A Few Notes on Sets of Probability Distributions

A person looks at the sky, and tries to assess what she expects about the weather in the near future; another person weighs the costs of a financial application against its expected gain. Such expectations can sometimes be translated into a single number: for instance, the expected return on the financial application may be exactly 100 monetary units. More often, one has less determinate assessments about expectations, such as “I expect a return between 30% and 40% on this investiment.” Or perhaps, “I’d rather bet that it will rain tomorrow than that I get heads in the next toss of this coin.” In general, assessments specify a set of probability distributions that represent the uncertainty about a particular situation. This initial chapter introduces some concepts and tools that are useful in dealing with such sets.