Prior Information and Subjective Probability

Abstract

As mentioned in Chapter 1, an important element of many decision problems is the prior information concerning O. It was stated that a convenient way to quantify such information is in terms of a probability distribution on 0. In this chapter, methods and problems involved in the construction of such probability distributions will be discussed. 3.1. Subjective Probability The first point that must be discussed is the meaning of probabilities concerning events (subsets) in 0. The classical concept of probability involves a long sequence of repetitions of a given situation. For example, saying that a fair coin has probability ~ of coming up heads, when flipped, means that, in a long series of independent flips of the coin, heads will occur about ~ of the time. Unfortunately, this frequency concept won't suffice when dealing with probabilities about O. For example, consider the problem of trying to determine 0, the proportion of smokers in the United States. What meaning does the statement P(0.3 < 0 < 0.35) = 0.5 have? Here o is simply some number we happen not to know. Clearly it is either in the interval (0.3, 0.35) or it is not. There is nothing random about it. As a second example, let () denote the unemployment rate for next year. It is somewhat easier here to think of 0 as random, since the future is uncertain, but how can PO % < () < 4 %) be interpreted in terms of a sequence of identical situations? The unemployment situation next year will be a unique, one-time event. The theory of subjective probability has been created to enable one to talk about probabilities when the frequency viewpoint does not apply. (Some J. O. Berger, Statistical Decision Theory and Bayesian Analysis