by the physical properties of the coin, such as its flat symmetric shape and its two sides.

•Subjectivist interpretation

The frequentist and the propensity views of probability are known as objectivist as they assume that the probability is an objective property of the physical world. In the subjectivist, also known as Bayesian interpretation, probability of an event is subjective to personal measure of the belief in that event occurring.

While the above three interpretations of probability are theoretical and subject to discussions and controversies in the domain of philosophy, they have serious implications on the practice of decision analysis. The first two views, known collectively as objectivist, are impractical for most real world decision problems. In the frequentist view, in order for a probability to be a meaningful measure of uncertainty, it is necessary that we deal with a process that is or at least can be imagined as repetitive in nature. While coin tosses provide such a process, uncertainty related to nuclear war is a rather hard case - there have been no nuclear wars in the past and even their repetition is rather hard to imagine. Obviously, for a sufficiently complex process, such as circumstances leading to a nuclear war, it is not easy to make an argument based on physical considerations. The subjectivist view gives us a tool for dealing with such problems and is the view embraced by decision analysis. The process of obtaining a subjective probability from a decision maker is known as probability elicitation.

The subjectivist view interprets probability as a measure of personal belief. It is legitimate in this view to believe that the probability of heads in a single coin toss is 0.3, just as it is legitimate to believe that it is 0.5 as long as one does not violate the axioms of probability, such as one stating that the sum of probabilities of an event and its complement is equal to 1.0. It is also legitimate to put a measure of uncertainty on the event of nuclear war. Furthermore, this measure, a personal belief in the event, can vary among various individuals. While this sounds perhaps like a little too much freedom, this view comes with a rule for updating probability in light of new observations, known as Bayes theorem. There exist limits theorems that prove that if Bayes theorem is used for updating the degree of belief, this degree of belief will converge to the limiting frequency regardless of the actual value of the initial degree of belief (as long as it is not extreme in the sense of being exactly zero or exactly one). While these theorems give guarantees in the infinity, convergence is fast in practice, and a reasonable prior belief makes convergence even faster.

The subjectivist view makes it natural to combine frequency data with expert judgment. Numerical probabilities can be extracted from databases, can be based on expert judgment, or a combination of both. Obtaining numbers for probabilistic and decision-theoretic models is not really difficult. The process of measuring the degree of belief is referred to as a probability assessment. Various decision-analytic methods are available for probability assessment.